strategic behavior in business and econ 3.2.1. static games of complete information: dominant...

Strategic Behavior in Business and Econ

3.2.1. Static Games of complete information: Dominant Strategies and Nash Equilibrium in pure and mixed

strategies- Some Games -


Outline

3.1. What is a Game ?3.1.1. The elements of a Game3.1.2 The Rules of the Game: Example3.1.3. Examples of Game Situations3.1.4 Types of Games

3.2. Solution Concepts3.2.1. Static Games of complete information: DominantStrategies and Nash Equilibrium in pure and mixed

strategies3.2.2. Dynamic Games of complete information: Backward

Induction and Subgame perfection


Solution concepts for this type of games

Equilibrium in Dominant Strategies When there is an “always winning” strategy

Equilibrium by elimination of Dominated StrategiesWhen there are “worse than” strategies

Nash EquilibriumWorks in any case• In pure strategies (players do not randomize)

• In mixed strategies (players do randomize)

Reminder


Reminder

If you have a Dominant Strategy use it, and expect your opponent to use it as well

If you have Dominated Strategies do not use any of them, and expect you opponent not to use them as well (eliminate them from the analysis of the game)

If there are neither Dominant Strategies nor Dominated Strategies, look for Nash Equilibria and play accordingly. Expect your opponent to play according to the Nash Equilibrium as well

If there are no Nash Equilibria in pure strategies, play at random (mixed strategy) to keep your opponent guessing

When playing randomly, do not follow any pattern, and try to discover patterns in your opponent's behavior


Some games

Hard to play games

Game of Chicken (original)Cold WarHawk-Dove Game

Hard to solve gamesStag hunt GameVolunteer's Dilemma

Hard to believe games• Traveler's Dilemma• Guess 2/3 of the average game Hard to analyze games

Hotelling Spatial Competition Game


Hard to play games

These are games that, although having a solution, they aredifficult to play if encountered in real life for they correspond to extreme situations

Game of Chicken (original)Cold WarHawk-Dove Game


Game of Chicken (original)

The game of Chicken models two drivers, both headed for a single lane bridge from opposite directions. The first to swerve away yields the bridge to the other. If neither player swerves, the result is a fatal head-on collision. It is presumed that the best thing for each driver is to stay straight while the other swerves (since the other is the "chicken" while a crash is avoided). Additionally, a crash is presumed to be the worst outcome for both players. This yields a situation where each player, in attempting to secure his best outcome, risks the worst.


Game of Chicken (original)


The environment of the game

Players: Driver 1 and Driver 2 Strategies: To swerve or To stay straight Payoffs: (to be defined)

The Rules of the Game

Timing of moves Simultaneous

Nature of conflict and interaction Conflict (anti-coordination)

Information conditions Symmetric


The payoffs

This is one of those games in which there is no monetarypayoff. Thus, we must define the payoffs in accordance to the nature of the game. For each player, we have that:

Best outcomeSecond BestThird BestWorst Outcome


The payoffs


Best outcome I stay straight while the other swerves (a)Second BestThird BestWorst Outcome


The payoffs


Best outcome I stay straight while the other swerves (a)Second Best We both swerve (b)Third BestWorst Outcome


The payoffs


Best outcome I stay straight while the other swerves (a)Second Best We both swerve (b)Third Best I swerve and the other stays straight (c)Worst Outcome


The payoffs


Best outcome I stay straight while the other swerves (a)Second Best We both swerve (b)Third Best I swerve and the other stays straight (c)Worst Outcome We both stay straight (crash !) (d)

Where a > b > c > d


The Game of Chicken (original)

Swerve Stay Straight

Swerve b, b c , a

a , c d , dStay Straight

Driver 1

Driver 2

Where a > b > c > d




Swerve b, b c , a


Driver 1

Driver 2

Where a > b > c > d

These games, were thevalue of the payoff doesn't

matter (only the order)are called Ordinal Games




Swerve b, b c , a


Driver 1

Driver 2

Where a > b > c > d

We could give any value toa, b, c, and d, as long as

they keep the ordera > b > c > d




Swerve b, b c , a


Driver 1

Driver 2

Where a > b > c > d

Thus, we can find the“best replies” as usual


In any solution, one of the drivers must play chicken Who is the chicken ? On the one side both player prefer the other to swerve On the other side the cost of a crash is so high compared

to the cost of being the chicken that it might make sense to swerve

It is mutually beneficial for the players to play different strategies (anti-coordination)

Random Strategies might make sense here, but are not possible to compute (the probabilities depend on the values of the payoffs, and these values are arbitrary here)


Bertrand Russell saw in chicken a metaphor for the nuclear stalemate. His 1959 book, Common Sense and Nuclear Warfare, not only describes the game but offers mordant comments on those who play the geopolitical version of it.

