in this case a player’s payoff is essentially independent of her opponents’ strategy profile (Proposition 2.3). We show in this paper that, when the number of strategies for every player is fixed, a Lipschitz constant of o (1 $_{/}$√$_{n}$$_{log}$$_{n}$$_{) is already sufficient to guarantee pure}$ ϵ -equilibrium (Theorem 2.5) and give an example for games with Lipschitz constant of order O (1 $_{/}$√$_{n}$$_{) with no such equilibrium (Theorem 2.8).}$ 

Our results give new perspective on the literature on large (with many players) games. Large games are used as a modeling apparatus that captures two intuitive aspects of in- teractions of many economic agents: Continuity and anonymity. Continuity means that an agent’s behavior has only small impact on the utility of other agents. Anonymity means that each agent’s utility depends only on the aggregate behavior of the other agents. A typical example is a congestion game with many players, such as drivers choosing among several roads: A driver’s utility depends on the other drivers’ choices only through the con- gestion they form on each road (anonymity), and each driver has a negligible impact on this congestion (continuity). 

One of the earliest models of strategic interaction with many players, due to Schmei- dler [20], is games with a continuum of players: The set of players is given by a non–atomic measure space; continuity is reflected by the fact that opponent’s strategy profiles which are equal almost everywhere yield the same payoff, and therefore a change in one opponent’s strategy does not affect a player’s payoff; and anonymity is reflected by the fact that oppo- nents’ strategy profile affect a player’s payoff only through its integral. Schmeidler shows that pure Nash equilibrium exists under these assumptions. Kalai [12] considers large fi- nite games under continuity and anonymity assumptions: Roughly speaking, for the special case of complete information games with deterministic types, a player’s payoff is a continuous function of her own strategy and of the empirical distribution of opponent’s types and strate- gies. Kalai’s result implies existence of pure ϵ -equilibrium in such games with sufficiently many players. Several subsequent papers [4, 6, 9, 10] relax some of Kalai’s assumptions and obtain similar results. 

