6. Anonymous games A game G is anonymous if all the players have the same strategy set and the payoff to each player i is not changed when two other players j and k exchange the strategies they play. In this section we show that for the class of anonymous games a much stronger result than Theorem 2.5 holds (assuming a fixed number of strategies). Namely, the Lipschitz constant required to guarantee existence of pure ϵ -equilibrium is independent of the number of players in the game: 6.1. Theorem. Let ϵ > 0 and m, n ∈ N . Then every anonymous game in L ( n, m, δ ) for δ = ϵ/ 2 m admits a pure ϵ –equilibrium. 6.2. Corollary. Fix m and let δ : N → [0 , 1] be such that δ ( n ) = o (1) . Then for every ϵ > 0 , there is N such that every game with n ≥ N players, m strategies for each player and Lipschitz constant smaller than δ ( n ) admits a pure ϵ -equilibrium. Before the proof we note that similar results to that of Theorem 6.1 appeared in previous papers: Rashid [17] was the first to prove an asymptotic (in the number of players) purifica- tion result for anonymous games; our proof follows the footsteps of his, the main difference being that we start with finite sets of strategies as a primitive while in Rashid’s formulation the strategy set of each player is the unit simplex. See also [3], [5, Lemma 5] and [7] for related results. Daskalakis and Papadimitriou [8, Theorem 2.1] proved that every anonymous game in L ( n, m, δ ) for δ = ϵ/O ( m $^{2}$) admits pure ϵ –equilibrium, and conjectured that the O ( m $^{2}$) can be improved to O ( m ); Theorem 6.1 confirms their conjecture. The difference between our proof and theirs can be roughly described as follows: Our approach is to build an auxiliary game in which the strategy set of each player is extended to the entire unit simplex, use Nash’s Theorem to obtain an equilibrium of the auxiliary game, and then use the Shapley-Folkman theorem to “purify” the equilibrium without changing the utilities too much. Daskalakis and Papadimitriou, on the other hand, extend the best-response function to a continuous 12