Characterization of optimal strategies in matrix games with convexity properties

This paper gives a full characterization of matrices with rows and columns having properties closely related to the (quasi-) convexity–concavity of functions. The matrix games described by such payoff matrices well approximate continuous games on the unit square with payoff functions F(x, y) concave in x for each y, and convex in y for each x. It is shown that the optimal strategies in such matrix games have a very simple structure and a search-procedure is given. The results have a very close relationship with the known theorem of Debreu and Glicksberg about the existence of a pure Nash equilibrium in n-person games.