The Diffusion Approximation of Stochastic Evolutionary Game Dynamics: Mean Effective Fixation Time and the Significance of the One-Third Law

Abstract

The one-third law introduced by Nowak et al. (Nature 428:646–650, 2004) for the Moran stochastic process has proven to be a robust criterion to predict when weak selection will favor a strategy invading a finite population. In this paper, we investigate fixation probability, mean effective fixation time, and average and expected fitnesses in the diffusion approximation of the stochastic evolutionary game. Our main results show that in two-strategy games with strict Nash equilibria A and B: (i) the one-third law means that, if selection favors strategy A when a single individual is using it initially, then one-third of the opponents one meets before fixation are A-individuals; and (ii) the average fitness of strategy A about the mean effective fixation time is larger than that of strategy B. The analysis reinforces the universal nature of the one-third law as of fundamental importance in models of selection. We also connect risk dominance of strategy A to its larger expected fitness with respect to the stationary distribution of the diffusion approximation that includes a small mutation rate between the two strategies.