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Existence of Equilibrium in Minimax Inequalities, Saddle Points, Fixed Points, and Games without Convexity Sets

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Listed:
  • Rabia Nessah

    (IÉSEG School of Management (LEM-CNRS))

  • Guoqiang Tian

    (Texas A&M University, USA)

Abstract

This paper characterizes the existence of equilibria in minimax inequalities without assuming any form of quasi-concavity of functions and convexity or compactness of choice sets. A new condition, called “local dominatedness property”, is shown to be necessary and further, under some mild continuity condition, sufficient for the existence of equilibrium. We then apply the basic result obtained in the paper to generalize the existing theorems on the existence of saddle points, fixed points, and coincidence points without convexity or compactness assumptions. As an application, we also characterize the existence of pure strategy Nash equilibrium in games with discontinuous and nonquasiconcave payoff functions and nonconvex and/or noncompact strategy spaces.

Suggested Citation

  • Rabia Nessah & Guoqiang Tian, 2008. "Existence of Equilibrium in Minimax Inequalities, Saddle Points, Fixed Points, and Games without Convexity Sets," Working Papers 2008-ECO-15, IESEG School of Management, revised Nov 2010.
    • Handle: RePEc:ies:wpaper:e200815
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    References listed on IDEAS

    as
    1. 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.
    2. 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 637-648, October.
    3. 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.
    4. M. B. Lignola, 1997. "Ky Fan Inequalities and Nash Equilibrium Points without Semicontinuity and Compactness," Journal of Optimization Theory and Applications, Springer, vol. 94(1), pages 137-145, July.
    5. 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.
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