week 14 game theory (jehle and reny, ch.7 mas-colell et al ......week 14 game theory (jehle and...
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Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
Week 14Game Theory
(Jehle and Reny, Ch.7Mas-Colell et al.,Ch.7-8
Watson)
Serçin �ahin
Y�ld�z Technical University
25 December 2012
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
Interdependence means that one person's behaviour a�ects
another person's well-being, either positively or negatively.
Situations of interdependence are called strategic settings
because, in order for a person to decide how best to behave,
he must consider how others around him choose their actions.
Games are formal descriptions of strategic settings. Thus,
game theory is a methodology of formally studying situations
of interdependence.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
The Penalty Kick Game
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
Game theory analysis consists of two major subbranches:
The noncooperative framework treats all of the agents' actionsas individual actions. An individual action is something that aperson decides on his own, independently of the other peoplepresent in the strategic environment.Analysing behaviour in models with joint actions requires adi�erent set of concepts from those used for noncooperativeenvironments; this alternative theory is called cooperative
game theory.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
There are three major tensions of strategic interaction that are
identi�ed by the theory:
1 the con�ict between individual and group interests2 strategic uncertainty3 ine�cient coordination
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
To describe a situation of strategic interaction with a game,
we need to know the following �ve elements that these
representations have in common:
1 A list of players: Who is involved?2 The rules of the game: A complete description of what the
players can do (their possible actions)3 A description of what the players know when they act4 A speci�cation of how the players' actions lead to outcomes:
For each possible set of actions by the players, what is theoutcome of the game?
5 A speci�cation of the players' preferences over outcomes
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
Example 1: Chess
1 Players: There are two players
2 Rules: The players alternate in moving pieces on the game
board, subject to rules about what moves can be made in any
given con�guration of the board
3 Information: players observe each other's moves, so each
knows the entire history of play as the game progresses
4 Outcomes: a player who captures the other player's king wins
the game; in certain situations a draw is decleared.
5 Preferences over outcomes: Players prefer winning over a draw
and a draw over losing.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
Example 2: Matching Pennies
1 Players: There are two players, denoted 1 and 2
2 Rules: Each player simultaneously puts a penny down, either
heads up or tails up.
3 Information: Both player observes his own and opponent's
penny at the same time.
4 Outcomes: If the two pennies match (either both heads up or
both tails up) player 1 pays 1 dollar to player 2; otherwise,
player 2 pays 1 dollar to player 1.
5 Preferences over outcomes: Player 1 has a preference for the
pennies do not match, and Player 2 has a preference for the
pennies do match.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
The Extensive Form
The extensive form relies on the conceptual apparatus known
as a game three.
Nodes represent places where something happens in the game
(such as a decision by one of the players)
Branches indicate the various actions that players can choose.
We represent nodes by solid circles and branches by arrows
connecting the nodes.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
The game starts with a node, and this node is called the initial
node.
Other nodes in the game are called decision nodes, because
players make decisions at these places in the game.
The other nodes are called end nodes; they represent
outcomes of the game - places where the game ends.
Each end node also corresponds to a unique path through the
tree, which is a way of getting from the initial node through
the tree by following branches in the direction of the arrows.
Each end node is also followed by a vector of payo�s.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
Matching Pennies Version B
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
It is common to use the term information set to specify the
players' information at decision nodes in the game.
Formally, the elements of an information set are a subset of a
particular player's decision nodes. The interpretation is that
when play has reached one of the decision nodes in the
information set and it is player's turn to move, she does not
know which of these nodes she is actually at.
If one player cannot distinguish between some nodes, then, his
lack of information is shown with a dashed line connecting
these nodes.
Every decision node is contained in an information set; some
information sets singleton -consist of only one node.
Extensive form games in which every information set is a
singleton are called games of perfect information.All other
games are called games with imperfect information.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
Perfect RecallPerfect recall means that a player does not forget what she once
knew, including her own actions.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
In an extensive form, we must draw one player's decision before
that of the other, but it is important to realize that this does not
necessarily correspond to the actual timing of the strategic setting.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
Up to this point, the outcome of a game has been a
deterministic function of the players' choices.
In many games, however, there is an element of chance.
This, too, can be captured in the extensive form representation
by including random moves of nature.
Its moves are assumed to be made according to a �xed
probability distribution.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
Gift Game
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
Mathematical Representation of Extensive Form:Formally, a game represented in extensive form, denoted by Γ ,
consists of the following items:
A �nite set of players, N.
