THE “CLASSIC” 2 x 2 SIMULTANEOUS CHOICE GAMES

Topic #4

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.

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.

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.

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.

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.

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).

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.

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.

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.

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).

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).

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.

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.