Strong Migration Limit for Games in Structured Populations: Applications to Dominance Hierarchy and Set Structure

In this paper, we deduce a condition for a strategy S1 to be more abundant on average at equilibrium under weak selection than another strategy S2 in a population structured into a finite number of colonies of fixed proportions as the population size tends to infinity. It is assumed that one individual reproduces at a time with some probability depending on the payoff received in pairwise interactions within colonies and between colonies and that the offspring replaces one individual chosen at random in the colony into which the offspring migrates. It is shown that an expected weighted average equilibrium frequency of S1 under weak symmetric strategy mutation between S1 and S2 is increased by weak selection if an expected weighted payoff of S1 near neutrality exceeds the corresponding expected weighted payoff of S2. The weights are given in terms of reproductive values of individuals in the different colonies in the neutral model. This condition for S1 to be favoured by weak selection is obtained from a strong migration limit of the genealogical process under neutrality for a sample of individuals, which is proven using a two-time scale argument. The condition is applied to games between individuals in colonies with linear or cyclic dominance and between individuals belonging to groups represented by subsets of a given set.