A Direct Proof of the Existence of Pure Strategy Equilibria in Large Generalized Games with Atomic Players

We consider a game with a continuum of players where at most a finite number of them are atomic. Objective functions are continuous and admissible strategies may depend on the actions chosen by atomic players and on aggregate information about the actions chosen by non-atomic players. When atomic players have convex sets of admissible strategies and quasi-concave objective functions, a pure strategy Nash equilibria always exists.