1 chapter 4 sequential games 2 extensive form games h h h t t t (1,2) (4,0) (2,1) any finite game of...
TRANSCRIPT
1
Chapter 4 Sequential Games
2
Extensive Form Games
H
H H
T
TT
(12) (40)(21) (21)
Any finite game of perfect information has a pure strategy Nash equilibrium It can be found by backward induction
Chess is a finite game of perfect information Therefore it is a ldquotrivialrdquo game from a game theoretic point of view
3
Extensive Form Games - Intro
bull A game can have complex temporal structurebull Information
ndash set of playersndash who moves when and under what circumstancesndash what actions are available when called upon to movendash what is known when called upon to movendash what payoffs each player receives
bull Foundation is a game tree
4
bull Big Monkey and Little Monkey eat warifruit which dangle from the extreme tip of a lofty branch of the waritree
bull A waritree produces only one fruit To get the warifruit at least one of the monkeys must climb the tree and shake the branch bearing the fruit until the fruit comes loose and falls to the ground
bull A warifruit is worth 10 calories of energy Climbing the tree uses 2 calories for Big Monkey but uses no energy for Little Monkey who is smaller If Little Monkey climbs the tree and shakes it down Big Monkey will eat 90 of the fruit (or 9 calories) before Little Monkey climbs back down and Little Monkey will get only 10 of the fruit (or 1 calorie)
bull If Big Monkey climbs the tree and Little Monkey waits Little Monkey will get 40 of the fruit and Big Monkey will get 60 If both monkeys climb the tree Big Monkey will get 70 of the fruit and Little Monkey will get 30 Assume each monkey is simply interested in maximizing his caloric intake
bull Each monkey can decide to climb the tree or wait at the bottom bull a What is likely to happen if Big Monkey makes his decision first bull b What is likely to happen if Little Monkey must decide firstbull c What if they both decide simultaneously
5
Fundamental Tools bull Big Monkey (BM) ndash Little Monkey (LM)bull Warifruit from waritree (only one per tree) = 10 Caloriesbull Climb the tree to get the fruitbull Cost to get the fruit
ndash 2 Calories for Big Monkeyndash zero for Little Monkey
bull Payoff ndash Both climb BM 7 Calories ndash LM 3 Caloriesndash BM climbs BM 6 Calories ndash LM 4 Caloriesndash LM climbs BM 9 Calories ndash LM 1 Calories
bull What will they do to maximize payoff taking into account cost
6
Fundamental Tools Extensive form games--Definition
bull An extensive form game G consists of ndash Playersndash Game treendash Payoffs
bull Terminal node t i(t) (payoffs)bull G has tree property only 1 path from root to any
terminal nodebull Occurrence of stochastic event fictitious player Nature probability assigned to each branch of which Nature is head node
7
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull 3 possibilities ndash BM decides first what to dondash LM decides first what to dondash Both decide simultaneously
bull BM decides first
Big Monkey
Little Monkey Little Monkey
w c
w c w c
00 91 44 53
8
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull Strategies ndash BM
bull Wait (w)bull Climb (c)
ndash LM Actions are ordered depending on (wc) of BMbull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)
bull A series of actions that fully define the behavior of a player = strategy
bull A strategy for a player is a complete plan of how to plan the game and prescribes his choices at every information set (in this case node)
9
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull LM decides first
ndash The strategies are conversed Utility=(LM BM)
Little Monkey
Big Monkey Big Monkey
w c
w c w c
00 44 19 35
10
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull They choose simultaneously
bull Information Set a set of nodes at which
bull The same player choosesbull The player choosing does not know which node represents the
actual choice node ndash represented by dotted line
Big Monkey
Little Monkeyw c
w c w c
00 91 44 53
c w
c 53 44
w 91 00
LM
BM
11
bull The key to representing information in a game tree is realizing the connection between nodes and history
bull If you know which node you have reached you know precisely the history of the play
bull To express uncertainty we use concept of information set (set of nodes you could be in at a given time)
12
Composition of information sets
bull Each decision node is in exactly one information set
bull all nodes of an information set must belong to same player
bull every node of an information set must have exactly the same set of available actions
bull If every information set of every player is a singleton we have a game of perfect information
13
Fundamental Tools Normal form games--Definition
The n-player normal form game consists of ndash Players i = 1hellipnndash A set Si of strategies for player i = 1hellipn
We call s = (s1 hellip sn) where si Si for i = 1hellipn a strategy profile for the game Each si is a strategy for player i
ndash A function i S for player i = 1hellipn where S is the set of strategy profiles so i(s) is player irsquos payoff when strategy profile s is chosen
14
Fundamental Tools Normal form games--Illustration
bull Another way to depict the BM-LM game (where BM chooses first) LM Actions are ordered depending on (wc) of BM
bull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
15
Fundamental Tools Normal form games--Illustration
bull Donrsquot get rid of weakly dominated as lose equilibrium
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
16
Sequential games
bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)
bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)
17
bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa
bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered
bull ndash We need to define the concept of an equilibrium in extensive form games
18
Problems with Nash equilibrium
bull Sequential nature of the game is lost when representing extensive form games in strategic form
bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do
bull Nash equilibrium does not distinguish between credible and non-credible threats
19
Solving sequential games
bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo
bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and
work backwards eliminating all but the optimal choice for the relevant player
(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)
20
21
Subgame
bull Its game tree is a branch of the original game tree
bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch
bull The payoff vectors are the same as in the original game
22
Subgame perfect equilibrium amp credible threats
bull Proper subgame = subtree (of the game tree) whose root is alone in its information set
bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in
every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play
23
bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida
bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response
bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island
bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces
bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief
24
Example Cuban Missile Crisis
Khrushchev
Kennedy
Arm
Retract
Fold
Nuke
-1 1
- 100 - 100
10 -10
Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)
Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible
25
Backwards induction
bull Start from the smallest subgames containing the terminal nodes of the game tree
bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the
player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game
Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played
bull Repeat until there are no action nodes left
26
(MDBK) payoff
27
The predation game
bull Nasty Guys is an incumbent firm producing bricks
bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market
bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo
bull What should SIC do
28
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
2
Extensive Form Games
H
H H
T
TT
(12) (40)(21) (21)
Any finite game of perfect information has a pure strategy Nash equilibrium It can be found by backward induction
Chess is a finite game of perfect information Therefore it is a ldquotrivialrdquo game from a game theoretic point of view
3
Extensive Form Games - Intro
bull A game can have complex temporal structurebull Information
ndash set of playersndash who moves when and under what circumstancesndash what actions are available when called upon to movendash what is known when called upon to movendash what payoffs each player receives
bull Foundation is a game tree
4
bull Big Monkey and Little Monkey eat warifruit which dangle from the extreme tip of a lofty branch of the waritree
bull A waritree produces only one fruit To get the warifruit at least one of the monkeys must climb the tree and shake the branch bearing the fruit until the fruit comes loose and falls to the ground
bull A warifruit is worth 10 calories of energy Climbing the tree uses 2 calories for Big Monkey but uses no energy for Little Monkey who is smaller If Little Monkey climbs the tree and shakes it down Big Monkey will eat 90 of the fruit (or 9 calories) before Little Monkey climbs back down and Little Monkey will get only 10 of the fruit (or 1 calorie)
bull If Big Monkey climbs the tree and Little Monkey waits Little Monkey will get 40 of the fruit and Big Monkey will get 60 If both monkeys climb the tree Big Monkey will get 70 of the fruit and Little Monkey will get 30 Assume each monkey is simply interested in maximizing his caloric intake
bull Each monkey can decide to climb the tree or wait at the bottom bull a What is likely to happen if Big Monkey makes his decision first bull b What is likely to happen if Little Monkey must decide firstbull c What if they both decide simultaneously
5
Fundamental Tools bull Big Monkey (BM) ndash Little Monkey (LM)bull Warifruit from waritree (only one per tree) = 10 Caloriesbull Climb the tree to get the fruitbull Cost to get the fruit
ndash 2 Calories for Big Monkeyndash zero for Little Monkey
bull Payoff ndash Both climb BM 7 Calories ndash LM 3 Caloriesndash BM climbs BM 6 Calories ndash LM 4 Caloriesndash LM climbs BM 9 Calories ndash LM 1 Calories
bull What will they do to maximize payoff taking into account cost
6
Fundamental Tools Extensive form games--Definition
bull An extensive form game G consists of ndash Playersndash Game treendash Payoffs
bull Terminal node t i(t) (payoffs)bull G has tree property only 1 path from root to any
terminal nodebull Occurrence of stochastic event fictitious player Nature probability assigned to each branch of which Nature is head node
7
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull 3 possibilities ndash BM decides first what to dondash LM decides first what to dondash Both decide simultaneously
bull BM decides first
Big Monkey
Little Monkey Little Monkey
w c
w c w c
00 91 44 53
8
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull Strategies ndash BM
bull Wait (w)bull Climb (c)
ndash LM Actions are ordered depending on (wc) of BMbull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)
bull A series of actions that fully define the behavior of a player = strategy
bull A strategy for a player is a complete plan of how to plan the game and prescribes his choices at every information set (in this case node)
9
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull LM decides first
ndash The strategies are conversed Utility=(LM BM)
Little Monkey
Big Monkey Big Monkey
w c
w c w c
00 44 19 35
10
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull They choose simultaneously
bull Information Set a set of nodes at which
bull The same player choosesbull The player choosing does not know which node represents the
actual choice node ndash represented by dotted line
Big Monkey
Little Monkeyw c
w c w c
00 91 44 53
c w
c 53 44
w 91 00
LM
BM
11
bull The key to representing information in a game tree is realizing the connection between nodes and history
bull If you know which node you have reached you know precisely the history of the play
bull To express uncertainty we use concept of information set (set of nodes you could be in at a given time)
12
Composition of information sets
bull Each decision node is in exactly one information set
bull all nodes of an information set must belong to same player
bull every node of an information set must have exactly the same set of available actions
bull If every information set of every player is a singleton we have a game of perfect information
13
Fundamental Tools Normal form games--Definition
The n-player normal form game consists of ndash Players i = 1hellipnndash A set Si of strategies for player i = 1hellipn
We call s = (s1 hellip sn) where si Si for i = 1hellipn a strategy profile for the game Each si is a strategy for player i
ndash A function i S for player i = 1hellipn where S is the set of strategy profiles so i(s) is player irsquos payoff when strategy profile s is chosen
14
Fundamental Tools Normal form games--Illustration
bull Another way to depict the BM-LM game (where BM chooses first) LM Actions are ordered depending on (wc) of BM
bull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
15
Fundamental Tools Normal form games--Illustration
bull Donrsquot get rid of weakly dominated as lose equilibrium
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
16
Sequential games
bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)
bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)
17
bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa
bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered
