Chapter Twenty-Eight Game Theory Game Theory Game theory models strategic behavior by agents who understand that their actions affect the actions of other agents. Some Applications of Game Theory The study of oligopolies (industries containing only a few firms) The study of cartels; e.g. OPEC

The study of externalities; e.g. using a common resource such as a fishery. The study of military strategies. What is a Game? A game consists of a set of players a set of strategies for each player the payoffs to each player for every possible list of strategy choices by the players. Two-Player Games A game with just two players is a two-player game.

We will study only games in which there are two players, each of whom can choose between only two strategies. An Example of a Two-Player Game The players are called A and B. Player A has two strategies, called Up and Down. Player B has two strategies, called Left and Right. The table showing the payoffs to both players for each of the four possible strategy combinations is the games payoff matrix.

An Example of a Two-Player Game Player B L R Player A U (3,9) (1,8) D (0,0) (2,1) This is the games payoff matrix. Player As payoff is shown first. Player Bs payoff is shown second.

An Example of a Two-Player Game Player B L R Player A U (3,9) (1,8) D (0,0) (2,1) This is the games payoff matrix. E.g. if A plays Up and B plays Right then As payoff is 1 and Bs payoff is 8. An Example of a Two-Player Game

Player B L R Player A U (3,9) (1,8) D (0,0) (2,1) This is the games payoff matrix. And if A plays Down and B plays Right then As payoff is 2 and Bs payoff is 1. An Example of a Two-Player Game Player B

L R Player A U (3,9) (1,8) D (0,0) (2,1) A play of the game is a pair such as (U,R) where the 1st element is the strategy chosen by Player A and the 2nd is the strategy chosen by Player B. An Example of a Two-Player Game Player B L R Player A

U (3,9) (1,8) D (0,0) (2,1) What plays are we likely to see for this game? An Example of a Two-Player Game Player B L R Player A U (3,9) (1,8)

D (0,0) (2,1) Is (U,R) a likely play? An Example of a Two-Player Game Player B L R Player A U (3,9) (1,8) D (0,0) (2,1) Is (U,R) a

likely play? If B plays Right then As best reply is Down since this improves As payoff from 1 to 2. So (U,R) is not a likely play. An Example of a Two-Player Game Player B L R Player A U (3,9) (1,8) D (0,0) (2,1) Is (D,R) a likely play?

An Example of a Two-Player Game Player B L R Player A U (3,9) (1,8) D (0,0) (2,1) Is (D,R) a likely play? If B plays Right then As best reply is Down. An Example of a Two-Player Game Player B

L R Player A U (3,9) (1,8) D (0,0) (2,1) Is (D,R) a likely play? If B plays Right then As best reply is Down. If A plays Down then Bs best reply is Right. So (D,R) is a likely play. An Example of a Two-Player Game Player B L

R Player A U (3,9) (1,8) D (0,0) (2,1) Is (D,L) a likely play? An Example of a Two-Player Game Player B L R Player A U

(3,9) (1,8) D (0,0) (2,1) Is (D,L) a likely play? If A plays Down then Bs best reply is Right, so (D,L) is not a likely play. An Example of a Two-Player Game Player B L R Player A U (3,9) (1,8)

D (0,0) (2,1) Is (U,L) a likely play? An Example of a Two-Player Game Player B L R Player A U (3,9) (1,8) D (0,0) (2,1) Is (U,L) a

likely play? If A plays Up then Bs best reply is Left. An Example of a Two-Player Game Player B L R Player A U (3,9) (1,8) D (0,0) (2,1) Is (U,L) a likely play? If A plays Up then Bs best reply is Left.

If B plays Left then As best reply is Up. So (U,L) is a likely play. Nash Equilibrium A play of the game where each strategy is a best reply to the other is a Nash equilibrium. Our example has two Nash equilibria; (U,L) and (D,R). An Example of a Two-Player Game Player B L R Player A U

(3,9) (1,8) D (0,0) (2,1) (U,L) and (D,R) are both Nash equilibria for the game. An Example of a Two-Player Game Player B L R Player A U (3,9) (1,8) D (0,0) (2,1)

(U,L) and (D,R) are both Nash equilibria for the game. But which will we see? Notice that (U,L) is preferred to (D,R) by both players. Must we then see (U,L) only? The Prisoners Dilemma To see if Pareto-preferred outcomes must be what we see in the play of a game, consider a famous second example of a two-player game called the Prisoners Dilemma. The Prisoners Dilemma Clyde Bonnie S

