Summary: 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 \(\frac1k\)-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 \(\frac1k\)-dominant equilibrium with arbitrarily high probability.