‘Since the nuclear stalemate became apparent, the Governments of East and West have adopted the policy which Mr. Dulles calls "brinkmanship." This is a policy adapted from a sport which, I am told, is practiced by some youthful degenerates. This sport is called "Chicken!" It is played by choosing a long straight road with a white line down the middle and starting two very fast cars towards each other from opposite ends. Each car is expected to keep the wheels of one side on the white line. As they approach each other, mutual destruction becomes more and more imminent. If one of them swerves from the white line before the other, the other, as he passes, shouts "Chicken!" and the one who has swerved becomes an object of contempt....’


‘As played by irresponsible boys, this game is considered decadent and immoral, though only the lives of the players are risked. But when the game is played by eminent statesmen, who risk not only their own lives but those of many hundreds of millions of human beings, it is thought on both sides that the statesmen on one side are displaying a high degree of wisdom and courage, and only the statesmen on the other side are reprehensible. This, of course, is absurd. Both are to blame for playing such an incredibly dangerous game. The game may be played without misfortune a few times, but sooner or laterit will come to be felt that loss of face is more dreadful than nuclear annihilation. The moment will come when neither side can face the derisive cry of "Chicken!" from the other side. When that moment is come, the statesmen of both sides will plunge the world into destruction.’

(From William Poundstone's “The Prisoner's Dilemma”)


Cold War (based on “Game theory and the Cuban missile crisis”

by Steven J. Brams)

This is a real life (historical) version of a Game of Chicken.It goes back to the 60's, when the USA and the USSR wereengaged in a so called “cold war”One of the episodes of the “cold war” was the Cuban missile crisis

The Cuban missile crisis was precipitated by a Soviet attempt in October 1962 to install medium-range and intermediate-range nuclear-armed ballistic missiles in Cuba that were capable of hitting a large portion of the United States. The goal of the United States was immediate removal of the Soviet missiles, and U.S. policy makers seriously considered two strategies to achieve this end


A naval blockade (B), or "quarantine" as it was euphemistically called, to prevent shipment of more missiles, possibly followed by stronger action to induce the Soviet Union to withdraw the missiles already installed.

A "surgical" air strike (A) to wipe out the missiles already installed, insofar as possible, perhaps followed by an invasion of the island.

The alternatives open to Soviet policy makers were:

Withdrawal (W) of their missiles. • Maintenance (M) of their missiles.



Players: USA and USSR Strategies: For USA: Blockade (B) or Air Strike (A)

For USSR: Withdrawal (W) or Maintenance (M) Payoffs: (to be defined)






The payoffs

This is “Game of Chicken”. Both would like the other to “swerve”. If they “stay straight”, the crash is the Nuclear War

USA USSRBest outcome (A,W) (B,M) 4Second Best (B,W) (B,W) 3Third Best (B,M) (A,W) 2Worst Outcome (A,M) (A,M) 1


Cuban Missile crisis

W M

BCompromise

3, 3

USSR wins

2 , 4 USA wins

4 , 2

Nuclear War

1 , 1 A

USA

USSRThe solution(s) is asbefore


The strategy choices, probable outcomes, and associated payoffs are a strong simplification of the crisis as it developed over a period of thirteen days.

Both sides considered more than the two alternatives listed, as well as several variations on each. The Soviets, for example, demanded withdrawal of American missiles from Turkey as a quid pro quo for withdrawal of their own missiles from Cuba, a demand publicly ignored by the United States.

Most observers of this crisis believed that the two superpowers were on a collision course

They also agree that neither side was eager to take any “irreversible step” (more on this next)


Although in one sense the United States "won" by getting the Soviets to withdraw their missiles, Premier Nikita Khrushchev of the Soviet Union at the same time extracted from President Kennedy a promise not to invade Cuba, which seems to indicate that the eventual outcome was a compromise of sorts

That is not the prediction of Game Theory ! Negotiation is the key to scape from a mutual bad outcome !


