THE “CLASSIC” 2 x 2 SIMULTANEOUS CHOICE GAMES

1. THE “CLASSIC” 2 x 2 SIMULTANEOUS CHOICE GAMES Topic #4

2. Two-Player Games • We now turn to two-player (proper) games, i.e., games between • two interested and rational players P1 vs. P2, rather than • one player P1 vs. an indifferent Nature. • The second player P2, • unlike like Nature but like P1, gets payoffs from the outcome of the games, and • like P1, makes rational choices aim at maximizing these payoffs. • In a 2 x 2 game, each player has just two strategies to choose from. • We look first at 2x2 games in which P1 and P2 either • choose their strategies simultaneously by “secret ballot,” or • If P1 chooses first and P2 chooses second, P2 must choose his strategy without knowing what strategy P1 has already chosen. • That is to say, in the manner of ‘Playing Games’ in class.

3. Matching Pennies: Coordination Version • We now need two payoff numbers in each cell, one for each player. • By convention: Row Player payoff, Column player payoff. • The Coordination version of Matching Pennies is a zero-conflict game; that is, • the players have identical payoff from each outcome. • But despite their identical interests, they face a coordination problem. • The decision principles previously identified are of no help.

4. Other Zero-Conflict Games • The top matrix show a non-problematic zero-conflict game. • Since there is a unique best outcome, the players face a coordination problem. • The bottom matrix shows a problematic zero-conflict game that results from an “embarrassment of riches.” • Since there are two outcome that are best, the player face a coordination problem. • This is strategically equivalent to Matching Pennies.

5. The Battle of the Sexes • A Coordination Game with Conflict of Interest. • Players have a common interest in coordinating, • but conflicting interest with respect to how to coordinate • Allowing pre-play communication may actually worsen the problem. • “Let’s both have fun doing what I want to do.” • Neither player has a dominant strategy. • Maximin does not help: • both strategies have the same security level. • If both player use maximax, they both get their minimum payoffs.

6. Matching Pennies: Total Conflict Version • Player 1 wants to “mix” while Player 2 wants to “match.” • This is a zero-sum game, because the payoffs in each cell add up to zero and, more generally, • there is no common interest between the players. • Once again our decision principles are of no help.

7. The D-Day Landings • The Zero-Sum Game between the Allies and the Germans. • Allies want to “mix”; Germans want to “match.” • This obviously is hugely oversimplified (and we’ll refine this a bit in the next topic). • Inspector vs. Evader (under an arms control regime).

8. Prisoner’s Dilemma • Two prisoners are held in jail separately and cannot communicate. • The District Attorney has evidence that they jointly committed a serious crime, for which the penalty is six years in prison. • However, this evidence is insufficient to convict either prisoner, in the absence of a confession by one implicating the other. • But the D. A. has other evidence sufficient to convict each prisoner (without any confession) on a less serious charge, for which the penalty is two years in prison. • The D. A. goes to each prisoner and offers the following deal (telling each the same offer has been made to the other): if you confess (implicating the other), I will take two years off whatever your sentence otherwise would be. • Will either prisoner accept the D. A.'s offer? Would they choose differently if they could communicate? • In the following PD payoff matrix, the negative payoffs indicate the number of years in prison.

9. Prisoner’s Dilemma (cont.) • The PD Game, like Battle of Sexes, is a variable-sum game, • that is, the players have a mixture of common and conflicting interests. • The PD payoff matrix itself appears to be unproblematic, in that the problem of strategic choice is “solved” by the Dominance Principle. • But the resulting outcome is worse for both players than if they both had chosen their dominated strategies.

10. Prisoner’s Dilemma (cont.) • Since the Dominance Principle produces the “inefficient” or “tragic” outcome, we know other Decision Principles do also. • “Confess” is both maximax and minimax, and • “confess” maximizes average payoffs and expected payoffs, regardless of the perceived probability that the other player confesses. • The Social Dilemma Game played in class was a multi-player generalization of the PD. • The PD serves as a very simple model of a stage in an arms race.

11. Nash Equilibrium • The PD outcome “both confess/defect” is a Nash Equilibrium, because • given the strategy of the other player, neither player has an incentive to change his strategy. • Put otherwise, each player’s strategy is a best reply to the other’s strategy. • In a coordination game (even with conflict of interest, i.e., the Battle of the Sexes), each coordinated outcome is an Nash equilibrium. • But in the total conflict version of Matching Pennies, there is no Nash equilibrium (in “pure” strategies).

12. The Game of Chicken • Two juvenile delinquents position their cars at opposite ends of a deserted stretch of road. • With their respective gang members and girl friends looking on, they drive towards each other at high speed, each straddling the center line. • The first driver to lose his nerve and swerve into his own lane to avoid a crash is revealed to be “chicken” and loses the game, while the other wins. • If both swerve, the outcome is a draw. • If both drive straight, the outcome is mutual disaster. • A payoff matrix for Chicken appears on the following slide. • These payoff numbers have no objective meaning (like years in prison in the PD matrix).

13. The Game of Chicken (cont.) • Neither player has a dominant strategy; for each player, “swerve” is the best reply to “straight” and “straight” is the best reply to “swerve.” • That is, the best choice for each player is to do the opposite of what the other player does, and each pair of opposite strategies is a (Nash) equilibrium. • But one equilibrium is a victory for Player 1 and the other for Player 2, with the victory going to the more reckless player.

14. The Game of Chicken (cont.) • The compromise outcome (both swerve) is the symmetric outcome best for both players, but • it is not an equilibrium, and • the apparent willingness of one player to swerve encourages the other player to drive straight. • In short, Chicken is a particularly nasty game. • The Game of Chicken underlies the theory of bargaining tactics presented in Topics #10-11.