Stochastic Adaptation in Finite Games Played by Heterogeneous Populations

Abstract

In this paper, I analyze stochastic adaptation in finite n-player games played by heterogeneous populations of myopic best repliers, better repliers and imitators. In each period, one individual from each of n populations, one for each player role, is drawn to play and chooses a pure strategy according to her personal learning rule after observing a sample from a finite history. With a small probability individuals also make a mistake and play a pure strategy at random. I prove that, for a sufficiently low ratio between the sample and history size, only pure-strategy profiles in certain minimal closed sets under better replies will be played with positive probability in the limit, as the probability of mistakes tends to zero. If, in addition, the strategy profiles in one such set have strictly higher payoffs than all other strategy profiles and the sample size is sufficiently large, then the strategies in this set will be played with probability one in the limit. Applied to 2x2 Coordination Games, the Pareto dominant equilibrium is selected for a sufficiently large sample size, but in all symmetric and many asymmetric games, the risk dominant equilibrium is selected for a sufficiently small sample size.Bounded rationality; Evolutionary game theory; Imitation; Better replies; Markov chain; Stochastic stability; Pareto dominance; Risk dominance