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