Polytopes and the existence of approximate equilibria in discontinuous games

Guilherme Carmona

Games and Economic Behavior, 2010, vol. 68, issue 1, 381-388

Abstract: Radzik [Radzik, T., 1991. Pure-strategy [epsilon]-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 [epsilon]-equilibria for all [epsilon]>0. Ziad [Ziad, A., 1997. Pure-strategy [epsilon]-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 , m[greater-or-equal, slanted]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 [epsilon]-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.

Keywords: Discontinuous; games; Upper; semi-continuity; Lower; semi-continuity; Quasi-concavity; Approximate; equilibrium (search for similar items in EconPapers)
Date: 2010
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (8)

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