Polyhedral Convexity and the Existence of Approximate Equilibria in Discontinuous Games

Radzik (1991) showed that two-player games on compact intervals of the real line have " { equilibria for all " > 0, provided that payo® functions are upper semicontinuous and strongly quasi-concave. In an attempt to generalize this theorem, Ziad (1997) stated that the same is true for n-player games on compact, convex subsets of Rm, m ¸ 1 provided that we strengthen the upper semicontinuity condition. We show that: 1. the action spaces need to be polyhedral in order for Ziad's ap- proach to work, 2. Ziad's strong upper semicontinuity condition is equivalent to some form of quasi-polyhedral concavity of players' value func- tions in simple games, and 3. Radzik's Theorem is a corollary of (the corrected) Ziad's result.