S C (-5,-5) (-30,-1) C (-1,-30) (-10,-10) What plays are we likely to see for this game? The Prisoners Dilemma Clyde Bonnie S S

C (-5,-5) (-30,-1) C (-1,-30) (-10,-10) If Bonnie plays Silence then Clydes best reply is Confess. The Prisoners Dilemma Clyde Bonnie S S C

(-5,-5) (-30,-1) C (-1,-30) (-10,-10) If Bonnie plays Silence then Clydes best reply is Confess. If Bonnie plays Confess then Clydes best reply is Confess. The Prisoners Dilemma Clyde Bonnie S S C

(-5,-5) (-30,-1) C (-1,-30) (-10,-10) So no matter what Bonnie plays, Clydes best reply is always Confess. Confess is a dominant strategy for Clyde. The Prisoners Dilemma Clyde Bonnie S S C (-5,-5)

(-30,-1) C (-1,-30) (-10,-10) Similarly, no matter what Clyde plays, Bonnies best reply is always Confess. Confess is a dominant strategy for Bonnie also. The Prisoners Dilemma Clyde Bonnie S S C (-5,-5)

(-30,-1) C (-1,-30) (-10,-10) So the only Nash equilibrium for this game is (C,C), even though (S,S) gives both Bonnie and Clyde better payoffs. The only Nash equilibrium is inefficient. Who Plays When? In both examples the players chose their strategies simultaneously. Such games are simultaneous play games. Who Plays When?

But there are games in which one player plays before another player. Such games are sequential play games. The player who plays first is the leader. The player who plays second is the follower. A Sequential Game Example Sometimes a game has more than one Nash equilibrium and it is hard to say which is more likely to occur. When such a game is sequential it is

sometimes possible to argue that one of the Nash equilibria is more likely to occur than the other. A Sequential Game Example Player B L R Player A U (3,9) (1,8) D (0,0) (2,1) (U,L) and (D,R) are both Nash equilibria when this game is played simultaneously and we have no way of deciding which equilibrium is more likely to occur.

A Sequential Game Example Player B L R Player A U (3,9) (1,8) D (0,0) (2,1) Suppose instead that the game is played sequentially, with A leading and B following. We can rewrite the game in its extensive form. A Sequential Game Example A

U D B L (3,9) A plays first. B plays second. B R L (1,8) (0,0) R (2,1) A Sequential Game Example

A U D B L (3,9) A plays first. B plays second. B R L (1,8) (0,0) R (2,1)

(U,L) is a Nash equilibrium. A Sequential Game Example A U D B L (3,9) A plays first. B plays second. B R L (1,8) (0,0)

R (2,1) (U,L) is a Nash equilibrium. (D,R) is a Nash equilibrium. Which is more likely to occur? A Sequential Game Example A U D B L (3,9) A plays first. B plays second. B R

L (1,8) (0,0) R (2,1) If A plays U then B plays L; A gets 3. A Sequential Game Example A U D B L (3,9) A plays first. B plays second.

B R L (1,8) (0,0) R (2,1) If A plays U then B plays L; A gets 3. If A plays D then B plays R; A gets 2. A Sequential Game Example A U D B L

(3,9) A plays first. B plays second. B R L (1,8) (0,0) R (2,1) If A plays U then B plays L; A gets 3. If A plays D then B plays R; A gets 2. So (U,L) is the likely Nash equilibrium. Pure Strategies Player B L

R Player A U (3,9) (1,8) D (0,0) (2,1) This is our original example once more. Suppose again that play is simultaneous. We discovered that the game has two Nash equilibria; (U,L) and (D,R). Pure Strategies Player B L R Player A

U (3,9) (1,8) D (0,0) (2,1) Player As has been thought of as choosing to play either U or D, but no combination of both; that is, as playing purely U or D. U and D are Player As pure strategies. Pure Strategies Player B L R Player A U (3,9) (1,8)

D (0,0) (2,1) Similarly, L and R are Player Bs pure strategies. Pure Strategies Player B L R Player A U (3,9) (1,8) D (0,0) (2,1)

Consequently, (U,L) and (D,R) are pure strategy Nash equilibria. Must every game have at least one pure strategy Nash equilibrium? Pure Strategies Player B L R Player A U (1,2) (0,4) D (0,5) (3,2)

Here is a new game. Are there any pure strategy Nash equilibria? Pure Strategies Player B L R Player A U (1,2) (0,4) D (0,5) (3,2)

Is (U,L) a Nash equilibrium? Pure Strategies Player B L R Player A U (1,2) (0,4) D (0,5) (3,2) Is (U,L) a Nash equilibrium? No. Is (U,R) a Nash equilibrium?