A set of actions, A, which includes all possible actions that
might potentially be taken at some point in the game. A need
not be �nite.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
A set of nodes or histories, X, where
X contains a distinguished element, x0, called the initial node,or empty historyeach x ∈ X\{x0} takes the form x = (a1, a2, ..., ak) for some�nitely many actions ai ∈ A, andif (a1, a2, ..., ak) ∈ X\{x0} for some k > 1, then(a1, a2, ..., ak−1) ∈ X\{x0}.A node, or history, is then simply a complete description of theactions that have been taken so far in the game. We shall usethe terms history and node interchangeably. For futurereference let
A(x) ≡ {a ∈ A | (x , a) ∈ X}denote the set of actions available to the player whose turn itis to move after the history x ∈ X\{x0}.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
A set of actions, A(x0) ⊆ A, and a probability distribution, π,on A(x0) to describe the role of chance in the game. Chance
always moves �rst, and just once, by randomly selecting an
action from A(x0) using the probability distribution π. Thus,(a1, a2, ..., ak) ∈ X\{x0} implies that ai ∈ A(x0) for i = 1 and
only i = 1.
A set of end nodes, E ≡ {x ∈ X | (x , a) /∈ X for all a ∈ A}.Each end node describes one particular complete play of the
game form beginning to end.
A function, ι : X\(E ∪ {x0}) −→ N that indicates whose turn
it is at each decision node in X. For future reference, let
Xi ≡ {x ∈ X\(E ∪ {x0}) | ι(x) = i}denote the set of decision nodes belonging to player i .
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
A partition, I , of the set of decision nodes, X\(E ∪ {x0}), suchthat if x and x
′are in the same element of the patition then
(i) ι(x) = ι(x′), and
(ii)A(x) = A(x′).
For future reference, let
Ii ≡ {I (x) | ι(x) = i , some x ∈ X\(E ∪ {x0})}denote the set of information sets belonging to player i .
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
For each i ∈ N, a von Neumann-Morgenstern payo� function
whose domain is the set of end nodes, ui :;E −→ <. Thisdescribes the payo� to each player for every possible complete
play of the game.
We write Γ =< N,A,X ,E , ι, π, I , (ui )i∈N >. If the sets of actions,
A, and nodes, X , are �nite, then Γ is called a �nite extensive form
game.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
The buyer and seller of the used car.1 Again there are two players, so N = {S ,B}, where S denotes
seller, and B , buyer.2 To keep things simple, assume that the seller, when choosing a
price, has only two choices: high and low. The set of actions
that might arise is A = { repair, don't repair, price high, price
low, accept, reject }.3 Because chance plays no role here, rather than give it a single
action, we simply eliminate chance from the analysis.4 A node in this game is, for example, x = { repair, price high }.
At this node, x , it is the buyer's turn to move, so that
ι(x) = B .5 Because at this node, the buyer is informed of the price chosen
by the seller, but not of the seller's repair decision, I (x) = {(repair, price high ), ( don't repair, price high )}. That is, whennode x is reached, the buyer is informed only that one of the
two histories in I (x) has occured; he is not informed of which
one, however.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
Strategy
Formally, a (pure) strategy for player i in an extensive form
game Γ is a function si : Ii −→ A, satisfying si (I (x)) ∈ A(x)for all x with ι(x) = i .
Let Si denote the set of (pure) strategies (also called strategy
space)for player i in Γ. That is, Si is a set of comprising each of
the possible strategies of player i in Γ. We shall assume that Γis a �nite game. Consequently, each of the sets Si is also �nite.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
Strategies in the Matching Pennies Version B
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
Strategies in the Matching Pennies Version B
Strategy 1(s1): Play H if player 1 plays H; play H if player 1
plays T.
Strategy 2(s2): Play H if player 1 plays H; play T if player 1
plays T.
Strategy 3(s3): Play T if player 1 plays H; play H if player 1
plays T.
Strategy 4(s4): Play T if player 1 plays H; play T if player 1
plays T.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
A strategy pro�le (or joint strategy) is a vector of strategies,
one for each player. In other words, a strategy pro�le describes
strategies for all of the players in the game. For example,
suppose we are studying a game with n players. Then a typical
strategy pro�le is a vector s = (s1, s2, s3, ..., sn) where si is thestrategy of player i , for i = 1, 2, 3, ..., n.
We will also sometimes separate the strategy pro�le s into the
strategy of player i and the strategies of the other players, we
write s = (si , s−i ). s−i is the (N − 1) vector of strategies for
players other than i : s−i = (s1, s2, ..., si−1, si+1, ..., sn).
Let S denote the set of strategy pro�les. Mathematically,
S = S1 × S2 × ...× Sn.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
The Strategic (Normal) Form GameA strategic (normal) form game is a tuple G = (Si , ui )
Ni=1 where for
each player i = 1, 2, ...N, Si is the set of strategies available to
player i , and ui : ×Nj=1Sj −→ R describes player i 's payo� as a
function of the strategies chosen by all players.
