Extreme selection unifies evolutionary game dynamics in finite and infinite populations

We show that when selection is extreme—the fittest strategy always reproduces or is mimicked—the unequivalence between evolutionary game dynamics in finite and infinite populations resolves, in the sense that the three generic outcomes—dominance, coexistence and mutual exclusion—emerge in well-mixed populations of any size, though the outcome remains size-dependent. We consider the simplest setting of a 2-players-2-strategies symmetric game and the two most common microscopic definitions of strategy spreading— the frequency-dependent Moran process and the imitation process by pairwise comparison—both in the case in which any intensity of selection is allowed. We show that of the seven different invasion and fixation scenarios that are generically possible in finite populations—fixation being more or less likely to occur and rapid compared to the neutral game—the three that are possible in large populations are the same three that occur for sufficiently strong selection: (1) invasion and quick fixation of one strategy, (2) mutual invasion and slow fixation of one strategy, (3) no invasion and no fixation. Moreover (and interestingly), in the limit of extreme selection, (2) becomes mutual invasion and no fixation, a case that is not possible for finite intensity of selection, but that better matches the deterministic case of coexistence. In the extreme selection limit, we also derive the large population deterministic limit of the two considered stochastic processes.