Polytopes and the existence of approximate equilibria in discontinuous games

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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 R-m, 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 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. (C) 2009 Elsevier Inc. All rights reserved.
Original languageEnglish
Pages (from-to)381-388
JournalGames And Economic Behavior
Volume68
Issue number1
DOIs
Publication statusPublished - 1 Jan 2010

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