A strategic form game is �nite if each player's strategy set contains
�nitely many elements.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
Matching Pennies Version B in the Normal Form
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
It is clear that for any extensive form representation of a game,
there is a unique normal form representation. The converse is not
true, however. Many di�erent extensive forms may be represented
by the same normal form.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
Beliefs
Mathematically, a belief of player i is a probability distribution
over the strategies of the other players. We denote such a
probability distribution ∆µ−i and write ∆µ−i ∈ ∆S−i , where∆S−i is the set of probability distributions over the strategies
of all the players except player i .
The belief of player i about the behaviour of player j is afunction µj ∈ ∆Sj such that, for each strategy sj ∈ Sj ofplayer j , µj(sj) is interpreted as the probability that player ithinks player j will play sj .As a probability distribution, µj has the property that
µj(sj) ≥ 0 for each sj ∈ Sj , and∑
sj∈Sj µj(sj) = 1.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
Mixed Strategies
A mixed strategy for a player is the act of selecting a strategy
according to a probability distribution.
For a �nite strategic form game G = (Si , ui )Ni=1, a mixed
strategy mi for player i is a probability distribution over Si .That is, mi : Si −→ [0, 1] assigns to each si ∈ Si theprobability, mi (si ), that si will be played.
We shall denote the set of mixed strategies for player i by Mi .
Consequently, Mi = {mi : Si −→ [0, 1] |∑
si∈Si mi (si ) = 1}.Suppose that player i has M pure strategies in set
Si = {s1i , s2i , ..., sMi}.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
Then, this set forms a unit simplex on the M-dimensional
Euclidian space and it is called the mixed extension of Si .∆(Si ) = {(m1i , ...,mMi ) ∈ RM : mji > 0 for all j = 1, ...,M
and∑M
j=1mji = 1}.Let M = ×N
i=1Mi denote the set of joint mixed strategies.
From now on, we shall drop the word "mixed" and simply call
m ∈ M a joint strategy and mi ∈ Mi a strategy for player i .
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
When players randomize over their pure strategies, the induced
outcome is itself random, leading to a probability distribution
over the terminal nodes of the game.
Let us assume that the players' payo�s are in fact von
Nuemann-Morgenstern utilities, and that they will behave to
maximise their expected utility.
If ui is a von Neumann-Morgenstern utility function on S, and
the strategy m ∈ M is played then player i 's expected utility is
ui (m) ≡∑
s∈S m1(s1)...mN(sN)ui (s)
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
Prisoners' Dilemma
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
Strictly Dominant Strategies
Let S = S1 × ...× SN denote the set of strategy pro�les. The
symbol, −i , denotes all players except player i. So, forexample, s−i denotes an element of S−i , which itself denotes
the set S1 × ...× Si−1 × Si+1 × ...× SN . Then we have the
following de�nition:
A strategy, si , for player i is strictly dominant if
ui (si , s−i ) > ui (si , s−i ) for all (si , s−i ) ∈ S with si 6= si .
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
Strictly Dominated Strategies
We should expect that player i will not play dominated
strategies, those for which there is some alternative strategy
that yields him a greater payo� regardless of what the other
players do.
Player i 's strategy si strictly dominates another of his
strategies si , for all s−i ∈ S−i . In this case, we also say that siis strictly dominated in S .
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
Iteratively Eleminating Strictly Dominated Strategies
Let S0i = Si for each player i , and for n ≥ 1, let Sn
i denote
those strategies of player i surviving after the nth round of
elimination. That is si ∈ Sni if si ∈ Sn−1
i is not strictly
dominated in Sn−1.
A strategy si for player i is iteratively strictly undominated in
S (or survives iterative elimination of strictly dominated
strategies) if si ∈ Sni , ∀n ≥ 1.
Once we have determined the set of undominated pure
strategies for player i , we need to consider which mixed
strategies are undominated.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
Weakly Dominated Strategies
So far, we have considered only notions of strict dominance.
Related notions of weak dominance are also available.
Player i 's strategy si weakly dominates another of his
strategies si , if
ui (si , s−i ) ≥ ui (si , s−i ) for all s−i ∈ S−i ,
with at least one strict inequality. In this case, we also say that
si is weakly dominated in S .
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
Iteratively Eliminating Weakly Dominated Strategies
Let W 0i = Si for each player i , and for n ≥ 1, let W n
i denote
those strategies of player i surviving after the nth round of
elimination of weakly dominated strategies.
That is, si ∈W ni if si ∈W n−1
i is not weakly dominated in
W n−1 = W n−11 × ...×W n−1
N .
A strategy si for player i is iteratively weakly undominated in S(or survives iterative elimination of weakly dominated
strategies) if si ∈W ni for all n ≥ 1.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
Best Response and Rationalizability
The set of rationalizable strategies consists precisely of those
strategies that may be played in a game where the structure of
the game and the players' rationality are common knowledge
among the players.