bull ndash We need to define the concept of an equilibrium in extensive form games
18
Problems with Nash equilibrium
bull Sequential nature of the game is lost when representing extensive form games in strategic form
bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do
bull Nash equilibrium does not distinguish between credible and non-credible threats
19
Solving sequential games
bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo
bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and
work backwards eliminating all but the optimal choice for the relevant player
(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)
20
21
Subgame
bull Its game tree is a branch of the original game tree
bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch
bull The payoff vectors are the same as in the original game
22
Subgame perfect equilibrium amp credible threats
bull Proper subgame = subtree (of the game tree) whose root is alone in its information set
bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in
every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play
23
bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida
bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response
bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island
bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces
bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief
24
Example Cuban Missile Crisis
Khrushchev
Kennedy
Arm
Retract
Fold
Nuke
-1 1
- 100 - 100
10 -10
Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)
Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible
25
Backwards induction
bull Start from the smallest subgames containing the terminal nodes of the game tree
bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the
player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game
Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played
bull Repeat until there are no action nodes left
26
(MDBK) payoff
27
The predation game
bull Nasty Guys is an incumbent firm producing bricks
bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market
bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo
bull What should SIC do
28
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
3
Extensive Form Games - Intro
bull A game can have complex temporal structurebull Information
ndash set of playersndash who moves when and under what circumstancesndash what actions are available when called upon to movendash what is known when called upon to movendash what payoffs each player receives
bull Foundation is a game tree
4
bull Big Monkey and Little Monkey eat warifruit which dangle from the extreme tip of a lofty branch of the waritree
bull A waritree produces only one fruit To get the warifruit at least one of the monkeys must climb the tree and shake the branch bearing the fruit until the fruit comes loose and falls to the ground
bull A warifruit is worth 10 calories of energy Climbing the tree uses 2 calories for Big Monkey but uses no energy for Little Monkey who is smaller If Little Monkey climbs the tree and shakes it down Big Monkey will eat 90 of the fruit (or 9 calories) before Little Monkey climbs back down and Little Monkey will get only 10 of the fruit (or 1 calorie)
bull If Big Monkey climbs the tree and Little Monkey waits Little Monkey will get 40 of the fruit and Big Monkey will get 60 If both monkeys climb the tree Big Monkey will get 70 of the fruit and Little Monkey will get 30 Assume each monkey is simply interested in maximizing his caloric intake
bull Each monkey can decide to climb the tree or wait at the bottom bull a What is likely to happen if Big Monkey makes his decision first bull b What is likely to happen if Little Monkey must decide firstbull c What if they both decide simultaneously
5
Fundamental Tools bull Big Monkey (BM) ndash Little Monkey (LM)bull Warifruit from waritree (only one per tree) = 10 Caloriesbull Climb the tree to get the fruitbull Cost to get the fruit
ndash 2 Calories for Big Monkeyndash zero for Little Monkey
bull Payoff ndash Both climb BM 7 Calories ndash LM 3 Caloriesndash BM climbs BM 6 Calories ndash LM 4 Caloriesndash LM climbs BM 9 Calories ndash LM 1 Calories
bull What will they do to maximize payoff taking into account cost
6
Fundamental Tools Extensive form games--Definition
bull An extensive form game G consists of ndash Playersndash Game treendash Payoffs
bull Terminal node t i(t) (payoffs)bull G has tree property only 1 path from root to any
terminal nodebull Occurrence of stochastic event fictitious player Nature probability assigned to each branch of which Nature is head node
7
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull 3 possibilities ndash BM decides first what to dondash LM decides first what to dondash Both decide simultaneously
bull BM decides first
Big Monkey
Little Monkey Little Monkey
w c
w c w c
00 91 44 53
8
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull Strategies ndash BM
bull Wait (w)bull Climb (c)
ndash LM Actions are ordered depending on (wc) of BMbull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)
bull A series of actions that fully define the behavior of a player = strategy
bull A strategy for a player is a complete plan of how to plan the game and prescribes his choices at every information set (in this case node)
9
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull LM decides first
ndash The strategies are conversed Utility=(LM BM)
Little Monkey
Big Monkey Big Monkey
w c
w c w c
00 44 19 35
10
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull They choose simultaneously
bull Information Set a set of nodes at which
bull The same player choosesbull The player choosing does not know which node represents the
actual choice node ndash represented by dotted line
Big Monkey
Little Monkeyw c
w c w c
00 91 44 53
c w
c 53 44
w 91 00
LM
BM
11
bull The key to representing information in a game tree is realizing the connection between nodes and history
bull If you know which node you have reached you know precisely the history of the play
bull To express uncertainty we use concept of information set (set of nodes you could be in at a given time)
12
Composition of information sets
bull Each decision node is in exactly one information set
bull all nodes of an information set must belong to same player
bull every node of an information set must have exactly the same set of available actions
bull If every information set of every player is a singleton we have a game of perfect information
13
Fundamental Tools Normal form games--Definition
The n-player normal form game consists of ndash Players i = 1hellipnndash A set Si of strategies for player i = 1hellipn
We call s = (s1 hellip sn) where si Si for i = 1hellipn a strategy profile for the game Each si is a strategy for player i
ndash A function i S for player i = 1hellipn where S is the set of strategy profiles so i(s) is player irsquos payoff when strategy profile s is chosen
14
Fundamental Tools Normal form games--Illustration
bull Another way to depict the BM-LM game (where BM chooses first) LM Actions are ordered depending on (wc) of BM
bull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
15
Fundamental Tools Normal form games--Illustration
bull Donrsquot get rid of weakly dominated as lose equilibrium
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
16
Sequential games
bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)
bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)
17
bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa
bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered
bull ndash We need to define the concept of an equilibrium in extensive form games
18
Problems with Nash equilibrium
bull Sequential nature of the game is lost when representing extensive form games in strategic form
bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do
bull Nash equilibrium does not distinguish between credible and non-credible threats
19
Solving sequential games
bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo
bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and
work backwards eliminating all but the optimal choice for the relevant player
(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)
20
21
Subgame
bull Its game tree is a branch of the original game tree
bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch
bull The payoff vectors are the same as in the original game
22
Subgame perfect equilibrium amp credible threats
bull Proper subgame = subtree (of the game tree) whose root is alone in its information set
bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in
every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play
23
bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida
bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response
bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island
bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces
bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief
24
Example Cuban Missile Crisis
Khrushchev
Kennedy
Arm
Retract
Fold
Nuke
-1 1
- 100 - 100
10 -10
Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)
Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible
25
Backwards induction
bull Start from the smallest subgames containing the terminal nodes of the game tree
bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the
player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game
Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played
bull Repeat until there are no action nodes left
26
(MDBK) payoff
27
The predation game
bull Nasty Guys is an incumbent firm producing bricks
bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market
bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo
bull What should SIC do
28
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
4
bull Big Monkey and Little Monkey eat warifruit which dangle from the extreme tip of a lofty branch of the waritree
bull A waritree produces only one fruit To get the warifruit at least one of the monkeys must climb the tree and shake the branch bearing the fruit until the fruit comes loose and falls to the ground
bull A warifruit is worth 10 calories of energy Climbing the tree uses 2 calories for Big Monkey but uses no energy for Little Monkey who is smaller If Little Monkey climbs the tree and shakes it down Big Monkey will eat 90 of the fruit (or 9 calories) before Little Monkey climbs back down and Little Monkey will get only 10 of the fruit (or 1 calorie)
bull If Big Monkey climbs the tree and Little Monkey waits Little Monkey will get 40 of the fruit and Big Monkey will get 60 If both monkeys climb the tree Big Monkey will get 70 of the fruit and Little Monkey will get 30 Assume each monkey is simply interested in maximizing his caloric intake
bull Each monkey can decide to climb the tree or wait at the bottom bull a What is likely to happen if Big Monkey makes his decision first bull b What is likely to happen if Little Monkey must decide firstbull c What if they both decide simultaneously
5
Fundamental Tools bull Big Monkey (BM) ndash Little Monkey (LM)bull Warifruit from waritree (only one per tree) = 10 Caloriesbull Climb the tree to get the fruitbull Cost to get the fruit
ndash 2 Calories for Big Monkeyndash zero for Little Monkey
bull Payoff ndash Both climb BM 7 Calories ndash LM 3 Caloriesndash BM climbs BM 6 Calories ndash LM 4 Caloriesndash LM climbs BM 9 Calories ndash LM 1 Calories
bull What will they do to maximize payoff taking into account cost
6
Fundamental Tools Extensive form games--Definition
bull An extensive form game G consists of ndash Playersndash Game treendash Payoffs
bull Terminal node t i(t) (payoffs)bull G has tree property only 1 path from root to any
terminal nodebull Occurrence of stochastic event fictitious player Nature probability assigned to each branch of which Nature is head node
7
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull 3 possibilities ndash BM decides first what to dondash LM decides first what to dondash Both decide simultaneously
bull BM decides first
Big Monkey
Little Monkey Little Monkey
w c
w c w c
00 91 44 53
8
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull Strategies ndash BM
bull Wait (w)bull Climb (c)
ndash LM Actions are ordered depending on (wc) of BMbull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)
bull A series of actions that fully define the behavior of a player = strategy
bull A strategy for a player is a complete plan of how to plan the game and prescribes his choices at every information set (in this case node)
9
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull LM decides first
ndash The strategies are conversed Utility=(LM BM)
Little Monkey
Big Monkey Big Monkey
w c
w c w c
00 44 19 35
10
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull They choose simultaneously
bull Information Set a set of nodes at which
bull The same player choosesbull The player choosing does not know which node represents the
actual choice node ndash represented by dotted line
Big Monkey
Little Monkeyw c
w c w c
00 91 44 53
c w
c 53 44
w 91 00
LM
BM
11
bull The key to representing information in a game tree is realizing the connection between nodes and history
bull If you know which node you have reached you know precisely the history of the play
bull To express uncertainty we use concept of information set (set of nodes you could be in at a given time)
12
Composition of information sets
bull Each decision node is in exactly one information set
bull all nodes of an information set must belong to same player
bull every node of an information set must have exactly the same set of available actions
bull If every information set of every player is a singleton we have a game of perfect information