“Irreversible steps”

One way to scape from a “game of chicken” is to take an“irreversible step”

What would happen if one of the two drivers (say Driver 1)defiantly rips off the steering wheel in full view of the otherdriver ?



Swerve b, b c , a


Driver 1

Driver 2The driver just eliminatedthe option of swerving !



Swerve b, b c , a


Driver 1

Driver 2Driver 1 just eliminatedthe option of swerving !




Driver 1

Driver 2Now, there is only onerational choice for

Driver 2


Swerve

a , cStay Straight

Driver 1

Driver 2And the game ends withthe preferred outcome

for Driver 1 !!!!!


Sometimes it makes sense to “dispose” of some strategies But it is very important that your opponent realizes you have done

so ! In Stanley Kubrick's Dr. Strangelove. the Russians sought to deter

American attack by building a "doomsday machine," a device that would trigger world annihilation if Russia was hit by nuclear weapons. However, the Russians failed to signal. They deployed their doomsday machine covertly !!


Hawk-Dove Game

This is version of a Game of Chicken that is very useful in evolutionary biology

The name "Hawk-Dove" refers to a situation in which two animals compete for a shared resource and the contestants can choose either conciliation or conflict.

V is the value of the contested resource, and C is the cost of an escalated fight. It is (almost always) assumed that the value of the resource is less than the cost of a fight is, i.e., C > V > 0

If the two animals behave in the same way, the split the resource.Otherwise, the animal playing Hawk gets the whole resource



Players: Animal 1 and Animal 2 Strategies: Dove (show you intention) or Hawk (attack) Payoffs: (see the table)






Dove Hawk

Dove 0 , V

V , 0 (V-C)/2 , (V-C)/2Hawk

Animal 1

Animal 2

V/2 , V/2


Dove Hawk

Dove 0 , V

V , 0 (V-C)/2 , (V-C)/2Hawk

Animal 1

Animal 2

V/2 , V/2

Notice, since V<C, V > V/2 > 0 > (V-C)/2 It's a Game of Chicken


Dove Hawk

Dove 0 , V

V , 0 (V-C)/2 , (V-C)/2Hawk

Animal 1

Animal 2

V/2 , V/2

Notice, since V<C, V > V/2 > 0 > (V-C)/2 It's a Game of Chicken

The solution(s) is asbefore


In this case, doesn't seem to be any way to take an “irreversible step”

In real life, some animals behave as doves while others Are Hawks (nature plays mixed strategies !)

This example set the basis for a extremely fruitful application of Game Theory to Evolutionary Biology

(John Maynard-Smith) And vice versa, Evolutionary Theory can be applied to

Game Theory !


Hard to solve games

These are social games, situations in which although players seeka common goal, their individualistic behavior leads to non desirableoutcomes

Stag hunt GameThe Volunteer's Dilemma


Stag hunt Game

This is a game which describes a conflict between safety and social cooperation. Jean-Jacques Rousseau described a situation in which two individuals go out on a hunt. Each can individually choose to hunt a stag or hunt a hare. Each player must choose an action without knowing the choice of the other. If an individual hunts a stag, he must have the cooperation of his partner in order to succeed. An individual can get a hare by himself, but a hare is worth less than a stag. This is taken to be an important analogy for social cooperation.



Players: Hunter 1 and Hunter 2 Strategies: Hunt Stag or Hunt Hare Payoffs: (to be defined)



Nature of conflict and interaction Cooperation



The payoffs

Again, this is a game with no monetary payoff, other than thevalue of a Stag or a Hare. We can define the payoffs in accordance to the nature of the game. For each player, we have that:

Best outcomeSecond BestWorst Outcome


The payoffs


Best outcome Both hunt Stag (a)Second BestWorst Outcome


The payoffs


Best outcome Both hunt Stag (a)Second Best I hunt Hare (b)Worst Outcome


The payoffs


Best outcome Both hunt Stag (a)Second Best I hunt Hare (b)Worst Outcome I hunt Stag while the other hunts hare (c)


