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 InfoArticle provided by Springer in its journal International Journal of Game Theory.
Volume (Year): 33 (2005)
Issue (Month): 2 (06)
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Web page: http://link.springer.de/link/service/journals/00182/index.htm
Other versions of this item:
- Guilherme Carmona, 2004. "On the Existence of Equilibria in Discontinuous Games: Three Counterexamples," Game Theory and Information, EconWPA 0402001, EconWPA.
- 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
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- Luciano I. de Castro, 2008. "Equilibria Existence in Regular Discontinuous Games," Discussion Papers, Northwestern University, Center for Mathematical Studies in Economics and Management Science 1463, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
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