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The Existence of Equilibria in Discontinuous and Nonconvex Games

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  • Rabia Nessah

    ()
    (IÉSEG School of Management (LEM-CNRS))

  • Guoqiang Tian

    (Texas A&M University, USA)

Abstract

This paper investigates the existence of pure strategy, dominant-strategy, and mixed strategy Nash equilibria in discontinuous and nonconvex games. We introduce a new notion of very weak continuity, called weak transfer continuity, which holds in a large class of discontinuous economic games and is easy to check. We show that it, together with the compactness of strategy space and the quasiconcavity of payoff functions, permits the existence of pure strategy Nash equilibria. Our equilibrium existence result neither implies nor is implied by the existing results in the literature such as those in Baye et al. [1993] and Reny [1999]. We provide sufficient conditions for weak transfer continuity by introducing notions of weak transfer upper continuity and weak transfer lower continuity. These conditions are satisfied in many economic games and are often quite simple to check. We also introduce the notion of weak dominant transfer upper continuity, and use it to study the existence of dominant strategy equilibria. We then generalize these results and those in Baye et al. [1993] and Reny [1999] without assuming any form of quasi-concavity of payoff functions or convexity of strategy spaces.

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Bibliographic Info

Paper provided by IESEG School of Management in its series Working Papers with number 2008-ECO-14.

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Length: 34 pages
Date of creation: Nov 2008
Date of revision: Mar 2010
Handle: RePEc:ies:wpaper:e200814

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Related research

Keywords: Nash equilibrium; dominant strategy equilibrium; discontinuity; nonquasiconcavity; nonconvexity and mixed strategy;

This paper has been announced in the following NEP Reports:

References

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  1. Paul Rothstein, 2007. "Discontinuous Payoffs, Shared Resources, and Games of Fiscal Competition: Existence of Pure Strategy Nash Equilibrium," Journal of Public Economic Theory, Association for Public Economic Theory, vol. 9(2), pages 335-368, 04.
  2. Vives, X., 1988. "Nash Equilibrium With Strategic Complementarities," UFAE and IAE Working Papers 107-88, Unitat de Fonaments de l'Anàlisi Econòmica (UAB) and Institut d'Anàlisi Econòmica (CSIC).
  3. Simon, Leo K. & Zame, William R., 1987. "Discontinous Games and Endogenous Sharing Rules," Department of Economics, Working Paper Series qt8n46v2wv, Department of Economics, Institute for Business and Economic Research, UC Berkeley.
  4. Robson, A.J., 1990. "An "informationally robust equilibrium" for two-person nonzero-sum games," Discussion Paper 1990-39, Tilburg University, Center for Economic Research.
  5. Simon, Leo K, 1987. "Games with Discontinuous Payoffs," Review of Economic Studies, Wiley Blackwell, vol. 54(4), pages 569-97, October.
  6. 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.
  7. Tian, Guoqiang & Zhou, Jianxin, 1995. "Transfer continuities, generalizations of the Weierstrass and maximum theorems: A full characterization," Journal of Mathematical Economics, Elsevier, vol. 24(3), pages 281-303.
  8. 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.
  9. Guilherme Carmona, 2004. "On the Existence of Equilibria in Discontinuous Games: Three Counterexamples," Game Theory and Information 0402001, EconWPA.
  10. Morgan, Jacqueline & Scalzo, Vincenzo, 2007. "Pseudocontinuous functions and existence of Nash equilibria," Journal of Mathematical Economics, Elsevier, vol. 43(2), pages 174-183, February.
  11. Nishimura, Kazuo & Friedman, James, 1981. "Existence of Nash Equilibrium in n Person Games without Quasi-Concavity," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 22(3), pages 637-48, October.
  12. Tian, Guoqiang, 1993. "Necessary and Sufficient Conditions for Maximization of a Class of Preference Relations," Review of Economic Studies, Wiley Blackwell, vol. 60(4), pages 949-58, October.
  13. Guoqiang Tian, 2009. "The Existence of Equilibria in Games with Arbitrary Strategy Spaces and Payoffs: A Full Characterization," Levine's Working Paper Archive 814577000000000160, David K. Levine.
  14. 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, vol. 60(4), pages 935-48, October.
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