Supplementary Information: payoffs and best responses in the game
The payoff
Focus on an individual that cooperates for m rounds before defecting. Suppose the opponent cooperates for n rounds before defecting. Then the payoff to the focal animal is:
mR + T ifmn
mR + Pifm = n
nR + Sifmn.
Suppose that in the resident population a proportion pn of individuals cooperate for n rounds before defecting (n = 0, 1, …, N). Then by the above, the fitness of the focal individual is
.
Should the focal individual cooperate for longer?
In this population it is better to cooperate for m rounds rather than m+1 if and only if . From the above we obtain
.(*)
Note that by assumption and .
Evolutionary stability: the backward induction argument
We now show that the only evolutionarily stable strategy in the game without maintenance of genetic variation by extrinsic factors is to defect on the first round.
Suppose that in the resident population for some n > 0. Then there exists an n0 in the range such thatand for nn0. Thus for , and hence by (*)
.
Thus the best response to the resident population involves defection with probability 1 on round n0. The resident population strategy is thus not a best response to itself and is hence not evolutionarily stable.
Conversely suppose that for all . Then for all , so that the resident population strategy is the unique best response to itself. Thus the unique evolutionarily stable strategy is to cooperate for zero rounds, i.e. to defect on the first round.
Best responses to selected resident strategies.
We now consider two cases in which the mean number of rounds of cooperation in the population satisfies . In each case we find the best response to this resident population strategy.
(a) Suppose that in the resident population all individuals cooperate for exactly n0 rounds before defecting. That is and for . Then by (*) , , and for . Thus W(n) has a strict maximum at ; i.e. the best response to the resident strategy is to cooperate for exactly rounds.
(b) Now suppose that the number of rounds of cooperation in the resident population is uniformly distributed between and . Then by (*) it can be seen that for we have if and only if . Thus the best response is to cooperate for rounds. Thus the best response is to cooperate for more rounds than the population mean of if .
Note that although the best response can exceed the mean for the resident population, there are population members that are more cooperative than under the best response strategy. Thus under the best response strategy the maximum number of rounds of cooperation is less than in the resident population. In the absence of an extrinsic factor (such as mutation) that maintains genetic variation natural selection would therefore lead to a reduction in the maximum number of rounds of cooperation in the population. Thus the above does not contradict the backwards induction argument. The result does, however, motivate why cooperation can evolve when there is an extrinsic factor maintaining genetic variation.
The case where the number of rounds of cooperation has a pseudo-normal distribution in the resident population (Figure 1 in the main text) is not easy to analyse analytically. As the figure shows, however, results are analogous to case (b) above in that increasing the spread of the distribution increases the best response.