Assortment and the evolution of cooperation in a Moran process with exponential fitness
Daniel Cooney, Carl Veller
We study the evolution of cooperation in a finite population interacting according to a simple model of like-with-like assortment. Evolution proceeds as a Moran process, and payoffs from the underlying cooperator-defector game are translated to positive fitnesses by an exponential transformation. The use of the exponential transformation, rather than the usual linear one, allows for a tractable characterization of the effect of assortment on the evolution of cooperation. We define two senses in which a greater degree of assortment can favour the evolution of cooperation, the first stronger than the second: (i) greater assortment increases, at all population states, the probability that the number of cooperators increases, relative to the probability that the number of defectors increases; and (ii) greater assortment increases the fixation probability of cooperation, relative to that of defection. We show that, even by the stronger definition, greater assortment favours the evolution of cooperation for many cooperative dilemmas of interest, including prisoners’ dilemmas, snowdrift games, and stag-hunt games. For other cooperative dilemmas, greater assortment favours cooperation by the weak definition, but not by the strong definition. Allen and Nowak (2015) have derived similar results for a Wright-Fisher process with assortment. Our results complement theirs, and extend them in two ways: First, while their results hold only for weak selection, our results hold for any strength of selection. Second, while their results apply only to the weak definition by which assortment favours cooperation, we derive results for the strong definition too.