Note that the formulation of the continuity assumption in Kalai’s paper already assumes anonymity. In this paper continuity is captured by the Lipschitz constant of the game, and is therefore not tied to anonymity. Kalai’s argument is based on self-purification : the realized profile of a mixed Nash equilibrium is, with high probability, a pure ϵ -equilibrium, and it has been a folk knowledge that the argument is based on continuity alone. This paper pinpoints the limit behavior of the Lipschitz constant required for the self-purification argument to hold and for pure equilibrium to exist. A common situation where the continuity assumption holds without the anonimity is network games [11, Sections 9.2-9.3]. In these games players are identified with nodes of a graph, and the payoff of each player depends in a Lipschitz continuous way on the empirical distribution of strategies of her neighbors. These games are not anonymous, but it follows from our result that if the set of neighbors of each player is sufficiently large relative to the total number of players, then a pure approximate equilibrium exists. Section 2 contains the necessary definitions and the main results. The proofs of the existence result and of its tightness are in Sections 3 and 4. In Section 5 we show that if the number of strategies of each player is unbounded, then it is not possible to improve the trivial bound of o (1 /n ) to guarantee pure ϵ -equilibrium existence. In Section 6 we consider the special case of anonymous games and show that for such games the required Lipschitz constant is o (1), that is independent of the number of players in the game, even though the self-purification argument fails. Appendix A shows that existence of pure approximate equilibria in Lipschitz games implies Nash’s theorem on existence of mixed equilibrium. 2. Main results An n –player game in normal form G is given by finite sets { A$_{i}$ } n i =1 of strategies and by payoff functions { f$_{i}$ : A → R } n i $_{=1}$, where A = ∏ n i $_{=1}$A$_{i}$ is the set of strategy profiles . A mixed strategy for player i is a probability distribution over A$_{i}$ . Nash equilibrium and ϵ –Nash equilibrium (in pure or mixed strategies) are defined as usual. 3