# Suppose that a single play of a prisoners’ dilemma has the following payoffs: In a large population.

Suppose that a single play of a prisoners’ dilemma has

the following payoffs:

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 “sets”

of n single plays of this dilemma (where n $ 2). The payoff to each player in

one whole set (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 2 p). Each member of the population plays sets of dilemmas

repeatedly, matched against a new, randomly chosen opponent for each new set. A

tit-for-tat player always begins each new set by cooperating on its first play.

(a) Show in a two-by-two table the payoffs to a player of

each type when, in one set of plays, each player meets an opponent of each of

the two types.

(b) Find the fitness (average payoff in one set against a

randomly chosen opponent) for a defector.

(c) Find the fitness for a tit-for-tat player.

(d) Use the answers to parts (b) and (c) to show that,

when p . (n 2 2)(n 21),the defector type has greater fitness and that, when p ,

(n 2 2)(n 21), the tit-for-tat type has greater fitness.

(e) 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

evolutionary stable?) Use a diagram with the fitness graphs to illustrate your

answer.

(f) In what sense does more repetition (larger values of

n) facilitate the evolution of cooperation?