A comment on: 'Learning, mutation, and long-run equilibria in games'

In a recent article in this journal, Kandori, Mailath, and Rob (1993) (KMR) study the Darwinian dynamics of a 2x2 symmetric game, played repeatedly within a finite population. KMR first provide a useful general theorem concerning the stationary distribution of strategies under Darwinian dynamics. They then divide the analysis of the 2x2 game into three cases : dominant strategy (DS) games (e.g., prisoners' dilemma), coordination (C) games, and games with no symmetric pure strategy equilibrium ( NP) (e.g., battle of the sexes). In each case, KMR claim that, as the rate of mutation vanishes, the stationary distribution of strategies converges to a symmetric Nash equilibrium. They emphasize C games, which have two symmetric Nash equilibria, and characterize the conditions under which the distribution converges to the risk dominant equilibrium. In this note, we argue that while their formal conclusions for C games are correct, their results for DS and NP games are valid only for large populations of players. In small populations, Darwinian dynamics may produce non-Nash outcomes in these two cases.