Characterizing Pure-strategy Equilibria in Large Games

In this paper, we consider a generalized large game model where the agent space is divided into countable subgroups and each player's payoff depends on her own action and the action distribution in each of the subgroups. Given the countability assumption on its action or payoff space or the Loeb assumption on its agent space, we show that that a given distribution is an equilibrium distribution if and only if for any (Borel) subset of actions the proportion of players in each group playing this subset of actions is no larger than the proportion of players in that group having a best response in this subset. Furthermore, we also present a counterexample showing that this characterization result does not hold for a more general setting