Evolutionary dynamics of stochastic games in set-structured populations.

In structured populations, the ecology of games may vary over neighborhoods. The effect of the ecological variations on population dynamics remains largely unknown. We here incorporate the ecological variations into the set-structured populations to explore the coevolutionary dynamics of the ecology and cooperation. Individuals of a population are distributed over sets. Interactions occur in the form of evolutionary games. When two individuals share more common sets, they play the weak prisoner's dilemma. Otherwise, they play the strong prisoner's dilemma. Both the set memberships and the strategy update in the evolutionary process. Changes in set memberships hold sway over the games to be played, which, in turn, influences the performance of strategies. Combining evolutionary set theory and random walks on graphs, we derived the conditions for cooperation to be selected under the weak selection limit. We find that a denser set-structured population increases the probability of individuals participating in a weak prisoner's dilemma, and thereby promoting the spread of cooperation. Properly modulating the population structure and the payoff feedback can further lower the critical benefit-cost ratio required for cooperation to be selected. Our results may help better understand the effects of ecological variations in enhancing cooperative behavior in set-structured populations.