PHONE: [972]-2-6584135 FAX: [972]-2-6513681 E-MAIL: ratio@math.huji.ac.il URL: http://www.ratio.huji.ac.il/ Abstract 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 cn logn steps for some constant c > 0). Second, Nash approximate equilibria are played by the dynamic with

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... a limit frequency of at least 1 − e −c n for some constant c > 0.