Chapter 6Chapter 6

© 2006 Thomson Learning/South-Western

Game Theory

2

Basic Concepts

All games have three basic elements: Players Strategies Payoffs

Players can make binding agreements in cooperative games, but can not in noncooperative games, which are studied in this chapter.

3

Players

A player is a decision maker and can be anything from individuals to entire nations.

Players have the ability to choose among a set of possible actions.

Games are often characterized by the fixed number of players.

Generally, the specific identity of a play is not important to the game.

4

Strategies

A strategy is a course of action available to a player.

Strategies may be simple or complex. In noncooperative games each player is

uncertain about what the other will do since players can not reach agreements among themselves.

5

Payoffs

Payoffs are the final returns to the players at the conclusion of the game.

Payoffs are usually measure in utility although sometimes measure monetarily.

In general, players are able to rank the payoffs from most preferred to least preferred.

Players seek the highest payoff available.

6

Equilibrium Concepts

In the theory of markets an equilibrium occurred when all parties to the market had no incentive to change his or her behavior.

When strategies are chosen, an equilibrium would also provide no incentives for the players to alter their behavior further.

The most frequently used equilibrium concept is a Nash equilibrium.

7

Nash Equilibrium

The most widely used approach to defining equilibrium in games is that proposed by Cournot and generalized in the 1950s by John Nash.

A Nash equilibrium is a set of strategies, one for each player, that are each best responses against one another.

8

Nash Equilibrium

In a two-player games, a Nash equilibrium is a pair of strategies (a*,b*) such that a* is an optimal strategy for A against b* and b* is an optimal strategy for B against A*. Players can not benefit from knowing the

equilibrium strategy of their opponents. Not every game has a Nash equilibrium,

and some games may have several.

9

The Prisoner’s Dilemma

The Prisoner’s Dilemma is a game in which the optimal outcome for the players is unstable.

The name comes from the following situation. Two people are arrested for a crime. The district attorney has little evidence but is

anxious to extract a confession.

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The Prisoner’s Dilemma

The DA separates the suspects and tells each, “If you confess and your companion doesn’t, I can promise you a six-month sentence, whereas your companion will get ten years. If you both confess, you will each get a three year sentence.”

Each suspect knows that if neither confess, they will be tried for a lesser crime and will receive two-year sentences.

11

The Prisoner’s Dilemma

The normal form (i.e. matrix) of the game is shown in Table 6-1. The confess strategy dominates for both

players so it is a Nash equilibria. However, an agreement to remain silent (not

to confess) would reduce their prison terms by one year each.

This agreement would appear to be the rational solution.

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TABLE 6-1: The Prisoner’s Dilemma

B Confess Silent

Confess

-3, -3

-1, -10 A Silent

-10, -1

-2, -2

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The Prisoner’s Dilemma: Extensive Form

The representation of the game as a tree is referred to as the extensive form.

Action proceeds from top to bottom.

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FIGURE 6-1: The Prisoner’s Dilemma: Extensive Form

.. .Confess

Confess Confess

Silent

Silent Silent

-3, -3 -10, -1 -1, -10 -2, -2

A

B B

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TABLE 6-2: Solving for Nash Equilibrium in Prisoner’s Dilemma Using the Underlining Method

B Confess Silent

Confess

-3, -3

-1, -10 A Silent

-10, -1

-2, -2

Step 1

16

TABLE 6-2: Solving for Nash Equilibrium in Prisoner’s Dilemma Using the Underlining Method

B Confess Silent

Confess

-3, -3

-1, -10 A Silent

-10, -1

-2, -2

Step 2

17

TABLE 6-2: Solving for Nash Equilibrium in Prisoner’s Dilemma Using the Underlining Method

B Confess Silent

Confess

-3, -3

-1, -10 A Silent

-10, -1

-2, -2

Step 3

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TABLE 6-2: Solving for Nash Equilibrium in Prisoner’s Dilemma Using the Underlining Method

B Confess Silent

Confess

-3, -3

-1, -10 A Silent

-10, -1

-2, -2

Step 4

19

TABLE 6-2: Solving for Nash Equilibrium in Prisoner’s Dilemma Using the Underlining Method

B Confess Silent

Confess

-3, -3

-1, -10 A Silent

-10, -1

-2, -2

Step 5

20

Dominant Strategies

A dominant strategy refers to the best response to any strategy chosen by the other player.

