Assortment by Group Founders Always Promotes the Evolution of Cooperation Under Global Selection But Can Oppose it Under Local Selection

We consider assortment of cooperators or defectors within groups by assuming that group founders express a preference for other group members that are identical-by-type to them. Assuming pairwise interactions within groups and payoffs determined by a game matrix as in a repeated Prisoner’s dilemma with strategies tit-for-tat (representing cooperation C) and always-defect (representing defection D), we show that the dynamics in an infinite population or a large finite population is described by a replicator equation or a diffusion approximation, respectively, as for random pairwise interactions in a well-mixed population with some effective game matrix involving coefficients of pairwise and tripletwise assortment. We deduce that an increase in the assortment level facilitates the evolution of cooperation based on stability properties in an infinite population as well as fixation probabilities in a finite population if selection is global so that groups contribute proportionally to their average payoffs, or if selection is local so that groups contribute equally, but in this case only if C is risk dominant over D. Actually, in the latter case, an increase in the assortment level is less conducive to cooperation, and even opposes the evolution of cooperation, if D is risk dominant over C. Moreover, we show that stochastic variability in the assortment level and/or the group size always facilitates the evolution of cooperation in the former case but not necessarily in the latter.