Stag hunt Game

Stag Hare

Stag a, a c , b

b , c b , b Hare

Hunter 1

Hunter 2

Where a > b > c


Stag hunt Game

Stag Hare

Stag a, a c , b

b , c b , b Hare

Hunter 1

Hunter 2

Where a > b > c

We could give any value toa, b, and c, as long as

they keep the ordera > b > c


Stag hunt Game

Stag Hare

Stag a, a c , b

b , c b , b Hare

Hunter 1

Hunter 2

Where a > b > c

We can find the“best replies” as usual


Stag hunt Game

Stag Hare

Stag a, a c , b

b , c b , b Hare

Hunter 1

Hunter 2

Where a > b > c

There are 2 equilibria,but one is better forthe two players than

the other


Stag hunt Game

Stag Hare

Stag a, a c , b

b , c b , b Hare

Hunter 1

Hunter 2

Where a > b > c

The equilibrium at whichboth player go for the

Stag is calledPayoff Dominant


Stag hunt Game

Stag Hare

Stag a, a c , b

b , c b , b Hare

Hunter 1

Hunter 2

Where a > b > c

It corresponds to thesocial cooperation

solution


Stag hunt Game

Stag Hare

Stag a, a c , b

b , c b , b Hare

Hunter 1

Hunter 2

Where a > b > c

The equilibrium at whichboth player go for the

Hare is calledRisk Dominant


Stag hunt Game

Stag Hare

Stag a, a c , b

b , c b , b Hare

Hunter 1

Hunter 2

Where a > b > c

It corresponds to theindividually safe

solution


Individually safe solution

Suppose that Hunter 1 is unsure about what Hunter 2 is goingto do. Suppose she thinks that Hunter 2 is going to go forStag with probability p

Stag (p) Hare (1-p)

Stag a, a c , b

b , c b , b Hare

Hunter 1

Hunter 2



Then, she (Hunter 1) must evaluate the Expected payoff ofeach of hear choices: E(Stag) vs. E(Hare)

Stag (p) Hare (1-p)

Stag a, a c , b

b , c b , b Hare

Hunter 1

Hunter 2



E(Stag) = p x a + (1-p) x c = c + p (a - c)E(Hare) = b

Stag (p) Hare (1-p)

Stag a, a c , b

b , c b , b Hare

Hunter 1

Hunter 2



Hence, Hunter 1 will choose Hare if E(Stag) < E(Hare), that is, if c + p(a – c) < b

Stag (p) Hare (1-p)

Stag a, a c , b

b , c b , b Hare

Hunter 1

Hunter 2



Solving for p: c + p(a – c) < b p(a – c) < (b – c)

(b - c)p <

(a - c)

Stag (p) Hare (1-p)

Stag a, a c , b

b , c b , b Hare

Hunter 1

Hunter 2



Thus, if Hunter 1 thinks that Hunter 2 will hunt Stag with a low probability

(b - c)p <

(a – c)Then her safer choice is tohunt Hare Stag (p) Hare (1-p)

Stag a, a c , b

b , c b , b Hare

Hunter 1

Hunter 2



Thus, if players are unsure about the behavior of the other, and they are rather pessimistic (they believe that the probability that the other goes for Stag are low),then the “individually safe”choice is to hunt Hare

Stag (p) Hare (1-p)

Stag a, a c , b

b , c b , b Hare

Hunter 1

Hunter 2



Example: Take a = 3, b = 2, and c = 1

Stag (p) Hare (1-p)

Stag 3, 3 1 , 2

2 , 1 2 , 2 Hare

Hunter 1

Hunter 2



Example: Take a = 3, b = 2, and c = 1Then,

(b - c) (2 – 1) 1p < = =

(a– c) (3 – 1) 2

Stag (p) Hare (1-p)

Stag 3, 3 1 , 2

2 , 1 2 , 2 Hare

Hunter 1

Hunter 2



Thus, if players believe that the other will hunt Stag witha probability lower than 50%, the “individually safe” choiceis to hunt Hare

Stag (p) Hare (1-p)

Stag 3, 3 1 , 2

2 , 1 2 , 2 Hare

Hunter 1

Hunter 2


This game shows that some mutually beneficial outcome may be hard to achieve if cooperation is required

