Best-response dynamics in a birth-death model of evolution in games

We consider a model of evolution with mutations as in Kandori et al. (1993) [Kandori, M., Mailath, G.J., Rob, R., 1993. Learning, mutation, and long run equilibria in games. Econometrica 61, 29-56], where agents follow best-response decision rules as in Sandholm (1998) [Sandholm, W., 1998. Simple and clever decision rules for a model of evolution. Economics Letters 61, 165-170]. Contrary to those papers, our model gives rise to a birth-death process, which allows explicit computation of the long-run probabilities of equilibria for given values of the mutation rate and the population size. We use this fact to provide a direct proof of the stochastic stability of risk-dominant equilibria as the mutation rate tends to zero, and illustrate the outcomes of the dynamics for positive mutation rates.