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On the Existence of Equilibria in Discontinuous Games: Three Counterexamples

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Guilherme Carmona

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Abstract

We 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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Paper provided by EconWPA in its series Game Theory and Information with number 0402001.

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Length: 11 pages
Date of creation: 09 Feb 2004
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Handle: RePEc:wpa:wuwpga:0402001

Note: Type of Document - pdf; prepared on winxp; to print on general; pages: 11; figures: 3. none
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Related research
Keywords: Discontinuous Games; Nash Equilibrium;

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Find related papers by JEL classification:
C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games

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References listed on IDEAS
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  1. Philip J. Reny, 1999. "On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games," Econometrica, Econometric Society, vol. 67(5), pages 1029-1056, September.
  2. Simon, Leo K, 1987. "Games with Discontinuous Payoffs," Review of Economic Studies, Blackwell Publishing, vol. 54(4), pages 569-97, October. [Downloadable!] (restricted)
  3. Dasgupta, Partha & Maskin, Eric, 1986. "The Existence of Equilibrium in Discontinuous Economic Games, I: Theory," Review of Economic Studies, Blackwell Publishing, vol. 53(1), pages 1-26, January. [Downloadable!] (restricted)
  4. Baye, Michael R & Tian, Guoqiang & Zhou, Jianxin, 1993. "Characterizations of the Existence of Equilibria in Games with Discontinuous and Non-quasiconcave Payoffs," Review of Economic Studies, Blackwell Publishing, vol. 60(4), pages 935-48, October. [Downloadable!] (restricted)
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Cited by:
(explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)

  1. Rabia Nessah & Guoqiang Tian, 2009. "The Existence of Equilibria in Discontinuous and Nonconvex Games," Working Papers 2008-ECO-14, IESEG School of Management. [Downloadable!]
  2. 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. [Downloadable!]
  3. Rabia Nessah & Guoqiang Tian, 2009. "Existence of Equilibria in Discontinuous and Nonconvex Games," Levine's Working Paper Archive 814577000000000206, David K. Levine. [Downloadable!]
  4. Paulo Klinger Monteiro & Frank H. Page Jr, 2005. "Uniform payoff security and Nash equilibrium in metric games," Cahiers de la Maison des Sciences Economiques b05086, Université Panthéon-Sorbonne (Paris 1). [Downloadable!]
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