Games of incomplete information and myopic equilibria

We consider a finitely defined game where the payoff for each player at each terminal point of the game is not a fixed quantity but varies according to probability distributions on the terminal points induced by the strategies chosen. We prove that if these payoffs have an upper-semi- continuous and convex valued structure then the game has an equilibrium. For this purpose the concept of a myopic equilibrium is introduced, a con- cept that generalizes that of a Nash equilibrium and applies to the games we consider. We answer in the affirmative a question posed by A. Neyman: if the payoffs of an infinitely repeated game of incomplete information on one side are a convex combination of the undiscounted payoffs and payoffs from a finite number of initial stages, does the game have an equilibrium?