Polytopes and the existence of approximate equilibria in discontinuous games

Radzik (1991) showed that, by strengthening the usual quasi-concavity assumption on players’ payoff functions, upper semi-continuous two-player games on compact intervals of the real line have ε-equilibria for all ε > 0. Ziad (1997) then stated that the same conclusion holds for n-player games on compact, convex subsets of Rm, m ≥ 1, provided that the upper semi-continuity condition is strengthened. Both Radzik’s and Ziad’s proofs rely crucially on the lower hemi-continuity of the ε-best reply correspondence. We show that: (1) in contrast to what is stated by Ziad, his conditions fail to be sufficient for the lower hemi-continuity of the approximate best-reply correspondence, (2) the approximate best-reply correspondence is indeed lower hemi-continuous if players’ action spaces are polytopes, and (3) with action spaces as polytopes, Ziad’s theorem can be stated so that it properly generalizes Radzik’s theorem. ∗I wish to thank Erik Balder, Paulo Bárcia, Paulo Côrte-Real, Branko Grünbaun, Gil Kalai, Armando Machado, Lúıs Vasconcelos and an anonymous referee for very helpful comments. Special thanks are due to an associate editor, whose comments greatly improved the paper. I thank also John Huffstot for editorial assistance. Any remaining error is, of course, my own. †Address: Universidade Nova de Lisboa, Faculdade de Economia, Campus de Campolide, 1099032 Lisboa, Portugal; Phone: (351) 21 38