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.
(This abstract was borrowed from another version of this item.)"> 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.
(This abstract was borrowed from another version of this item.)">
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On the existence of equilibria in discontinuous games: three counterexamples

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  • Guilherme Carmona
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.
(This abstract was borrowed from another version of this item.)

Suggested Citation

  • Guilherme Carmona, 2005. "On the existence of equilibria in discontinuous games: three counterexamples," International Journal of Game Theory, Springer;Game Theory Society, vol. 33(2), pages 181-187, June.
    • Handle: RePEc:spr:jogath:v:33:y:2005:i:2:p:181-187
    DOI: 10.1007/s001820400187
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    1. Partha Dasgupta & Eric Maskin, 1986. "The Existence of Equilibrium in Discontinuous Economic Games, I: Theory," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 53(1), pages 1-26.
    2. 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.
    3. Leo K. Simon, 1987. "Games with Discontinuous Payoffs," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 54(4), pages 569-597.
    4. Michael R. Baye & Guoqiang Tian & Jianxin Zhou, 1993. "Characterizations of the Existence of Equilibria in Games with Discontinuous and Non-quasiconcave Payoffs," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 60(4), pages 935-948.
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    Cited by:

    1. 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.
    2. Guilherme Carmona, 2006. "Polyhedral convexity and the existence of approximate equilibria in discontinuous games," Nova SBE Working Paper Series wp488, Universidade Nova de Lisboa, Nova School of Business and Economics.
    3. Philip J. Reny, 2016. "Nash equilibrium in discontinuous games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 61(3), pages 553-569, March.
    4. Pavlo Prokopovych & Nicholas C. Yannelis, 2012. "On Uniform Conditions for the Existence of Mixed Strategy Equilibria," Discussion Papers 48, Kyiv School of Economics.
    5. Paulo Klinger Monteiro & Frank H. Page Jr., 2005. "Uniform payoff security and Nash equilibrium in metric games," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00197491, HAL.
    6. Luciano I. de Castro, 2008. "Equilibria Existence in Regular Discontinuous Games," Discussion Papers 1463, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    7. Luciano Castro, 2011. "Equilibrium existence and approximation of regular discontinuous games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 48(1), pages 67-85, September.
    8. Philip J. Reny, 2020. "Nash Equilibrium in Discontinuous Games," Annual Review of Economics, Annual Reviews, vol. 12(1), pages 439-470, August.
    9. Pavlo Prokopovych, 2008. "A Short Proof of Reny's Existence Theorem for Payoff Secure Games," Discussion Papers 12, Kyiv School of Economics.
    10. Nessah, Rabia & Tian, Guoqiang, 2008. "Existence of Equilibria in Discontinuous Games," MPRA Paper 41206, University Library of Munich, Germany, revised Mar 2010.
    11. Capraro, Valerio & Scarsini, Marco, 2013. "Existence of equilibria in countable games: An algebraic approach," Games and Economic Behavior, Elsevier, vol. 79(C), pages 163-180.
    12. Rabia Nessah & Guoqiang Tian, 2008. "The Existence of Equilibria in Discontinuous and Nonconvex Games," Working Papers 2008-ECO-14, IESEG School of Management, revised Mar 2010.
    13. Oriol Carbonell-Nicolau, 2011. "The Existence of Perfect Equilibrium in Discontinuous Games," Games, MDPI, vol. 2(3), pages 1-22, July.
    14. Pavlo Prokopovych, 2011. "On equilibrium existence in payoff secure games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 48(1), pages 5-16, September.
    15. Monteiro, Paulo Klinger & Page Jr, Frank H., 2007. "Uniform payoff security and Nash equilibrium in compact games," Journal of Economic Theory, Elsevier, vol. 134(1), pages 566-575, May.

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    More about this item

    Keywords

    discontinuous games; payoff security; reciprocal upper semicontinuity; nash equilibrium;
    All these keywords.

    JEL classification:

    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games

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