Best-reply dynamics in large binary-choice anonymous games

We consider small-influence anonymous games with a large number of players n where every player has two actions. For this class of games we present a best-reply dynamic with the following two properties. First, the dynamic reaches Nash approximate equilibria fast (in at most cnlogn steps for some constant c>0). Second, Nash approximate equilibria are played by the dynamic with a limit frequency of at least 1−e−c′n for some constant c′>0.