To maximize the payo� that you expect to obtain -which we
assume is the mark of rational behaviour- you should select the
strategy that yields the greatest expected payo� against your
belief. Such a strategy is called a best response.
Formally,suppose player i has a belief µ−i ∈ ∆S−i about thestrategies played by the other players. Player i 's strategysi ∈ Si is a best response if
ui (si , µ−i ) ≥ ui (s′i , µ−i )
for every s′i ∈ Si .
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
In a �nite game, every belief has at least one best response.
For each belief µ−i of player i , we denote the set of best
responses by BRi (µ−i ).In this way, the belief rationalizes
playing the strategy. Furthermore, each player should assign
positive probability only to strategies of the other players that
can be similarly rationalized.
Clearly, a player should not play a strategy that is never a best
response.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
As in the case of strictly dominated strategies, common
knowledge of rationality and the game's structure implies that
we can iterate the deletion of strategies that are never best
response.
Equally important, the strategies that remain after this
iterative deletion are the strategies that a rational player can
justify, or rationalize.The set of strategies surviving this
iterative deletion process can be said to be precisely the set of
strategies that can be played by rational players in a game in
which the players' rationality and the structure of the game are
common knowledge. They are known as rationalizable
strategies.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
Dominance and Best Response Compared
We have said that the set of rationalizable strategies is no
larger than the set remaining after iterative deletion of strictly
dominated strategies.
It turns out, however, that for the case of two-player games
(N = 2), these two sets are identical because in two-player
games a (mixed) strategy mi is a best response to some
strategy choice of a player's rival whenever mi is not strictly
dominated. Then since this process is equivalent to iterated
dominance, we can just perform the iterative deletion of
dominated strategies.
With more than two players, however, there can be strategies
that are never a best response and yet are not strictly
dominated.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
Nash Equilibrium
In the strategic setting, just as in the demand-supply setting,
regularities in behaviour that can be rationally sustained will
be called equilibria.
If the agreement is to play strategy pro�le s and si /∈ BRi (s−i )for some player i , then this player has no incentive to abide by
the agreement and will chose a strategy that is di�erent from
si . Thus, a Nash equilibrium describes behaviour that can be
rationally sustained.
Formally, given a strategic form game G = (Si , ui )Ni=1, the
strategy pro�le s ∈ S is a pure strategy Nash equilibrium of Gif for each player
i , ui (s) ≥ ui (si , s−i )
for all si ∈ Si .
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
A compact restatement of the de�nition of a Nashequilibrium
Lets de�ne for each player i , a pure-strategy best-response
correspondence ψi : S → Si which speci�es for every strategy
pro�le m−i ∈ M−i by player i 's opponents a set
BRi (m−i ) ⊂ Si of player i 's pure strategies which are best
responses.
Now we form a new correspondence ψ by forming the
Cartesian product of the n personal best response
correspondences ψi . We de�ne for every strategy pro�le s ∈ S ,
ψ(s) = ×i∈Nψi (s)
Therefore, we see that ψ is a correspondence itself from the
space of strategy pro�les into the space of strategy pro�les;
i.e., ψ : S → S .
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
In a Nash equilibrium, each player i 's strategy si is a best
response to the other players' strategies si . Because this
inclusion must hold for all players, we have
(s1, ..., sn) ∈ ×i∈Nψi (s) = ψ(s)
Then, a Nash equilibrium pro�le is a �xed point of the best
response correspondence ψ. This logic is reversible: any �xed
point of the best response correspondence is a Nash
equilibrium pro�le.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
Mixed Strategy Nash Equilibrium
It is straightforward to extend the de�nition of Nash
equilibrium to games in which we allow the players to
randomize over their pure strategies.
Given a �nite strategic form game G = (Si , ui )Ni=1, a strategy
pro�le m ∈ M is a Nash equilibrium of G if for each player i ,ui (m) ∈ M is a Nash equilibrium of G if for each player i ,
ui (m) ≥ ui (mi , m−i ) for all mi ∈ Mi .
A necessary and su�cient condition for mixed strategy pro�le
m to be a Nash equilibrium of game G is that each player,
given the distribution of strategies played by his opponents, is
indi�erent among all the pure strategies that he plays with
positive probability.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
Existence of Nash EquilibriumEvery �nite strategic form game possesses at least one Nash
equilibrium.
Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings
References
Jehle and Reny, 2011, Advanced Microeconomic Theory, Third
Edition, Prentice Hall
Mas Colell, Whinston and Green, 1995, Microeconomic
Theory, Oxford University Press
Watson, 2002, Strategy: An Introduction to Game Theory,
W.W.Norton and Company