Chapter 6 © 2006 Thomson Learning/South-Western Game Theory

Overview 1. Games – concepts 2. Prisoners’ Dilemma 3. Mixed Strategies 4. Multiple Equilibria 5. Tragedy of Commons 2

1.Games - concepts 3

4 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.

5 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 player is not important to the game.

6 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.

7 Payoffs Payoffs are the final returns to the players at the conclusion of the game. Payoffs are usually measured in utility although sometimes also monetarily. In general, players are able to rank the payoffs from most preferred to least preferred. Players seek the highest payoff available.

8 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.

9 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.

2. Prisoners’ Dilemma 10

11 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. Every game has a Nash equilibrium, and some games may have several.

12 The Prisoner’s Dilemma The Prisoner’s Dilemma is a game in which the optimal outcome for the players is unstable – there is a temptation to deviate. 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.

13 The Prisoner’s Dilemma The DA separates the suspects and tells each, “If you confess and your companion doesn’t, I can promise you reduced (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.

14 TABLE 6-1: The Prisoner’s Dilemma

15 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 equilibrium. 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.

16 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.

17 Figure 6-1: The Prisoner’s Dilemma: Extensive Form... Confess Silent -3, -3-10, -1-1, -10-2, -2 A BB

18 Table 6-2: Solving for Nash Equilibrium in Prisoner’s Dilemma Using the Underlining Method Step 1

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

20 Table 6-2: Solving for Nash Equilibrium in Prisoner’s Dilemma Using the Underlining Method Step 3

21 Table 6-2: Solving for Nash Equilibrium in Prisoner’s Dilemma Using the Underlining Method Step 4

22 Table 6-2: Solving for Nash Equilibrium in Prisoner’s Dilemma Using the Underlining Method Step 5

23 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.

3. Mixed Strategies 24

25 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.

26 Table 6-3: Matching Pennies Game in Normal Form

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

28 Table 6-4: Solving for Pure-Strategy Nash Equilibrium in Matching Pennies Game 1/2

29 Solving for Mixed-Strategy Equilibrium Expected payoffs for A: (1/4)(1)+(1/4)(-1)+(1/4)(-1)+(1/4)(1)=0 and similarly for B. So, both playing Heads and Tails with equal chance is the mixed-strategy Nash equilibrium. If there is a strategy that produces higher payoff, then this is NOT Nash equilibrium. Assume B plays H with ½ and T with ½ probability. If A plays only H or only T (or anything in between) A’s payoffs will still be 0.

30 Solving for Mixed-Strategy Equilibrium Show that this is the only mixed-strategy equilibrium (no other probabilities would work). Assume B plays H with 1/3 and T with 2/3. Expected payoff for A from playing H (1/3)(1)+(2/3)(-1)=-1/3 And from playing T Examples of mixed strategies:

31 4. Multiple Equilibria

32 Multiple Equilibria TABLE 6-5: Battle of the Sexes in Normal Form Both prefer to be together rather than apart. She prefers ballet, he boxing.

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

34 Solving for Mixed Strategy Nash Equilibrium Equilibrium probabilities do not end up to be equal for each action: w – probability that wife chooses ballet 1-w- that she does not (chooses boxing) h – probability that husband chooses ballet 1-h – he chooses boxing Goal is to compute equilibrium w and h.

Solving for Mixed Strategy Nash Equilibrium w and h are any of the values between (0,1) and we can not use matrix Need to find 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. Compute wife’s BRF: find w that max her payoff for each of husband’s strategies

For a given h she can: prefer to play Ballet and then her best response is w=1 prefer to play Boxing and her best response is w=0,or be indifferent and her best response is tie between all the values (0,1) Husband plays Ballet with probability h and Boxing 1-h. Best Response Function

37 TABLE 6-7: Computing the Wife’s Best Response to the Husband’s Mixed Strategy (h)(2) + (1 – h)(0) = 2h (h)(0) + (1 – h)(1) = 1 - h She prefers Ballet if 2h>1-h or h>1/3, w=1 She prefers Boxing if h<1/3, w=0 She is indifferent when h=1/3 and her best response is w(0,1)

38 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

39 The Problem of Multiple Equilibria Which equilibrium to choose (will happen)? 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.

5. Tragedy of Commons 40

41 Tragedy of Commons Benefit A gets from each sheep  Total benefit A gets  Marginal benefit of additional sheep

Tragedy of Commons(Continuous Actions)  Marginal benefit=Marginal cost=0 and solve for s A

43 Nash equilibrium B’s best-response function SASA 120 FIGURE 6-8: Best-Response Functions in the Tragedy of the Commons SBSB A’s best-response function

44 Tragedy of Commons Equations for the Tragedy of Commons After Equilibria are Shifted:

45 Nash equilibrium shifts B’s best-response function SASA FIGURE 6-9: Shift in Equilibrium When A’s Benefit Increases SBSB A’s best-response function shifts out