When a player has a dominant strategy in a game, there is good reason to predict that this is how the player will play the game.

21

Mixed Strategies

A mixed strategy refers to when the player randomly selects from several possible actions.

By contrast, the strategies in which a player chooses one action or another with certainty are called pure strategies.

22

TABLE 6-3: Matching Pennies Game in Normal Form

B Heads Tails

Heads

1, -1

-1, 1 A Tails

-1, 1

1, -1

23

FIGURE 6-2: Matching Pennies Game in Extensive Form

.. .

Heads

Heads Heads

Tails

Tails Tails

1, -1 -1, 1 -1, 1 1, -1

A

B B

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TABLE 6-4: Solving for Pure-Strategy Nash Equilibrium in Matching Pennies Game

B Heads Tails

Heads

1 , -1

-1, 1 A Tails

-1, 1

1 , -1

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TABLE 6-5: Battle of the Sexes in Normal Form

B (Husband) Ballet Boxing

Ballet 2, 1 0, 0 A (Wife) Boxing 0, 0 1, 2

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FIGURE 6-3: Battle of the Sexes Game in Extensive Form

.. .

Ballet

Ballet Ballet

Boxing

Boxing Boxing

2, 1 0, 0 0, 0 1, 2

A (Wife)

B (Husband)B (Husband)

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TABLE 6-6: Solving for Pure-Strategy Nash Equilibria in Battle of the Sexes

B (Husband) Ballet Boxing

Ballet 2, 1 0, 0 A (Wife) Boxing 0, 0 1, 2

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Best-Response Function

The function which gives the payoff-maximizing choice for one player in each of a continuum of actions of the other player is referred to as the best-response function.

29

TABLE 6-7: Computing the Wife’s Best Response to the Husband’s Mixed Strategy

B (Husband) Ballet h Boxing 1-h

Ballet Box 1 2, 1

Box2 0, 0 A (Wife)

Boxing Box 3 0, 0

Box 4 1, 2

(h)(2) + (1 – h)(0) = 2h

(h)(0) + (1 – h)(1) = 1 - h

30

FIGURE 6-4: Best-Response Functions Allowing Mixed Strategies in the Battle of the Sexes

w

h

12/3

1

1/3

.

.

Pure-strategy Nash equilibrium (both play Ballet)

Mixed-strategy Nash equilibrium

Pure-strategy Nash equilibrium (both play Boxing)

.

Wife’s best-response function

Husband’s best-response function

31

The Problem of Multiple Equilibria

A rule that selects the highest total payoff would not distinguish between two pure-strategy equilibria.

To select between these, one might follow T. Schelling’s suggestion and look for a focal point…a logical outcome on which to coordinate, based on information outside the game.

32

TABLE 6-8: Husband’s Contingent Strategies

Contingent Strategy Strategy written equivalently in conditional format

Always go to Ballet

Follow his wife

Do the opposite

Always go to Boxing

Ballet / Ballet, Ballet / Boxing

Ballet / Ballet, Boxing / Boxing

Boxing / Ballet, Ballet / Boxing

Boxing / Ballet, Boxing / Boxing

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TABLE 6-9: Sequential Version of the Battle of the Sexes in Normal Form

B (Husband)

Ballet / Ballet Ballet / Ballet

Ballet / Boxing Boxing / Boxing Boxing / Ballet Boxing / Ballet Ballet / Boxing Boxing / Boxing

Ballet 2, 1 2, 1 0, 0 0, 0 A (Wife) Boxing 0, 0 1, 2 0, 0 1, 2

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FIGURE 6-5: Sequential Version of the Battle of the Sexes in Extensive Form

.. .

