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Polyhedral Convexity and the Existence of Approximate Equilibria in Discontinuous Games

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  • Carmona, Guilherme

Abstract

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.

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Bibliographic Info

Paper provided by Universidade Nova de Lisboa, Faculdade de Economia in its series FEUNL Working Paper Series with number wp488.

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Length: 30 pages
Date of creation: 2006
Date of revision:
Handle: RePEc:unl:unlfep:wp488

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  1. Radzik, Tadeusz, 1991. "Pure-strategy [epsiv]-Nash equilibrium in two-person non-zero-sum games," Games and Economic Behavior, Elsevier, vol. 3(3), pages 356-367, August.
  2. Ziad, Abderrahmane, 1997. "Pure-Strategy [epsiv]-Nash Equilibrium inn-Person Nonzero-Sum Discontinuous Games," Games and Economic Behavior, Elsevier, vol. 20(2), pages 238-249, August.
  3. Guilherme Carmona, 2004. "On the Existence of Equilibria in Discontinuous Games: Three Counterexamples," Game Theory and Information 0402001, EconWPA.
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