A Comment on "Learning, Mutation, and Long-Run Equilibria in Games"
评论KMR关于达尔文动力学下策略稳态分布的结论，指出在2x2博弈中，对于占优策略博弈和无对称纯策略均衡博弈，其结论仅在大规模群体中成立，小规模群体可能产生非纳什结果。
mutation.) Kandori, Mailath, and Rob (henceforth KMR) first provide a useful general theorem concerning the stationary distribution of strategies under Darwinian dynamics. They then divide the analysis of the 2 x 2 game into three cases: dominant strategy games (e.g., prisoners' dilemma), coordination games, and games with no symmetric pure strategy equilibrium (e.g., battle of the sexes). We refer to these as DS, C, and NP games. 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. Section 1 summarizes the KMR model, and Section 2 provides examples of 2 x 2 games in which Darwinian dynamics generate non-Nash outcomes. A theorem in Section 3 describes the Darwinian equilibrium of any 2 x 2 game.