On the Elimination of Dominated Strategies in Stochastic Models of Evolution With Large Populations

Abstract

A stochastic myopic best-reply dynamics is said to have property (W), for a given number of players $n$, if every pure weakly dominated strategy in every $n$-player game is eliminated in the long-run distribution of play induced by the dynamics. In this paper I give a necessary and sufficient condition that a dynamics has to satisfy in order for it to have property (W). The key determinant is found to be the sensitivity of the learning-rate to small payoff differences, inherent in the dynamics. If this sensitivity is higher than a certain cut-off, which depends on the number of players, then the dynamics satisfies property (W). If it is equal to or below that cut-off, then the dynamics does not satisfy property (W).