Suppose that a single play of a prisoners' dilemma has the following payoffs:

Player 2

Cooperate Defect

Cooperate. 3, 3. 1, 4

Player 1 Defect. 4, 1. 2, 2

In a large population in which each member's behavior is genetically determined, each player will be either a defector (that is, always defects in any play of a prisoners' dilemma game) or a tit-for-tat player (in multiple rounds of a prisoners' dilemma, she cooperates on the first play, and on any subsequent play she does whatever her opponent did on the preceding play). Pairs of randomly chosen players from this population will play rounds consisting of n single plays of this dilemma (where n 2). The payoff to each player in one whole round (of n plays) is the sum of her payoffs in the n plays. Let the population proportion of defectors be p and the proportion of tit-for-tat players be (1 p). Each member of the population plays rounds of the dilemma repeatedly, matched against a new, randomly chosen opponent for each new round. A tit-for-tat player always begins each new round by cooperating on the first play.

Show in a two-by-two table the payoffs to a player of each type when, in one round of plays, each player meets an opponent of each of the two types.

Find the fitness (average payoff in one round against a randomly chosen opponent) for a defector.

Find the fitness for a tit-for-tat player.

Use the answers to parts (b) and (c) to show that, when p > (n 2)/(n 1), the defector type has greater fitness and that, when p < (n 2)/(n 1), the tit-for-tat type has greater fitness.

If evolution leads to a gradual increase in the proportion of the fitter type in the population, what are the possible eventual equilibrium outcomes of this process for the population described in this exercise? (That is, what are the possible equilibria, and which are evolutionarily stable?) Use a diagram with the fitness graphs to illustrate your answer.

In what sense does more repetition (larger values of n) facilitate the evolution of cooperation?