Effective Game Matrix and Inclusive Payoff in Group-Structured Populations

We show that a matrix game within groups in a finite-island model is effectively equivalent to a matrix game in a well-mixed population. The effective game matrix is a sum of interaction effects minus competition effects, weighted by identity measures involving up to three individuals. These identity measures are computed in the absence of selection but depend on the selection regime and the dispersal pattern: differential viability or fertility, hard selection or soft selection, uniform dispersal or local extinction followed by uniform recolonization. Hard selection, which allows for group selection, understood as differential contributions of groups, reduces competition within groups compared to soft selection. Moreover, the reduction is more pronounced in the case of uniform dispersal than in the case of local extinction. Fertility differences add competition effects between an individual and itself. A personal inclusive payoff can be defined from the effective game matrix and used to predict the increase or decrease in frequency of a mutant strategy. However, this personal inclusive payoff is generally frequency-dependent and its mean does not necessarily increase over time.