The more pessimistic people is about the “social behavior” of others, the harder is to achieve cooperation

Other examples of Stag hunt games are:Two individuals who must row a boatRaising funds for a public facilityThe hunting practices of orca (known as carousel feeding)

are an example of a stag hunt. Here orcas cooperatively corral large schools of fish to the surface and stun them by hitting them with their tails. Since this requires that fish not have any mechanism for escape, it requires the cooperation of many orcas


The Volunteer's Dilemma

The Volunteer's Dilemma models the situation in which a public (common) good must be provided.While all the players benefit from the consumption of the public good, each of them prefers the others to pay for itThink of an scenario in which the electricity has gone out for a two flat apartment building. The two inhabitants know that the electricity company will fix the problem as long as at least one person calls to notify them, at some cost. If no one volunteers, the worst possible outcome is obtained for all participants. If any one person elects to volunteer, the other benefits by not doing so



Players: Person 1 and Person 2 Strategies: Call or Don't call Payoffs: (in the table)



Nature of conflict and interaction Cooperation-Conflict




Call Don't Call

Call 0, 0 0 , 1

1 , 0 -10 , -10 Don'tCall

Person 1

Person 2



Call Don't Call

Call 0, 0 0 , 1

1 , 0 -10 , -10 Don'tCall

Person 1

Person 2We have two equilibria


This is similar to a Game of Chicken in that the two players prefer the other to Call

The difference is that Calling doesn't make you the “Chicken”

Also, both Calling is not the best outcome for the two of them

This example illustrates how difficult is the provision ofa public good.Although everybody likes to have a City Park, all preferthe other to pay for it


Hard to believe games

In some cases, the prediction of Game Theory seem to be highly unintuitive. These are games that, if actually played, people doesn'tbehave as expected

The Traveler's DilemmaGuess 2/3 of the average game


The Traveler's Dilemma

An airline loses two suitcases belonging to two different travelers. Both suitcases happen to be identical and contain identical antiques. An airline manager tasked to settle the claims of both travelers explains that the airline is liable for a maximum of $100 per suitcase, and in order to determine an honest appraised value of the antiques the manager separates both travelers so they can't confer, and asks them to write down the amount of their value at no less than $2 and no larger than $100.


He also tells them that if both write down the same number, he will treat that number as the true dollar value of both suitcases and reimburse both travelers that amount. However, if one writes down a smaller number than the other, this smaller number will be taken as the true dollar value, and both travelers will receive that amount along with a bonus/malus: $2 extra will be paid to the traveler who wrote down the lower value and a $2 deduction will be taken from the person who wrote down the higher amount. The challenge is: what strategy should both travelers follow to decide the value they should write down



Players: Traveler 1 and Traveler 2 Strategies: Any (integer) number between 2 and 100 Payoffs: (in the table)



Nature of conflict and interaction Conflict




Traveler 2

Traveler 1



Traveler 2

Traveler 1

We can proceed by elimination of Dominated Strategies



Traveler 2

Traveler 1

This is the solution !



Traveler 2

Traveler 1

If we look for the Nash Equilibrium instead . . .



Traveler 2

Traveler 1

We draw the “Best Replies” (Red Circles)



Traveler 2

Traveler 1

The unique Nash Equilibrium is (2, 2) !


All “game theoretical” solutions have (2, 2) as the final prediction of the game

Experimental evidence shows that players choose quantities close to 100


Guess 2/3 of the average game

Guess 2/3 of the average is a game where several people guess what 2/3 of the average of their guesses will be, and where the numbers are restricted to the real numbers between 0 and 100. The winner is the one closest to the 2/3 average.

0 100


Guess 2/3 of the average game

Example: Three players play the game and the choices are: c

1 = 42, c

2 = 12, c

3 = 23

0 10042

c1

12

c2

23

c3

The average is ( 12 + 23 + 42 ) / 3 = 77 / 3 = 25.666 Thus, 2/3 of the average is 2/3 of 77/3 = 154/9 = 17.11111


17.1111 25.666

0 10042

c1

12

c2

23

c3

c1 – 17.111 = 42 – 17.111 = 24.888

c2 – 17.111 = 12 – 17.111 = -5.111111

c3 – 17.111 = 23 – 17.111 = 5.888

In this example, player 2 is the winner with c2 = 12



Players: Player 1, Player 2, Player 3, ... Strategies: Any number between 0 and 100

• Payoffs: $1 if win, 0 otherwise. If there is a tie,the dollar is split among the winners



Nature of conflict and interaction Conflict


(with respect to previous examples, this game features more than 2 players and an infinite number of strategies !)


