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On the Existence of Equilibria in Discontinuous Games: Three Counterexamples

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  • Carmona, Guilherme

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.

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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 wp438.

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Length: 11 pages
Date of creation: 2003
Date of revision:
Handle: RePEc:unl:unlfep:wp438

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References

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  1. Simon, Leo K, 1987. "Games with Discontinuous Payoffs," Review of Economic Studies, Wiley Blackwell, Wiley Blackwell, vol. 54(4), pages 569-97, October.
  2. Dasgupta, Partha & Maskin, Eric, 1986. "The Existence of Equilibrium in Discontinuous Economic Games, I: Theory," Review of Economic Studies, Wiley Blackwell, Wiley Blackwell, vol. 53(1), pages 1-26, January.
  3. Philip J. Reny, 1999. "On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games," Econometrica, Econometric Society, Econometric Society, vol. 67(5), pages 1029-1056, September.
  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, Wiley Blackwell, Wiley Blackwell, vol. 60(4), pages 935-48, October.
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Cited by:
  1. Paulo Klinger Monteiro & Frank H. Page Jr, 2005. "Uniform payoff security and Nash equilibrium in metric games," Cahiers de la Maison des Sciences Economiques, Université Panthéon-Sorbonne (Paris 1) b05086, Université Panthéon-Sorbonne (Paris 1).
  2. Nessah, Rabia & Tian, Guoqiang, 2008. "Existence of Equilibria in Discontinuous Games," MPRA Paper 41206, University Library of Munich, Germany, revised Mar 2010.
  3. Monteiro, Paulo Klinger & Page Jr, Frank H., 2007. "Uniform payoff security and Nash equilibrium in compact games," Journal of Economic Theory, Elsevier, Elsevier, vol. 134(1), pages 566-575, May.
  4. Oriol Carbonell-Nicolau, 2011. "The Existence of Perfect Equilibrium in Discontinuous Games," Games, MDPI, Open Access Journal, MDPI, Open Access Journal, vol. 2(3), pages 235-256, July.
  5. Rabia Nessah & Guoqiang Tian, 2008. "The Existence of Equilibria in Discontinuous and Nonconvex Games," Working Papers, IESEG School of Management 2008-ECO-14, IESEG School of Management, revised Mar 2010.
  6. Pavlo Prokopovych, 2011. "On equilibrium existence in payoff secure games," Economic Theory, Springer, Springer, vol. 48(1), pages 5-16, September.
  7. Philip J. Reny, 2013. "Nash Equilibrium in Discontinuous Games," Working Papers, Becker Friedman Institute for Research In Economics 2013-004, Becker Friedman Institute for Research In Economics.
  8. Luciano Castro, 2011. "Equilibrium existence and approximation of regular discontinuous games," Economic Theory, Springer, Springer, vol. 48(1), pages 67-85, September.
  9. Prokopovych, Pavlo & Yannelis, Nicholas C., 2014. "On the existence of mixed strategy Nash equilibria," Journal of Mathematical Economics, Elsevier, vol. 52(C), pages 87-97.
  10. 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.
  11. Rabia Nessah & Guoqiang Tian, 2009. "Existence of Equilibria in Discontinuous and Nonconvex Games," Levine's Working Paper Archive 814577000000000206, David K. Levine.
  12. repec:hal:journl:halshs-00197491 is not listed on IDEAS
  13. Carmona, Guilherme, 2006. "Polyhedral Convexity and the Existence of Approximate Equilibria in Discontinuous Games," FEUNL Working Paper Series, Universidade Nova de Lisboa, Faculdade de Economia wp488, Universidade Nova de Lisboa, Faculdade de Economia.
  14. Pavlo Prokopovych & Nicholas C. Yannelis, 2012. "On Uniform Conditions for the Existence of Mixed Strategy Equilibria," Discussion Papers 48, Kyiv School of Economics.
  15. Capraro, Valerio & Scarsini, Marco, 2013. "Existence of equilibria in countable games: An algebraic approach," Games and Economic Behavior, Elsevier, Elsevier, vol. 79(C), pages 163-180.

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