13
Fundamental Tools Normal form games--Definition
The n-player normal form game consists of ndash Players i = 1hellipnndash A set Si of strategies for player i = 1hellipn
We call s = (s1 hellip sn) where si Si for i = 1hellipn a strategy profile for the game Each si is a strategy for player i
ndash A function i S for player i = 1hellipn where S is the set of strategy profiles so i(s) is player irsquos payoff when strategy profile s is chosen
14
Fundamental Tools Normal form games--Illustration
bull Another way to depict the BM-LM game (where BM chooses first) LM Actions are ordered depending on (wc) of BM
bull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
15
Fundamental Tools Normal form games--Illustration
bull Donrsquot get rid of weakly dominated as lose equilibrium
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
16
Sequential games
bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)
bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)
17
bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa
bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered
bull ndash We need to define the concept of an equilibrium in extensive form games
18
Problems with Nash equilibrium
bull Sequential nature of the game is lost when representing extensive form games in strategic form
bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do
bull Nash equilibrium does not distinguish between credible and non-credible threats
19
Solving sequential games
bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo
bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and
work backwards eliminating all but the optimal choice for the relevant player
(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)
20
21
Subgame
bull Its game tree is a branch of the original game tree
bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch
bull The payoff vectors are the same as in the original game
22
Subgame perfect equilibrium amp credible threats
bull Proper subgame = subtree (of the game tree) whose root is alone in its information set
bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in
every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play
23
bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida
bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response
bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island
bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces
bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief
24
Example Cuban Missile Crisis
Khrushchev
Kennedy
Arm
Retract
Fold
Nuke
-1 1
- 100 - 100
10 -10
Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)
Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible
25
Backwards induction
bull Start from the smallest subgames containing the terminal nodes of the game tree
bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the
player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game
Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played
bull Repeat until there are no action nodes left
26
(MDBK) payoff
27
The predation game
bull Nasty Guys is an incumbent firm producing bricks
bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market
bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo
bull What should SIC do
28
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
5
Fundamental Tools bull Big Monkey (BM) ndash Little Monkey (LM)bull Warifruit from waritree (only one per tree) = 10 Caloriesbull Climb the tree to get the fruitbull Cost to get the fruit
ndash 2 Calories for Big Monkeyndash zero for Little Monkey
bull Payoff ndash Both climb BM 7 Calories ndash LM 3 Caloriesndash BM climbs BM 6 Calories ndash LM 4 Caloriesndash LM climbs BM 9 Calories ndash LM 1 Calories
bull What will they do to maximize payoff taking into account cost
6
Fundamental Tools Extensive form games--Definition
bull An extensive form game G consists of ndash Playersndash Game treendash Payoffs
bull Terminal node t i(t) (payoffs)bull G has tree property only 1 path from root to any
terminal nodebull Occurrence of stochastic event fictitious player Nature probability assigned to each branch of which Nature is head node
7
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull 3 possibilities ndash BM decides first what to dondash LM decides first what to dondash Both decide simultaneously
bull BM decides first
Big Monkey
Little Monkey Little Monkey
w c
w c w c
00 91 44 53
8
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull Strategies ndash BM
bull Wait (w)bull Climb (c)
ndash LM Actions are ordered depending on (wc) of BMbull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)
bull A series of actions that fully define the behavior of a player = strategy
bull A strategy for a player is a complete plan of how to plan the game and prescribes his choices at every information set (in this case node)
9
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull LM decides first
ndash The strategies are conversed Utility=(LM BM)
Little Monkey
Big Monkey Big Monkey
w c
w c w c
00 44 19 35
10
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull They choose simultaneously
bull Information Set a set of nodes at which
bull The same player choosesbull The player choosing does not know which node represents the
actual choice node ndash represented by dotted line
Big Monkey
Little Monkeyw c
w c w c
00 91 44 53
c w
c 53 44
w 91 00
LM
BM
11
bull The key to representing information in a game tree is realizing the connection between nodes and history
bull If you know which node you have reached you know precisely the history of the play
bull To express uncertainty we use concept of information set (set of nodes you could be in at a given time)
12
Composition of information sets
bull Each decision node is in exactly one information set
bull all nodes of an information set must belong to same player
bull every node of an information set must have exactly the same set of available actions
bull If every information set of every player is a singleton we have a game of perfect information
13
Fundamental Tools Normal form games--Definition
The n-player normal form game consists of ndash Players i = 1hellipnndash A set Si of strategies for player i = 1hellipn
We call s = (s1 hellip sn) where si Si for i = 1hellipn a strategy profile for the game Each si is a strategy for player i
ndash A function i S for player i = 1hellipn where S is the set of strategy profiles so i(s) is player irsquos payoff when strategy profile s is chosen
14
Fundamental Tools Normal form games--Illustration
bull Another way to depict the BM-LM game (where BM chooses first) LM Actions are ordered depending on (wc) of BM
bull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
15
Fundamental Tools Normal form games--Illustration
bull Donrsquot get rid of weakly dominated as lose equilibrium
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
16
Sequential games
bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)
bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)
17
bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa
bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered
bull ndash We need to define the concept of an equilibrium in extensive form games
18
Problems with Nash equilibrium
bull Sequential nature of the game is lost when representing extensive form games in strategic form
bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do
bull Nash equilibrium does not distinguish between credible and non-credible threats
19
Solving sequential games
bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo
bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and
work backwards eliminating all but the optimal choice for the relevant player
(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)
20
21
Subgame
bull Its game tree is a branch of the original game tree
bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch
bull The payoff vectors are the same as in the original game
22
Subgame perfect equilibrium amp credible threats
bull Proper subgame = subtree (of the game tree) whose root is alone in its information set
bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in
every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play
23
bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida
bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response
bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island
bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces
bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief
24
Example Cuban Missile Crisis
Khrushchev
Kennedy
Arm
Retract
Fold
Nuke
-1 1
- 100 - 100
10 -10
Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)
Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible
25
Backwards induction
bull Start from the smallest subgames containing the terminal nodes of the game tree
bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the
player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game
Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played
bull Repeat until there are no action nodes left
26
(MDBK) payoff
27
The predation game
bull Nasty Guys is an incumbent firm producing bricks
bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market
bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo
bull What should SIC do
28
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
6
Fundamental Tools Extensive form games--Definition
bull An extensive form game G consists of ndash Playersndash Game treendash Payoffs
bull Terminal node t i(t) (payoffs)bull G has tree property only 1 path from root to any
terminal nodebull Occurrence of stochastic event fictitious player Nature probability assigned to each branch of which Nature is head node
7
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull 3 possibilities ndash BM decides first what to dondash LM decides first what to dondash Both decide simultaneously
bull BM decides first
Big Monkey
Little Monkey Little Monkey
w c
w c w c
00 91 44 53
8
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull Strategies ndash BM
bull Wait (w)bull Climb (c)
ndash LM Actions are ordered depending on (wc) of BMbull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)
bull A series of actions that fully define the behavior of a player = strategy
bull A strategy for a player is a complete plan of how to plan the game and prescribes his choices at every information set (in this case node)
9
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull LM decides first
ndash The strategies are conversed Utility=(LM BM)
Little Monkey
Big Monkey Big Monkey
w c
w c w c
00 44 19 35
10
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull They choose simultaneously
bull Information Set a set of nodes at which
bull The same player choosesbull The player choosing does not know which node represents the
actual choice node ndash represented by dotted line
Big Monkey
Little Monkeyw c
w c w c
00 91 44 53
c w
c 53 44
w 91 00
LM
BM
11
bull The key to representing information in a game tree is realizing the connection between nodes and history
bull If you know which node you have reached you know precisely the history of the play
bull To express uncertainty we use concept of information set (set of nodes you could be in at a given time)
12
Composition of information sets
bull Each decision node is in exactly one information set
bull all nodes of an information set must belong to same player
bull every node of an information set must have exactly the same set of available actions
bull If every information set of every player is a singleton we have a game of perfect information
13
Fundamental Tools Normal form games--Definition
The n-player normal form game consists of ndash Players i = 1hellipnndash A set Si of strategies for player i = 1hellipn
We call s = (s1 hellip sn) where si Si for i = 1hellipn a strategy profile for the game Each si is a strategy for player i
ndash A function i S for player i = 1hellipn where S is the set of strategy profiles so i(s) is player irsquos payoff when strategy profile s is chosen
14
Fundamental Tools Normal form games--Illustration
bull Another way to depict the BM-LM game (where BM chooses first) LM Actions are ordered depending on (wc) of BM
bull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
15
Fundamental Tools Normal form games--Illustration
bull Donrsquot get rid of weakly dominated as lose equilibrium
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
16
Sequential games
bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)
bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)
17
bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa
bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered
bull ndash We need to define the concept of an equilibrium in extensive form games
18
Problems with Nash equilibrium
bull Sequential nature of the game is lost when representing extensive form games in strategic form
bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do
bull Nash equilibrium does not distinguish between credible and non-credible threats
19
Solving sequential games
bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo
bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and
work backwards eliminating all but the optimal choice for the relevant player
(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)
20
21
Subgame
bull Its game tree is a branch of the original game tree
bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch
bull The payoff vectors are the same as in the original game
22
Subgame perfect equilibrium amp credible threats
bull Proper subgame = subtree (of the game tree) whose root is alone in its information set
bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in
every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play
23
bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida
bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response
bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island
bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces
bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief
24
Example Cuban Missile Crisis
Khrushchev
Kennedy
Arm
Retract
Fold
Nuke
-1 1
- 100 - 100
10 -10
Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)
Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible
25
Backwards induction
bull Start from the smallest subgames containing the terminal nodes of the game tree
bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the
player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game
Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played
bull Repeat until there are no action nodes left
26
(MDBK) payoff
27
The predation game
bull Nasty Guys is an incumbent firm producing bricks
bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market
bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo
bull What should SIC do
28
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
7
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull 3 possibilities ndash BM decides first what to dondash LM decides first what to dondash Both decide simultaneously
bull BM decides first
Big Monkey
Little Monkey Little Monkey
w c
w c w c
00 91 44 53
8
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull Strategies ndash BM
bull Wait (w)bull Climb (c)
ndash LM Actions are ordered depending on (wc) of BMbull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)
bull A series of actions that fully define the behavior of a player = strategy
bull A strategy for a player is a complete plan of how to plan the game and prescribes his choices at every information set (in this case node)
9
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull LM decides first
ndash The strategies are conversed Utility=(LM BM)
Little Monkey
Big Monkey Big Monkey
w c
w c w c
00 44 19 35
10
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull They choose simultaneously
bull Information Set a set of nodes at which
bull The same player choosesbull The player choosing does not know which node represents the
actual choice node ndash represented by dotted line
Big Monkey
Little Monkeyw c
w c w c
00 91 44 53
c w
c 53 44
w 91 00
LM
BM
11
bull The key to representing information in a game tree is realizing the connection between nodes and history
bull If you know which node you have reached you know precisely the history of the play
bull To express uncertainty we use concept of information set (set of nodes you could be in at a given time)
12
Composition of information sets
bull Each decision node is in exactly one information set
bull all nodes of an information set must belong to same player
bull every node of an information set must have exactly the same set of available actions
bull If every information set of every player is a singleton we have a game of perfect information
13
Fundamental Tools Normal form games--Definition
The n-player normal form game consists of ndash Players i = 1hellipnndash A set Si of strategies for player i = 1hellipn
We call s = (s1 hellip sn) where si Si for i = 1hellipn a strategy profile for the game Each si is a strategy for player i
ndash A function i S for player i = 1hellipn where S is the set of strategy profiles so i(s) is player irsquos payoff when strategy profile s is chosen
14
Fundamental Tools Normal form games--Illustration
bull Another way to depict the BM-LM game (where BM chooses first) LM Actions are ordered depending on (wc) of BM
bull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
15
Fundamental Tools Normal form games--Illustration
bull Donrsquot get rid of weakly dominated as lose equilibrium
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
16
Sequential games
bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)
bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)
17
bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa
bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered
bull ndash We need to define the concept of an equilibrium in extensive form games
18
Problems with Nash equilibrium
bull Sequential nature of the game is lost when representing extensive form games in strategic form
bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do
bull Nash equilibrium does not distinguish between credible and non-credible threats
19
Solving sequential games
bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo
bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and
work backwards eliminating all but the optimal choice for the relevant player
(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)
20
21
Subgame
bull Its game tree is a branch of the original game tree
bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch
bull The payoff vectors are the same as in the original game
22
Subgame perfect equilibrium amp credible threats
bull Proper subgame = subtree (of the game tree) whose root is alone in its information set
bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in
every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play
23
bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida
bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response
bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island
bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces
bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief
24
Example Cuban Missile Crisis
Khrushchev
Kennedy
Arm
Retract
Fold
Nuke
-1 1
- 100 - 100
10 -10
Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)
Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible
25
Backwards induction
bull Start from the smallest subgames containing the terminal nodes of the game tree
bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the
player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game
Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played
bull Repeat until there are no action nodes left
26
(MDBK) payoff
27
The predation game
bull Nasty Guys is an incumbent firm producing bricks
bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market
bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo
bull What should SIC do
28
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
8
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull Strategies ndash BM
bull Wait (w)bull Climb (c)
ndash LM Actions are ordered depending on (wc) of BMbull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)
bull A series of actions that fully define the behavior of a player = strategy
bull A strategy for a player is a complete plan of how to plan the game and prescribes his choices at every information set (in this case node)
9
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull LM decides first
ndash The strategies are conversed Utility=(LM BM)
Little Monkey
Big Monkey Big Monkey
w c
w c w c
00 44 19 35
10
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull They choose simultaneously
bull Information Set a set of nodes at which
bull The same player choosesbull The player choosing does not know which node represents the
actual choice node ndash represented by dotted line
Big Monkey
Little Monkeyw c
w c w c
00 91 44 53
c w
c 53 44
w 91 00
LM
BM
11
bull The key to representing information in a game tree is realizing the connection between nodes and history
bull If you know which node you have reached you know precisely the history of the play
bull To express uncertainty we use concept of information set (set of nodes you could be in at a given time)
12
Composition of information sets
bull Each decision node is in exactly one information set
bull all nodes of an information set must belong to same player
bull every node of an information set must have exactly the same set of available actions
bull If every information set of every player is a singleton we have a game of perfect information
13
Fundamental Tools Normal form games--Definition
The n-player normal form game consists of ndash Players i = 1hellipnndash A set Si of strategies for player i = 1hellipn
We call s = (s1 hellip sn) where si Si for i = 1hellipn a strategy profile for the game Each si is a strategy for player i
ndash A function i S for player i = 1hellipn where S is the set of strategy profiles so i(s) is player irsquos payoff when strategy profile s is chosen
14
Fundamental Tools Normal form games--Illustration
bull Another way to depict the BM-LM game (where BM chooses first) LM Actions are ordered depending on (wc) of BM
bull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
15
Fundamental Tools Normal form games--Illustration
bull Donrsquot get rid of weakly dominated as lose equilibrium
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
16
Sequential games
bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)
bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)
17
bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa
bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered
bull ndash We need to define the concept of an equilibrium in extensive form games
18
Problems with Nash equilibrium
bull Sequential nature of the game is lost when representing extensive form games in strategic form
bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do
bull Nash equilibrium does not distinguish between credible and non-credible threats
19
Solving sequential games
bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo
bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and
work backwards eliminating all but the optimal choice for the relevant player
(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)
20
21
Subgame
bull Its game tree is a branch of the original game tree
bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch
bull The payoff vectors are the same as in the original game
22
Subgame perfect equilibrium amp credible threats
bull Proper subgame = subtree (of the game tree) whose root is alone in its information set
bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in
every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play
23
bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida
bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response
bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island
bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces
bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief
24
Example Cuban Missile Crisis
Khrushchev
Kennedy
Arm
Retract
Fold
Nuke
-1 1
- 100 - 100
10 -10
Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)
Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible
25
Backwards induction
bull Start from the smallest subgames containing the terminal nodes of the game tree
bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the
player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game
Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played
bull Repeat until there are no action nodes left
26
(MDBK) payoff
27
The predation game
bull Nasty Guys is an incumbent firm producing bricks
bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market
bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo
bull What should SIC do
28
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
9
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull LM decides first
ndash The strategies are conversed Utility=(LM BM)
Little Monkey
Big Monkey Big Monkey
w c
w c w c
00 44 19 35
10
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull They choose simultaneously
bull Information Set a set of nodes at which
bull The same player choosesbull The player choosing does not know which node represents the
actual choice node ndash represented by dotted line
Big Monkey
Little Monkeyw c
w c w c
00 91 44 53
c w
c 53 44
w 91 00
LM
BM
11
bull The key to representing information in a game tree is realizing the connection between nodes and history
bull If you know which node you have reached you know precisely the history of the play
bull To express uncertainty we use concept of information set (set of nodes you could be in at a given time)
12
Composition of information sets
bull Each decision node is in exactly one information set
bull all nodes of an information set must belong to same player
bull every node of an information set must have exactly the same set of available actions
bull If every information set of every player is a singleton we have a game of perfect information
13
Fundamental Tools Normal form games--Definition
The n-player normal form game consists of ndash Players i = 1hellipnndash A set Si of strategies for player i = 1hellipn
We call s = (s1 hellip sn) where si Si for i = 1hellipn a strategy profile for the game Each si is a strategy for player i
ndash A function i S for player i = 1hellipn where S is the set of strategy profiles so i(s) is player irsquos payoff when strategy profile s is chosen
14
Fundamental Tools Normal form games--Illustration
bull Another way to depict the BM-LM game (where BM chooses first) LM Actions are ordered depending on (wc) of BM
bull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
15
Fundamental Tools Normal form games--Illustration
bull Donrsquot get rid of weakly dominated as lose equilibrium
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
16
Sequential games
bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)
bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)
17
bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa
bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered
bull ndash We need to define the concept of an equilibrium in extensive form games
18
Problems with Nash equilibrium
bull Sequential nature of the game is lost when representing extensive form games in strategic form
bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do
bull Nash equilibrium does not distinguish between credible and non-credible threats
19
Solving sequential games
bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo
bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and
work backwards eliminating all but the optimal choice for the relevant player
(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)
20
21
Subgame
bull Its game tree is a branch of the original game tree
bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch
bull The payoff vectors are the same as in the original game
22
Subgame perfect equilibrium amp credible threats
bull Proper subgame = subtree (of the game tree) whose root is alone in its information set
bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in
every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play
23
bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida
bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response
bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island
bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces
bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief
24
Example Cuban Missile Crisis
Khrushchev
Kennedy
Arm
Retract
Fold
Nuke
-1 1
- 100 - 100
10 -10
Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)
Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible
25
Backwards induction
bull Start from the smallest subgames containing the terminal nodes of the game tree
bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the
player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game
Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played
bull Repeat until there are no action nodes left
26
(MDBK) payoff
27
The predation game
bull Nasty Guys is an incumbent firm producing bricks
bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market
bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo
bull What should SIC do
28
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
10
Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
bull They choose simultaneously
bull Information Set a set of nodes at which
bull The same player choosesbull The player choosing does not know which node represents the
actual choice node ndash represented by dotted line
Big Monkey
Little Monkeyw c
w c w c
00 91 44 53
c w
c 53 44
w 91 00
LM
BM
11
bull The key to representing information in a game tree is realizing the connection between nodes and history
bull If you know which node you have reached you know precisely the history of the play
bull To express uncertainty we use concept of information set (set of nodes you could be in at a given time)
12
Composition of information sets
bull Each decision node is in exactly one information set
bull all nodes of an information set must belong to same player
bull every node of an information set must have exactly the same set of available actions
bull If every information set of every player is a singleton we have a game of perfect information
13
Fundamental Tools Normal form games--Definition
The n-player normal form game consists of ndash Players i = 1hellipnndash A set Si of strategies for player i = 1hellipn
We call s = (s1 hellip sn) where si Si for i = 1hellipn a strategy profile for the game Each si is a strategy for player i
ndash A function i S for player i = 1hellipn where S is the set of strategy profiles so i(s) is player irsquos payoff when strategy profile s is chosen
14
Fundamental Tools Normal form games--Illustration
bull Another way to depict the BM-LM game (where BM chooses first) LM Actions are ordered depending on (wc) of BM
bull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
15
Fundamental Tools Normal form games--Illustration
bull Donrsquot get rid of weakly dominated as lose equilibrium
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
16
Sequential games
bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)
bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)
17
bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa
bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered
bull ndash We need to define the concept of an equilibrium in extensive form games
18
Problems with Nash equilibrium
bull Sequential nature of the game is lost when representing extensive form games in strategic form
bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do
bull Nash equilibrium does not distinguish between credible and non-credible threats
19
Solving sequential games
bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo
bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and
work backwards eliminating all but the optimal choice for the relevant player
(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)
20
21
Subgame
bull Its game tree is a branch of the original game tree
bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch
bull The payoff vectors are the same as in the original game
22
Subgame perfect equilibrium amp credible threats
bull Proper subgame = subtree (of the game tree) whose root is alone in its information set
bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in
every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play
23
bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida
bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response
bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island
bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces
bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief
24
Example Cuban Missile Crisis
Khrushchev
Kennedy
Arm
Retract
Fold
Nuke
-1 1
- 100 - 100
10 -10
Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)
Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible
25
Backwards induction
bull Start from the smallest subgames containing the terminal nodes of the game tree
bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the
player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game
Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played
bull Repeat until there are no action nodes left
26
(MDBK) payoff
27
The predation game
bull Nasty Guys is an incumbent firm producing bricks
bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market
bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo
bull What should SIC do
28
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
11
bull The key to representing information in a game tree is realizing the connection between nodes and history
bull If you know which node you have reached you know precisely the history of the play
bull To express uncertainty we use concept of information set (set of nodes you could be in at a given time)
12
Composition of information sets
bull Each decision node is in exactly one information set
bull all nodes of an information set must belong to same player
bull every node of an information set must have exactly the same set of available actions
bull If every information set of every player is a singleton we have a game of perfect information
13
Fundamental Tools Normal form games--Definition
The n-player normal form game consists of ndash Players i = 1hellipnndash A set Si of strategies for player i = 1hellipn
We call s = (s1 hellip sn) where si Si for i = 1hellipn a strategy profile for the game Each si is a strategy for player i
ndash A function i S for player i = 1hellipn where S is the set of strategy profiles so i(s) is player irsquos payoff when strategy profile s is chosen
14
Fundamental Tools Normal form games--Illustration
bull Another way to depict the BM-LM game (where BM chooses first) LM Actions are ordered depending on (wc) of BM
bull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
15
Fundamental Tools Normal form games--Illustration
bull Donrsquot get rid of weakly dominated as lose equilibrium
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
16
Sequential games
bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)
bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)
17
bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa
bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered
bull ndash We need to define the concept of an equilibrium in extensive form games
18
Problems with Nash equilibrium
bull Sequential nature of the game is lost when representing extensive form games in strategic form
bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do
bull Nash equilibrium does not distinguish between credible and non-credible threats
19
Solving sequential games
bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo
bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and
work backwards eliminating all but the optimal choice for the relevant player
(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)
20
21
Subgame
bull Its game tree is a branch of the original game tree
bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch
bull The payoff vectors are the same as in the original game
22
Subgame perfect equilibrium amp credible threats
bull Proper subgame = subtree (of the game tree) whose root is alone in its information set
bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in
every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play
23
bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida
bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response
bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island
bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces
bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief
24
Example Cuban Missile Crisis
Khrushchev
Kennedy
Arm
Retract
Fold
Nuke
-1 1
- 100 - 100
10 -10
Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)
Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible
25
Backwards induction
bull Start from the smallest subgames containing the terminal nodes of the game tree
bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the
player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game
Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played
bull Repeat until there are no action nodes left
26
(MDBK) payoff
27
The predation game
bull Nasty Guys is an incumbent firm producing bricks
bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market
bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo
bull What should SIC do
28
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
12
Composition of information sets
bull Each decision node is in exactly one information set
bull all nodes of an information set must belong to same player
bull every node of an information set must have exactly the same set of available actions
bull If every information set of every player is a singleton we have a game of perfect information
13
Fundamental Tools Normal form games--Definition
The n-player normal form game consists of ndash Players i = 1hellipnndash A set Si of strategies for player i = 1hellipn
We call s = (s1 hellip sn) where si Si for i = 1hellipn a strategy profile for the game Each si is a strategy for player i
ndash A function i S for player i = 1hellipn where S is the set of strategy profiles so i(s) is player irsquos payoff when strategy profile s is chosen
14
Fundamental Tools Normal form games--Illustration
bull Another way to depict the BM-LM game (where BM chooses first) LM Actions are ordered depending on (wc) of BM
bull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
15
Fundamental Tools Normal form games--Illustration
bull Donrsquot get rid of weakly dominated as lose equilibrium
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
16
Sequential games
bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)
bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)
17
bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa
bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered
bull ndash We need to define the concept of an equilibrium in extensive form games
18
Problems with Nash equilibrium
bull Sequential nature of the game is lost when representing extensive form games in strategic form
bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do
bull Nash equilibrium does not distinguish between credible and non-credible threats
19
Solving sequential games
bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo
bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and
work backwards eliminating all but the optimal choice for the relevant player
(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)
20
21
Subgame
bull Its game tree is a branch of the original game tree
bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch
bull The payoff vectors are the same as in the original game
22
Subgame perfect equilibrium amp credible threats
bull Proper subgame = subtree (of the game tree) whose root is alone in its information set
bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in
every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play
23
bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida
bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response
bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island
bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces
bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief
24
Example Cuban Missile Crisis
Khrushchev
Kennedy
Arm
Retract
Fold
Nuke
-1 1
- 100 - 100
10 -10
Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)
Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible
25
Backwards induction
bull Start from the smallest subgames containing the terminal nodes of the game tree
bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the