Ballet

Ballet Ballet

Boxing

Boxing Boxing

2, 1 0, 0 0, 0 1, 2

A (Wife)

B (Husband) B (Husband)

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TABLE 6-10: Solving for Nash Equilibria in the Sequential Version of the Battle of the Sexes

B (Husband)

Ballet / Ballet Ballet / Ballet

Ballet / Boxing Boxing / Boxing Boxing / Ballet Boxing / Ballet Ballet / Boxing Boxing / Boxing

Ballet Nash equilibrium 1

2, 1

Nash equilibrium 2

2, 1

0, 0

0, 0 A (Wife) Boxing

0, 0

1, 2

0, 0

Nash equilibrium 3

1, 2

36

Subgame-Perfect Equilibrium

Game theory offers a formal way of selecting the reasonable Nash equilibria in sequential games using the concept of subgame-perfect equilibrium.

A proper subgame consists of the part of the game tree including an initial decision not connected to another in an oval and everything branching out below it.

37

FIGURE 6-6: Proper Subgames in the Battle of the Sexes

.. .

Ballet

Ballet Ballet

Boxing

Boxing Boxing

2, 1 0, 0 0, 0 1, 2

A (Wife)

B (Husband) B (Husband)

Simultaneous Version

38

FIGURE 6-6 (cont.): Proper Subgames in the Battle of the Sexes

.. .

Ballet

Ballet Ballet

Boxing

Boxing Boxing

2, 1 0, 0 0, 0 1, 2

A (Wife)

B (Husband) B (Husband)

Sequential Version

39

Backward Induction

A shortcut to finding the subgame-perfect equlibrium directly is to use backward induction.

Backward induction solves for the equilibrium by working backwards from the end of the game to the beginning.

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FIGURE 6-7: Backward Induction in the Sequential Battle of the Sexes

.. .

Ballet

Ballet Ballet

Boxing

Boxing Boxing

2, 1 0, 0 0, 0 1, 2

A (Wife)

B (Husband) B (Husband)

(Next Slide)

41

FIGURE 6-7: Backward Induction in the Sequential Battle of the Sexes

.

. .Ballet

plays Ballet

Boxing

plays Boxing2, 1 1, 2

A (Wife)

B (Husband) B (Husband)

42

Indefinite Time Horizon

Use the following version of the Prisoners’ Dilemma: The game is played in the first period for

certain, but for how many more periods after that the game is played is uncertain.

Let r be the probability the game is repeated for another period.

(1 – r) is the probability the repitions stop for good.

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Indefinite Time Horizon

In equilibrium, both players play Silent and each earns –2 each period the game is played, implying a player’s expected payoff over the course of the game is

(-2)(1 + r + r2 + r3 + . . .) 6.1

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Indefinite Time Horizon

If a player cheats and plays Confess, the cheater earns –1 in that period, but then both play Confess every period and from then on, each earning –3 for each period, for a total expected payoff of

-1 + (-3)(r + r2 + r3 + . . .) 6.2

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Indefinite Time Horizon

For cooperation to be a subgame-perfect equilibrium, (6.1) must exceed (6.2). Adding 2 to both expressions, and then adding 3(r + r2 + r3 + . . .) to both expressions, (6.1) exceeds (6.2) if

r + r2 + r3 + . . . > 1 6.3

46

Continuous Actions

Equations for the Tragedy of Commons:

4.6120 BA ss

5.6)120( BAA sss

6.62120 BA ss

7.62

60 BA

ss

8.62

60 AB

ss

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Nash equilibrium

B’s best-responsefunction

SA120

FIGURE 6-8: Best-Response Functions in the Tragedy of the Commons

SB

120

60

60

40

40

A’s best-responsefunction

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Continuous Actions

Equations for the Tragedy of Commons After Equilibria are Shifted:

9.6132 BA ss

10.62

66 BA

ss

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Nash equilibriumshifts

B’s best-responsefunction

SA

FIGURE 6-9: Shift in Equilibrium When A’s Benefit Increases

SB

48

36

40

40

A’s best-responsefunction shifts out