Guess 2/3 of the average game (Analysis)

Although this game doesn't admit a “table” representation, it can be solved by elimination of Dominated Strategies

Notice first that for whatever choices of a number by the players, the average can NEVER be larger than 100

Therefore, 2/3 of the average can NEVER be above (2/3) x 100 = 66.67

Therefore, I should NEVER choose a number above 66.67 That is, the choice of any number above 66.67 is a

Dominated Strategy



Although this game doesn't admit a “table” representation, it can be solved by elimination of Dominated Strategies

Notice first that for whatever choices of a number by the players, the average can NEVER be larger than 100

Therefore, 2/3 of the average can NEVER be above (2/3) x 100 = 66.67• Therefore, I should NEVER choose a number above 66.67• That is, the choice of any number above 66.67 is a Dominated Strategy

0 100 66.67

Dominated Strategies



Hence, the players will never choose a number above 66.67 and then the average can NEVER be larger than 66.67

Therefore, 2/3 of the average can NEVER be above (2/3) x 66.67 = 44.45

Therefore, I should never choose a number above 44.45 That is, the choice of any number above 44.45 is now a

Dominated Strategy

0 100 66.67







Dominated Strategy

0 100 66.67


44.45






Dominated Strategy

0 100 66.67


44.4529.64



And by continuing the elimination of Dominated Strategies, the only choice that survives is to choose 0 as your guess of 2/3 of the average of all the number chosen by the players

0 100 66.67


44.4529.64



The predicted outcome is that all the players will choose 0.Then, the average will be 0, and 2/3 of the average willalso be 0. That is, all players win.

Experimental evidence goes against this theoretical predictionIn an Internet contest with 19.196 people playing and a $865 prize,the winning value was 21.6


Hard to analyze games

Most games that represent real business and/or economics scenarios often involve a large number of strategies, manyplayers, and complicated rules. In such cases a simple tablerepresentation is not possible and the search for equilibriarequires other techniques

• Hotelling Spatial Competition Game


• Hotelling Spatial Competition Game

Suppose that there are two competing shops selling the same indistinguishable product at the same price, and that they must be located along the length of a street running north and south. Each shop owner wants to locate his shop such that he maximizes his own market share by drawing the largest number of customers. Customers are spread equally along the street. Suppose, finally, that each customer will always choose the nearest shop

1 2



• Players: Shop 1 and Shop 2• Strategies: Any location within the length of the street• Payoffs: For each shop, all the customers that are closer to it than to the competitor


• Timing of moves Simultaneous• Nature of conflict and interaction Conflict • Information conditions Symmetric


What will be the location of the shops ?

1 2


First Guess . . .

1 2


But, what would you do if you were Shop 1

1 2


You get more costumers than before !

1 2


What will be the “best reply” by Shop 2 ?

1 2


What will be the “best reply” by Shop 2 ?

12


12

You get more costumers than before !


1 2

If this dynamics continues, the unique stable point is


1 2

The two shops locate at the middle of the street !


• This is an example of “spatial clustering” of business

• There are real examples of this• Car Dealers• Oriental rug stores• Computer (electronics) districts• Michigan Avenue

• But this also may apply to other characteristics of businesses:

• Quality• Sweetness

• Even to politics !


This model can also explain the common complaint that, for instance, the presidential candidates of the two American political parties are "practically the same". Once each candidate is confirmed during primaries, they are usually established within their own partisan camps. The remaining undecided electorate resides in the middle of the political spectrum, and there is a tendency for the candidates to "rush for the middle" in order to appeal to this crucial bloc. Like the paradigmatic example, the assumption is that people will choose the least distant option, (in this case, the distance is ideological) and that the most votes can be had by being directly in the center.


Primaries


Nominees


Final campaign


What would happen if . . .


What would happen if . . .

This is know as the Median Voter Theory, very popular inmodern Political Science

strategic behavior in business and econ 3.2.1. static games of complete information: dominant...

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