player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game
Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played
bull Repeat until there are no action nodes left
26
(MDBK) payoff
27
The predation game
bull Nasty Guys is an incumbent firm producing bricks
bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market
bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo
bull What should SIC do
28
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
13
Fundamental Tools Normal form games--Definition
The n-player normal form game consists of ndash Players i = 1hellipnndash A set Si of strategies for player i = 1hellipn
We call s = (s1 hellip sn) where si Si for i = 1hellipn a strategy profile for the game Each si is a strategy for player i
ndash A function i S for player i = 1hellipn where S is the set of strategy profiles so i(s) is player irsquos payoff when strategy profile s is chosen
14
Fundamental Tools Normal form games--Illustration
bull Another way to depict the BM-LM game (where BM chooses first) LM Actions are ordered depending on (wc) of BM
bull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
15
Fundamental Tools Normal form games--Illustration
bull Donrsquot get rid of weakly dominated as lose equilibrium
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
16
Sequential games
bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)
bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)
17
bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa
bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered
bull ndash We need to define the concept of an equilibrium in extensive form games
18
Problems with Nash equilibrium
bull Sequential nature of the game is lost when representing extensive form games in strategic form
bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do
bull Nash equilibrium does not distinguish between credible and non-credible threats
19
Solving sequential games
bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo
bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and
work backwards eliminating all but the optimal choice for the relevant player
(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)
20
21
Subgame
bull Its game tree is a branch of the original game tree
bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch
bull The payoff vectors are the same as in the original game
22
Subgame perfect equilibrium amp credible threats
bull Proper subgame = subtree (of the game tree) whose root is alone in its information set
bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in
every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play
23
bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida
bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response
bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island
bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces
bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief
24
Example Cuban Missile Crisis
Khrushchev
Kennedy
Arm
Retract
Fold
Nuke
-1 1
- 100 - 100
10 -10
Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)
Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible
25
Backwards induction
bull Start from the smallest subgames containing the terminal nodes of the game tree
bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the
player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game
Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played
bull Repeat until there are no action nodes left
26
(MDBK) payoff
27
The predation game
bull Nasty Guys is an incumbent firm producing bricks
bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market
bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo
bull What should SIC do
28
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
14
Fundamental Tools Normal form games--Illustration
bull Another way to depict the BM-LM game (where BM chooses first) LM Actions are ordered depending on (wc) of BM
bull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
15
Fundamental Tools Normal form games--Illustration
bull Donrsquot get rid of weakly dominated as lose equilibrium
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
16
Sequential games
bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)
bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)
17
bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa
bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered
bull ndash We need to define the concept of an equilibrium in extensive form games
18
Problems with Nash equilibrium
bull Sequential nature of the game is lost when representing extensive form games in strategic form
bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do
bull Nash equilibrium does not distinguish between credible and non-credible threats
19
Solving sequential games
bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo
bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and
work backwards eliminating all but the optimal choice for the relevant player
(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)
20
21
Subgame
bull Its game tree is a branch of the original game tree
bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch
bull The payoff vectors are the same as in the original game
22
Subgame perfect equilibrium amp credible threats
bull Proper subgame = subtree (of the game tree) whose root is alone in its information set
bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in
every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play
23
bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida
bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response
bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island
bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces
bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief
24
Example Cuban Missile Crisis
Khrushchev
Kennedy
Arm
Retract
Fold
Nuke
-1 1
- 100 - 100
10 -10
Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)
Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible
25
Backwards induction
bull Start from the smallest subgames containing the terminal nodes of the game tree
bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the
player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game
Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played
bull Repeat until there are no action nodes left
26
(MDBK) payoff
27
The predation game
bull Nasty Guys is an incumbent firm producing bricks
bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market
bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo
bull What should SIC do
28
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
15
Fundamental Tools Normal form games--Illustration
bull Donrsquot get rid of weakly dominated as lose equilibrium
LM
BMcc cw wc ww
w 91 91 00 00
c 53 44 53 44
16
Sequential games
bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)
bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)
17
bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa
bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered
bull ndash We need to define the concept of an equilibrium in extensive form games
18
Problems with Nash equilibrium
bull Sequential nature of the game is lost when representing extensive form games in strategic form
bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do
bull Nash equilibrium does not distinguish between credible and non-credible threats
19
Solving sequential games
bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo
bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and
work backwards eliminating all but the optimal choice for the relevant player
(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)
20
21
Subgame
bull Its game tree is a branch of the original game tree
bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch
bull The payoff vectors are the same as in the original game
22
Subgame perfect equilibrium amp credible threats
bull Proper subgame = subtree (of the game tree) whose root is alone in its information set
bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in
every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play
23
bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida
bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response
bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island
bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces
bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief
24
Example Cuban Missile Crisis
Khrushchev
Kennedy
Arm
Retract
Fold
Nuke
-1 1
- 100 - 100
10 -10
Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)
Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible
25
Backwards induction
bull Start from the smallest subgames containing the terminal nodes of the game tree
bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the
player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game
Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played
bull Repeat until there are no action nodes left
26
(MDBK) payoff
27
The predation game
bull Nasty Guys is an incumbent firm producing bricks
bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market
bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo
bull What should SIC do
28
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
16
Sequential games
bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)
bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)
17
bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa
bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered
bull ndash We need to define the concept of an equilibrium in extensive form games
18
Problems with Nash equilibrium
bull Sequential nature of the game is lost when representing extensive form games in strategic form
bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do
bull Nash equilibrium does not distinguish between credible and non-credible threats
19
Solving sequential games
bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo
bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and
work backwards eliminating all but the optimal choice for the relevant player
(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)
20
21
Subgame
bull Its game tree is a branch of the original game tree
bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch
bull The payoff vectors are the same as in the original game
22
Subgame perfect equilibrium amp credible threats
bull Proper subgame = subtree (of the game tree) whose root is alone in its information set
bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in
every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play
23
bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida
bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response
bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island
bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces
bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief
24
Example Cuban Missile Crisis
Khrushchev
Kennedy
Arm
Retract
Fold
Nuke
-1 1
- 100 - 100
10 -10
Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)
Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible
25
Backwards induction
bull Start from the smallest subgames containing the terminal nodes of the game tree
bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the
player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game
Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played
bull Repeat until there are no action nodes left
26
(MDBK) payoff
27
The predation game
bull Nasty Guys is an incumbent firm producing bricks
bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market
bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo
bull What should SIC do
28
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
17
bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa
bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered
bull ndash We need to define the concept of an equilibrium in extensive form games
18
Problems with Nash equilibrium
bull Sequential nature of the game is lost when representing extensive form games in strategic form
bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do
bull Nash equilibrium does not distinguish between credible and non-credible threats
19
Solving sequential games
bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo
bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and
work backwards eliminating all but the optimal choice for the relevant player
(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)
20
21
Subgame
bull Its game tree is a branch of the original game tree
bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch
bull The payoff vectors are the same as in the original game
22
Subgame perfect equilibrium amp credible threats
bull Proper subgame = subtree (of the game tree) whose root is alone in its information set
bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in
every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play
23
bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida
bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response
bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island
bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces
bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief
24
Example Cuban Missile Crisis
Khrushchev
Kennedy
Arm
Retract
Fold
Nuke
-1 1
- 100 - 100
10 -10
Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)
Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible
25
Backwards induction
bull Start from the smallest subgames containing the terminal nodes of the game tree
bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the
player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game
Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played
bull Repeat until there are no action nodes left
26
(MDBK) payoff
27
The predation game
bull Nasty Guys is an incumbent firm producing bricks
bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market
bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo
bull What should SIC do
28
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
18
Problems with Nash equilibrium
bull Sequential nature of the game is lost when representing extensive form games in strategic form
bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do
bull Nash equilibrium does not distinguish between credible and non-credible threats
19
Solving sequential games
bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo
bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and
work backwards eliminating all but the optimal choice for the relevant player
(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)
20
21
Subgame
bull Its game tree is a branch of the original game tree
bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch
bull The payoff vectors are the same as in the original game
22
Subgame perfect equilibrium amp credible threats
bull Proper subgame = subtree (of the game tree) whose root is alone in its information set
bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in
every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play
23
bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida
bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response
bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island
bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces
bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief
24
Example Cuban Missile Crisis
Khrushchev
Kennedy
Arm
Retract
Fold
Nuke
-1 1
- 100 - 100
10 -10
Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)
Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible
25
Backwards induction
bull Start from the smallest subgames containing the terminal nodes of the game tree
bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the
player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game
Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played
bull Repeat until there are no action nodes left
26
(MDBK) payoff
27
The predation game
bull Nasty Guys is an incumbent firm producing bricks
bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market
bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo
bull What should SIC do
28
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
19
Solving sequential games
bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo
bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and
work backwards eliminating all but the optimal choice for the relevant player
(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)
20
21
Subgame
bull Its game tree is a branch of the original game tree
bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch
bull The payoff vectors are the same as in the original game
22
Subgame perfect equilibrium amp credible threats
bull Proper subgame = subtree (of the game tree) whose root is alone in its information set
bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in
every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play
23
bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida
bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response
bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island
bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces
bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief
24
Example Cuban Missile Crisis
Khrushchev
Kennedy
Arm
Retract
Fold
Nuke
-1 1
- 100 - 100
10 -10
Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)
Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible
25
Backwards induction
bull Start from the smallest subgames containing the terminal nodes of the game tree
bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the
player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game
Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played
bull Repeat until there are no action nodes left
26
(MDBK) payoff
27
The predation game
bull Nasty Guys is an incumbent firm producing bricks
bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market
bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo
bull What should SIC do
28
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
20
21
Subgame
bull Its game tree is a branch of the original game tree
bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch
bull The payoff vectors are the same as in the original game
22
Subgame perfect equilibrium amp credible threats
bull Proper subgame = subtree (of the game tree) whose root is alone in its information set
bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in
every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play
23
bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida
bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response
bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island
bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces
bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief
24
Example Cuban Missile Crisis
Khrushchev
Kennedy
Arm
Retract
Fold
Nuke
-1 1
- 100 - 100
10 -10
Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)
Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible
25
Backwards induction
bull Start from the smallest subgames containing the terminal nodes of the game tree
bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the
player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game
Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played
bull Repeat until there are no action nodes left
26
(MDBK) payoff
27
The predation game
bull Nasty Guys is an incumbent firm producing bricks
bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market
bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo
bull What should SIC do
28
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
21
Subgame
bull Its game tree is a branch of the original game tree
bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch
bull The payoff vectors are the same as in the original game
22
Subgame perfect equilibrium amp credible threats
bull Proper subgame = subtree (of the game tree) whose root is alone in its information set
bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in
every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play
23
bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida
bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response
bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island
bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces
bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief
24
Example Cuban Missile Crisis
Khrushchev
Kennedy
Arm
Retract
Fold
Nuke
-1 1
- 100 - 100
10 -10
Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)
Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible
25
Backwards induction
bull Start from the smallest subgames containing the terminal nodes of the game tree
bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the
player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game
Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played
bull Repeat until there are no action nodes left
26
(MDBK) payoff
27
The predation game
bull Nasty Guys is an incumbent firm producing bricks
bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market
bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo
bull What should SIC do
28
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
22
Subgame perfect equilibrium amp credible threats
bull Proper subgame = subtree (of the game tree) whose root is alone in its information set
bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in
every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play
23
bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida
bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response
bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island
bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces
bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief
24
Example Cuban Missile Crisis
Khrushchev
Kennedy
Arm
Retract
Fold
Nuke
-1 1
- 100 - 100
10 -10
Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)
Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible
25
Backwards induction
bull Start from the smallest subgames containing the terminal nodes of the game tree
bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the
player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game
Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played
bull Repeat until there are no action nodes left
26
(MDBK) payoff
27
The predation game
bull Nasty Guys is an incumbent firm producing bricks
bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market
bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo
bull What should SIC do
28
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
23
bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida
bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response
bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island
bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces
bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief
24
Example Cuban Missile Crisis
Khrushchev
Kennedy
Arm
Retract
Fold
Nuke
-1 1
- 100 - 100
10 -10
Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)
Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible
25
Backwards induction
bull Start from the smallest subgames containing the terminal nodes of the game tree
bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the
player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game
Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played
bull Repeat until there are no action nodes left
26
(MDBK) payoff
27
The predation game
bull Nasty Guys is an incumbent firm producing bricks
bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market
bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo
bull What should SIC do
28
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
24
Example Cuban Missile Crisis
Khrushchev
Kennedy
Arm
Retract
Fold
Nuke
-1 1
- 100 - 100
10 -10
Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)
Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible
25
Backwards induction
bull Start from the smallest subgames containing the terminal nodes of the game tree
bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the
player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game
Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played
bull Repeat until there are no action nodes left
26
(MDBK) payoff
27
The predation game
bull Nasty Guys is an incumbent firm producing bricks
bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market
bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo
bull What should SIC do
28
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
25
Backwards induction
bull Start from the smallest subgames containing the terminal nodes of the game tree
bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the
player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game
Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played
bull Repeat until there are no action nodes left
26
(MDBK) payoff
27
The predation game
bull Nasty Guys is an incumbent firm producing bricks
bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market
bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo
bull What should SIC do
28
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
26
(MDBK) payoff
27
The predation game
bull Nasty Guys is an incumbent firm producing bricks
bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market
bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo
bull What should SIC do
28
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
27
The predation game
bull Nasty Guys is an incumbent firm producing bricks
bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market
bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo
bull What should SIC do
28
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
28
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
29
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
30
The predation game
SIC = -10 NG = -10
SIC = 30 NG = 30
Fight
Donrsquot fight
NGEnter
SIC
Donrsquot enter
SIC=0 NG=100
So the equilibrium is
SIC will enter
NG will not fight
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
31
Credible commitments
bull When Cortes arrived in Mexico he ordered that his ships should be burnt
bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo
home
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
32
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
Think of Cortes trying to motivate his own soldiers
Fight Hard
Be careful
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
33
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo
Fight Hard
Be careful
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
34
C
Burn ships
Keep Ships
S
S Be careful
Fight Hard
C = 100 S = 0
C = -100 S = -100
C = 100 S = 0
C = 0 S = 10
So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave
Fight Hard
Be careful
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
35
Hold up
bull Hold up occurs if one party has to incur sunk costs before they bargain with another party
bull For example hardware manufacturers and software developersndash Hardware manufacturers want software
manufacturers to make applications for their hardware
ndash But most of the cost of software is sunkndash So if bargain after the software is designed the
hardware manufacturer can seize most of the benefits
