© 2006 Thomson Learning/South-Western

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 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.

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 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.

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 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.

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 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.

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 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.

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 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.

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 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.

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 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 DA separates the suspects and tells each, “If you confess and your companion doesn’t, I can promise you a one-year 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.

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 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 61: The Prisoner’s Dilemma
A
Confess
Silent
Confess
-3, -3
-10, -1
B
Silent
-1, -10
-2, -2

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 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 61: The Prisoner’s Dilemma:
Extensive Form
A
.
.
Confess
B
Silent
.
B
Confess Silent Confess Silent
-3, -3 -10, -1 -1, -10 -2, -2

Table 6-2: Solving for Nash Equilibrium in
Prisoner’s Dilemma Using the Underlining
Method
Step 1
A
Confess
Silent
Confess
-3, -3
-10, -1
B
Silent
-1, -10
-2, -2
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Table 6-2: Solving for Nash Equilibrium in
Prisoner’s Dilemma Using the Underlining
Method
Step 2
Confess
A
Silent
Confess
-3, -3
-10, -1
B
Silent
-1, -10
-2, -2
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Table 6-2: Solving for Nash Equilibrium in
Prisoner’s Dilemma Using the Underlining
Method
Step 3
Confess
A
Silent
Confess
-3, -3
-10, -1
B
Silent
-1, -10
-2, -2
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Table 6-2: Solving for Nash Equilibrium in
Prisoner’s Dilemma Using the Underlining
Method
Step 4
Confess
A
Silent
Confess
-3, -3
-10, -1
B
Silent
-1, -10
-2, -2
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Table 6-2: Solving for Nash Equilibrium in
Prisoner’s Dilemma Using the Underlining
Method
Step 5
Confess
A
Silent
Confess
-3, -3
-10, -1
B
Silent
-1, -10
-2, -2
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 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.

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 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 .

A
Heads
Tails
Heads
1, -1
-1, 1
B
Tails
-1, 1
1, -1
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Figure 6-2: Matching Pennies Game in Extensive Form
A
.
.
Heads
B
Tails
.
B
Heads Tails Heads Tails
1, -1 -1, 1 -1, 1 1, -1

Table 6-4: Solving for Pure-Strategy Nash
Equilibrium in Matching Pennies Game
A
Heads
Tails
Heads
1 , -1
-1, 1
B
Tails
-1, 1
1 , -1
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TABLE 6-5: Battle of the Sexes in
Normal Form
A (Wife)
Ballet
Boxing
B (Husband)
Ballet Boxing
2, 1 0, 0
0, 0 1, 2

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Figure 6-3: Battle of the Sexes Game in Extensive Form
2, 1
Ballet
.
Ballet
B (Husband)
Boxing Ballet
Boxing
.
B (Husband)
Boxing
0, 0 0, 0 1, 2

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TABLE 6-6: Solving for Pure-Strategy
Nash Equilibria in Battle of the Sexes
A (Wife)
Ballet
Boxing
B (Husband)
Ballet Boxing
2, 1
0, 0
0, 0
1, 2

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

TABLE 67: Computing the Wife’s Best
Response to the Husband’s Mixed Strategy
A (Wife)
B (Husband)
Ballet h Boxing 1-h
Ballet Box 1
2, 1
Box2
0, 0
Boxing Box 3
0, 0
Box 4
1, 2
(h)(2) + (1 – h)(0)
= 2h
(h)(0) + (1 – h)(1)
= 1 - h
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Figure 6-4: Best-Response Functions
Allowing Mixed Strategies in the Battle of the Sexes
1 h
Husband’s best-response function
.
Pure-strategy
Nash equilibrium
(both play Ballet)
Wife’s bestresponse function
1/3
.
Pure-strategy
Nash equilibrium
(both play Boxing)
2/3
.
Mixed-strategy
Nash equilibrium
1 w

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 A rule that selects the highest total payoff would not distinguish between two purestrategy 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.