Poor convexity and Nash equilibria in games

This paper considers two-person non-zero-sum games on the unit square with payoff functions having a new property called poor convexity. This property describes “something between” the classical convexity and quasi-convexity. It is proved that various types of such games have Nash equilibria with a very simple structure, consisting of the players’ mixed strategies with at most two-element supports. Since poor convexity is a basic notion in the paper, also a theory of poorly convex functions is also developed.