Almost global convergence to p-dominant equilibrium

A population of players repeatedly plays an n strategy symmetric game. Players update their strategies by sampling the behavior of k opponents and playing a best response to the distribution of strategies in the sample. Suppose the game possesses a 1 k -dominant strategy which is initially played by a positive fraction of the population. Then if the population size is large enough, play converges to the 1 k -dominant equilibrium with arbitrarily high probability.