Two Timescales in Stochastic Evolutionary Games

Convergence of discrete-time Markov chains with two timescales is a powerful tool to study stochastic evolutionary games in subdivided populations. Focusing on linear games within demes, convergence to a continuous-time, continuous-state-space diffusion process for the strategy frequencies as the unit of time increases to infinity with either the size of the demes or their number yields a strong-migration limit. The same limit is obtained for a linear game in a well-mixed population with effective payoffs that depend on the reproductive values of the demes and identity measures between interacting individuals and competitors within the same demes. The effective game matrix is almost the same under global (hard) selection and local (soft) selection if migration is not too strong. Moreover, the fixation probability of a strategy introduced as a single mutant is given by a formula that can be calculated in the case of a small population-scaled intensity of selection. The first-order effect of selection on this probability, which extends the one-third law of evolution, can also be obtained directly by summing the successive expected changes in the mutant type frequency that involve expected coalescence times of ancestral lines under neutrality. These can be approached by resorting to the existence of two timescales in the genealogical process. On the other hand, keeping the population size fixed but increasing the unit of time to infinity with the inverse of the intensity of migration, convergence to a continuous-time Markov chain after instantaneous initial transitions can be used to obtain a low-migration limit that depends on fixation probabilities within demes in the absence of migration. In the limit of small uniform dispersal, the fixation probability of a mutant strategy in the whole population exceeds its initial frequency if it is risk-dominant over the other strategy with respect to the average payoffs in pairwise interactions in all demes. Finally, introducing recurrent mutation from one strategy to the other, the low-mutation limit takes the inverse of the mutation rate as unit of time. As this rate goes to zero, the average abundance of a strategy in the long run is determined by the fixation probabilities of both strategies when introduced as single mutants in a deme chosen at random.