Polytopes and the existence of approximate equilibria in discontinuous games

Radzik [Radzik, T., 1991. Pure-strategy ε-Nash equilibrium in two-person non-zero-sum games. Games Econ. Behav. 3, 356-367] 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 [Ziad, A., 1997. Pure-strategy ε-Nash equilibrium in n-person nonzero-sum discontinuous games. Games Econ. Behav. 20, 238-249] 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.