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
36
Holdup in equilibrium no-one designs software
payoffs = (software nintendo)
SoftwareDesigner
Nintendo(-$50000 $250000)
($100000 $100000)
(0 0)
design
Donrsquot design
Bargain hard (= Pay low price)
Bargainldquosoftrdquo
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
37
Strategies in extensive form
A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move
Key pointsndashIt is not sufficient to specify responses only at
those action nodes that are arrived at via some particular sequence of plausible play
ndashA strategy must prescribe an action at any action node where that player moves node where that player
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
38
Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset
of the nodes in a game tree belonging to player P such that
- All iI belong to P- For ijI there is no path from
i to j- All iI have the same number
of outgoing edges
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
39
Sequential Prisonerrsquos Dilemma
dotted line means P2 doesnrsquot know which state he is in
P1
P2 P2
Confess
Confess Confess Deny
Deny
Deny
(-5-5) (0-10) (-100) (-1-1)
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
40
bull With perfect information ndash each information set is a singleton (as you always know which state you are in)
bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)
bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path
bull A Nash equilbrium in sequential game (perfect or imperfect)
bull U(si s-i) gtU(si s-i) for all i
bull Note there can be two different strategy profiles which have the same path
bull Every path from the root to a terminal node is supported by at least one strategy profile
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
41
Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]
P1
P2 P2
A
(03) (41) (10)
B C
D E F
(21)G
L
LrsquoRrdquo
R
Rrsquo Lrdquo
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
42
Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)
P1
P2 P2
A
(10) (01) (22)
B C
D E F
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
43
Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses
can be solved by backward induction for each quantity q1 the follower chooses its best response q2
i (q1 q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the market is q
ci is the marginal cost of the production of the product by firm i
That is the profit for each firm is
i (q1 q2) = qi[A-q1-q2 -ci]
Example 412 Stackelberg Duopoly Model
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
44
Solving by backwards induction
bull This is a two person game sequential game with two stages and perfect information
bull Find best response for each choice of q1
2 (q1 q2) = max q2[A-q1-q2 ndashc2]
2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]
= -2q2 +A-q1-c2
Second derivative = -2
So the maximizer is (A-q1-c2)2
2
2
q
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
45
Continuing
bull Thus firm 1 should anticipate this result and choose q1 to maximize
1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1
2 + (A+c2-2c1)q1)
bull = -q1 +12(A+c2-2c1)
bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)
1
1
q
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
46
Type of games
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
47
subgame perfect equilibrium
bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree
bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
48
Imperfect information
bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game
bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
49
bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior
bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
50
Bob and Betty
bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes
bull If she chooses dishes then Bob chooses to Go Out or Cook
bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree
bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
51
Note use normal form game to pick what Betty should do at CD
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
52
bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame
bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes
BobOut bull The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path
bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium
bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium
bull Are there any other strategy profiles that will support Path Two
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
53
The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium
At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium
But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)
Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node
But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
54
bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One
bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile
bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium
bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats
bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
55
In Class ExerciseAsk the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first
What happens if we delay
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
56
bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
57
Prove or disprove
bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
58
Sequential Monopolist View
What are Nash equilibriaAre they subgame perfect
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
59
Thought Question
bull How do we change a game to our advantage
bull Use commitment threats and promises to change the nature of a game
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
60
Commitment
bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)
ndash Reputation leads to credibility
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
61
Commitment ndash An Example
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
62
Commitment ndash An ExampleFor those that intend to teachhellip
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
NE
Tough 3 2 1 1
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
63
Commitment ndash An ExampleBut if we announce we are tough
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2 1 1
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
64
Commitment ndash An ExampleGet different NE
STUDENT
TE
AC
HE
R
Punctual Late
Weak 4 3 2 4
Tough 3 2NE
1 1
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
65
Strategic Moves and Threats
bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat
bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to
be successful but is not a sufficient conditionndash CAREFUL
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
66
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
67
Trade Negotiation
Japan
US Open Closed
Open 4 3 3 4
Closed 2 1 1 2
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
68
Changing the game ndash A threat
bull A threat -- ldquoWe will close our market if you close yoursrdquo
bull And a promise ndash ldquo We will open market if you open yoursrdquo
bull Effectively reduces Japans options tondash If Japan stays open then the US stays open
giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)
ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
69
Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly
to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders
bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible
bull If Japanese market is already open then threat is part of deterrence strategy
bull If Japanese market is closed then threat is part of compellence strategy
bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
70
Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash
banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics
bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat
bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
71
Prisoners Dilemma ndash Promises to Keep
bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
72
Promises or Threats
bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises
bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
73
Countering Threatsbull Irrationality So nuts that any threats
will not have an effect on your behavior
bull Cut of communication so threats donrsquot reach you
bull Open escape routes for enemy thus tempting them to renege on threats
bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
74
Credibility Devices
bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility
Reducing the freedom of action through ndash automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades
ndash burning bridges ndash cutting off communication so nobody can argue with you
regarding your threat
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
75
Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing
players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and
foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
76
Solving Extensive Form Games
bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria
bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form
bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
77
Seltenrsquos Game
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
78
bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there
bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out
bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
79
bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play
bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
80
Little Horsey
bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
81
Where are the NE
bull Convert to normal formbull Use standard techniques
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
82
Giving Gifts
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
83
bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
84
bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted
bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1
bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing
bull Letrsquos construct the strategic form of this game
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
85
Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift
The action GN refers to ldquogive if get game theory not give if star trekrdquo
(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
86
bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
87
bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set
bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1
bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
88
bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)
bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift
bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game
bull The belief will be a probability distribution over the nodes
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
89
bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached
bull A continuation game refers to the information set and all nodes that follow from that information set
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
90
bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ
bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
91
bull Player irsquos expected utility in the continuation game starting at node x then is
bull Ui(σ|x) = zP(z|σx)ui(z)
bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-
92
bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q
bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1
bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold
bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set
bull In other words the unique sequentially rational strategy is to choose Y with certainty
- Chapter 4 Sequential Games
- Extensive Form Games
- Extensive Form Games - Intro
- Slide 4
- Fundamental Tools
- Fundamental Tools Extensive form games--Definition
- Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Composition of information sets
- Fundamental Tools Normal form games--Definition
- Fundamental Tools Normal form games--Illustration
- Slide 15
- Sequential games
- Slide 17
- Problems with Nash equilibrium
- Solving sequential games
- Slide 20
- Subgame
- Subgame perfect equilibrium amp credible threats
- Slide 23
- Example Cuban Missile Crisis
- Backwards induction
- Slide 26
- The predation game
- Slide 28
- Slide 29
- Slide 30
- Credible commitments
- Slide 32
- Slide 33
- Slide 34
- Hold up
- Holdup in equilibrium no-one designs software payoffs = (software nintendo)
- Strategies in extensive form
- Slide 38
- Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
- Slide 40
- Slide 41
- Slide 42
- Solving by backwards induction
- Continuing
- Type of games
- subgame perfect equilibrium
- Imperfect information
- Slide 49
- Bob and Betty
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- In Class Exercise
- Slide 56
- Prove or disprove
- Sequential Monopolist View
- Thought Question
- Commitment
- Commitment ndash An Example
- Commitment ndash An Example For those that intend to teachhellip
- Commitment ndash An Example But if we announce we are tough
- Commitment ndash An Example Get different NE
- Strategic Moves and Threats
- Trade Negotiation
- Slide 67
- Changing the game ndash A threat
- Trade Relations ndash Threats in Action
- Threats in action (cont)
- Prisoners Dilemma ndash Promises to Keep
- Promises or Threats
- Countering Threats
- Credibility Devices
- Slide 75
- Solving Extensive Form Games
- Seltenrsquos Game
- Slide 78
- Slide 79
- Little Horsey
- Where are the NE
- Slide 82
- Slide 83
- Slide 84
- Slide 85
- Slide 86
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Slide 91
- Slide 92
-