On the existence of equilibria in discontinuous games: three counterexamples

Guilherme Carmona

Nova SBE Working Paper Series from Universidade Nova de Lisboa, Nova School of Business and Economics

Abstract: We study whether we can weaken the conditions given in Reny [4] and still obtain existence of pure strategy Nash equilibria in quasiconcave normal form games, or, at least, existence of pure strategy "!equilibria for all " > 0. We show by examples that there are: 1. quasiconcave, payoff secure games without pure strategy "!equilibria for small enough " > 0 (and hence, without pure strategy Nash equilibria), 2. quasiconcave, reciprocally upper semicontinuous games without pure strategy "!equilibria for small enough " > 0, and 3. payoff secure games whose mixed extension is not payoff secure. The last example, due to Sion and Wolfe [6], also shows that nonquasiconcave games that are payoff secure and reciprocally upper semicontinuous may fail to have mixed strategy equilibria.

Pages: 11 pages
Date: 2003
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Related works:
Journal Article: On the existence of equilibria in discontinuous games: three counterexamples (2005)
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