On the Existence of Equilibria in Discontinuous Games: Three Counterexamples
AbstractWe study whether we can weaken the conditions given in Reny (1999) and still obtain existence of pure strategy Nash equilibria in quasiconcave normal form games, or, at least, existence of pure strategy $\varepsilon-$equilibria for all epsilon>0. We show by examples that there are: (1) quasiconcave, payoff secure games without pure strategy epsilon-equilibria for small enough epsilon>0 (and hence, without pure strategy Nash equilibria), (2) quasiconcave, reciprocally upper semicontinuous games without pure strategy epsilon-equilibria for small enough epsilon>0, and (3) payoff secure games whose mixed extension is not payoff secure. The last example, due to Sion and Wolfe (1957), also shows that non-quasiconcave games that are payoff secure and reciprocally upper semicontinuous may fail to have mixed strategy equilibria.
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Bibliographic InfoPaper provided by EconWPA in its series Game Theory and Information with number 0402001.
Length: 11 pages
Date of creation: 09 Feb 2004
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Discontinuous Games; Nash Equilibrium;
Other versions of this item:
- Guilherme Carmona, 2005. "On the existence of equilibria in discontinuous games: three counterexamples," International Journal of Game Theory, Springer, vol. 33(2), pages 181-187, 06.
- Carmona, Guilherme, 2003. "On the Existence of Equilibria in Discontinuous Games: Three Counterexamples," FEUNL Working Paper Series wp438, Universidade Nova de Lisboa, Faculdade de Economia.
- C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games
This paper has been announced in the following NEP Reports:
- NEP-ALL-2004-02-15 (All new papers)
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- Carmona, Guilherme, 2006. "Polyhedral Convexity and the Existence of Approximate Equilibria in Discontinuous Games," FEUNL Working Paper Series wp488, Universidade Nova de Lisboa, Faculdade de Economia.
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"Uniform payoff security and Nash equilibrium in metric games,"
Cahiers de la Maison des Sciences Economiques
b05086, Université Panthéon-Sorbonne (Paris 1).
- Paulo Klinger Monteiro & Frank H. Page Jr., 2005. "Uniform payoff security and Nash equilibrium in metric games," UniversitÃ© Paris1 PanthÃ©on-Sorbonne (Post-Print and Working Papers) halshs-00197491, HAL.
- repec:hal:journl:halshs-00197491 is not listed on IDEAS
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