Pure Strategies Player B L R Player A U (1,2) (0,4) D (0,5) (3,2) Is (U,L) a Nash equilibrium? No. Is (U,R) a Nash equilibrium? No. Is (D,L) a Nash equilibrium?

Pure Strategies Player B L R Player A U (1,2) (0,4) D (0,5) (3,2) Is (U,L) a Nash equilibrium? No. Is (U,R) a Nash equilibrium? No. Is (D,L) a Nash equilibrium? No.

Is (D,R) a Nash equilibrium? Pure Strategies Player B L R Player A U (1,2) (0,4) D (0,5) (3,2) Is (U,L) a Nash equilibrium? Is (U,R) a Nash equilibrium?

Is (D,L) a Nash equilibrium? Is (D,R) a Nash equilibrium? No. No. No. No. Pure Strategies Player B L R Player A U (1,2) (0,4) D

(0,5) (3,2) So the game has no Nash equilibria in pure strategies. Even so, the game does have a Nash equilibrium, but in mixed strategies. Mixed Strategies Instead of playing purely Up or Down, Player A selects a probability distribution (U,1-U), meaning that with probability U Player A will play Up and with probability 1U will play Down. Player A is mixing over the pure strategies Up and Down. The probability distribution (U,1-U) is a mixed strategy for Player A.

Mixed Strategies Similarly, Player B selects a probability distribution (L,1-L), meaning that with probability L Player B will play Left and with probability 1-L will play Right. Player B is mixing over the pure strategies Left and Right. The probability distribution (L,1-L) is a mixed strategy for Player B. Mixed Strategies Player B

Player A L R U (1,2) (0,4) D (0,5) (3,2) This game has no pure strategy Nash equilibria but it does have a Nash equilibrium in mixed strategies. How is it

computed? Mixed Strategies Player B U,U Player A D,1-U L,L R,1-L (1,2) (0,4) (0,5) (3,2) Mixed Strategies

Player B U,U Player A D,1-U L,L R,1-L (1,2) (0,4) (0,5) (3,2) If B plays Left her expected payoff is 2 U 5 ( 1 U ) Mixed Strategies

Player B U,U Player A D,1-U L,L R,1-L (1,2) (0,4) (0,5) (3,2) If B plays Left her expected payoff is 2 U 5(1 U ). If B plays Right her expected payoff is 4 U 2(1 U ).

Mixed Strategies Player B U,U Player A D,1-U L,L R,1-L (1,2) (0,4) (0,5) (3,2) If 2 U 5(1 U ) 4 U 2(1 U ) then B would play only Left. But there are no

Nash equilibria in which B plays only Left. Mixed Strategies Player B U,U Player A D,1-U L,L R,1-L (1,2) (0,4) (0,5) (3,2) If 2 U 5(1 U ) 4 U 2(1 U ) then

B would play only Right. But there are no Nash equilibria in which B plays only Right. Mixed Strategies Player B U,U Player A D,1-U L,L R,1-L (1,2) (0,4) (0,5) (3,2)

So for there to exist a Nash equilibrium, B must be indifferent between playing Left or Right; i.e. 2 U 5(1 U ) 4 U 2(1 U ) Mixed Strategies Player B U,U Player A D,1-U L,L R,1-L (1,2) (0,4) (0,5) (3,2)

So for there to exist a Nash equilibrium, B must be indifferent between playing Left or Right; i.e. 2 U 5(1 U ) 4 U 2(1 U ) U 3 / 5. Mixed Strategies Player B 3 U, 5 Player A 2 D, 5 L,L R,1-L (1,2)

(0,4) (0,5) (3,2) So for there to exist a Nash equilibrium, B must be indifferent between playing Left or Right; i.e. 2 U 5(1 U ) 4 U 2(1 U ) U 3 / 5. Mixed Strategies Player B 3 U, 5 Player A 2 D, 5 L,L

R,1-L (1,2) (0,4) (0,5) (3,2) Mixed Strategies Player B 3 U, 5 Player A 2 D, 5 L,L

R,1-L (1,2) (0,4) (0,5) (3,2) If A plays Up his expected payoff is 1 L 0 (1 L ) L . Mixed Strategies Player B 3 U, 5 Player A 2

D, 5 L,L R,1-L (1,2) (0,4) (0,5) (3,2) If A plays Up his expected payoff is 1 L 0 (1 L ) L . If A plays Down his expected payoff is 0 L 3 (1 L ) 3(1 L ).

