A direct proof of the existence of pure strategy equilibria in large generalized games with atomic players (2010)
Consider a game with a continuum of players where only a finite number of them are atomic. Objective functions and admissible strategies may dependon the actions chosen by atomic players and on aggregate information about theactions chosen by non-atomic players. Only atomic players are required to haveconvex sets of admissible strategies and quasi-concave objective functions. In thiscontext, we prove the existence of pure strategy Nash equilibria, a result that ex-tends Rath (1992, Theorem 2) to generalized games and gives a direct proof of aspecial case of Balder (1999, Theorem 2.1). Our proof has the merit of being simple,based only on standard fixed point arguments and finite dimensional real analysis.