Mixed Strategies Player B 3 U, 5 Player A 2 D, 5 L,L R,1-L (1,2) (0,4) (0,5) (3,2)

If L 3(1 L ) then A would play only Up. But there are no Nash equilibria in which A plays only Up. Mixed Strategies Player B 3 U, 5 Player A 2 D, 5 L,L R,1-L (1,2) (0,4)

(0,5) (3,2) If L 3(1 L ) then A would play only Down. But there are no Nash equilibria in which A plays only Down. Mixed Strategies Player B 3 U, 5 Player A 2 D, 5 L,L R,1-L

(1,2) (0,4) (0,5) (3,2) So for there to exist a Nash equilibrium, A must be indifferent between playing Up or Down; i.e. L 3(1 L ) Mixed Strategies Player B 3 U, 5 Player A 2 D, 5

L,L R,1-L (1,2) (0,4) (0,5) (3,2) So for there to exist a Nash equilibrium, A must be indifferent between playing Up or Down; i.e. L 3(1 L ) L 3 / 4. Mixed Strategies Player B 3 U,

5 Player A 2 D, 5 3 L, 4 1 R, 4 (1,2) (0,4) (0,5) (3,2) So for there to exist a Nash equilibrium, A must be indifferent between playing Up or

Down; i.e. L 3(1 L ) L 3 / 4. Mixed Strategies Player B 3 U, 5 Player A 2 D, 5 3 L, 4 1 R, 4 (1,2) (0,4)

(0,5) (3,2) So the games only Nash equilibrium has A playing the mixed strategy (3/5, 2/5) and has B playing the mixed strategy (3/4, 1/4). Mixed Strategies Player B 3 U, 5 Player A 2 D, 5 3 L, 4

1 R, 4 (1,2) 9/20 (0,4) (0,5) (3,2) The payoffs will be (1,2) with probability 3 3 9 5 4 20 Mixed Strategies Player B 3

U, 5 Player A 2 D, 5 3 L, 4 1 R, 4 (1,2) 9/20 (0,4) 3/20 (0,5) (3,2)

The payoffs will be (0,4) with probability 3 1 3 5 4 20 Mixed Strategies Player B 3 U, 5 Player A 2 D, 5 3 L, 4 (1,2) 9/20

(0,5) 6/20 1 R, 4 (0,4) 3/20 (3,2) The payoffs will be (0,5) with probability 2 3 6 5 4 20 Mixed Strategies Player B 3 U, 5 Player A

2 D, 5 3 L, 4 (1,2) 9/20 (0,5) 6/20 1 R, 4 (0,4) 3/20 (3,2) 2/20 The payoffs will be (3,2) with probability 2 1 2

5 4 20 Mixed Strategies Player B 3 U, 5 Player A 2 D, 5 3 L, 4 (1,2) 9/20 (0,5) 6/20

1 R, 4 (0,4) 3/20 (3,2) 2/20 Mixed Strategies Player B 3 U, 5 Player A 2 D, 5 3 L, 4

1 R, 4 (1,2) (0,4) 9/20 3/20 (0,5) (3,2) 6/20 2/20 As expected Nash equilibrium payoff is 9 1 20 3 0 20 6 0

20 2 3 20 3 . 4 Mixed Strategies Player B 3 U, 5 Player A 2 D, 5

3 L, 4 1 R, 4 (1,2) (0,4) 9/20 3/20 (0,5) (3,2) 6/20 2/20 As expected Nash equilibrium payoff is 9 1 20 3 0

20 6 0 20 9 2 20 3 4 20 6 5 20 2 3

20 3 . 4 2 2 20 16 . 5 Bs expected Nash equilibrium payoff is

How Many Nash Equilibria? A game with a finite number of players, each with a finite number of pure strategies, has at least one Nash equilibrium. So if the game has no pure strategy Nash equilibrium then it must have at least one mixed strategy Nash equilibrium.