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Existence of solution of minimax inequalities, equilibria in games and fixed points without convexity and compactness assumptions

Author

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  • R. Nessah

    (LEM - Lille - Economie et Management - Université de Lille, Sciences et Technologies - CNRS - Centre National de la Recherche Scientifique)

  • G. Tian

Abstract

This paper characterizes the existence of equilibria in minimax inequalities without assuming any form of quasiconcavity 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 non-quasiconcave payoff functions and nonconvex and/or noncompact strategy spaces.

Suggested Citation

  • R. Nessah & G. Tian, 2013. "Existence of solution of minimax inequalities, equilibria in games and fixed points without convexity and compactness assumptions," Post-Print hal-00785017, HAL.
    • Handle: RePEc:hal:journl:hal-00785017
    DOI: 10.1007/s10957-012-0176-5
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    Cited by:

    1. Courtois, Pierre & Nessah, Rabia & Tazdaït, Tarik, 2017. "Existence and computation of Berge equilibrium and of two refinements," Journal of Mathematical Economics, Elsevier, vol. 72(C), pages 7-15.
    2. F. Fakhar & H. R. Hajisharifi & Z. Soltani, 2023. "Noncoercive and noncontinuous equilibrium problems: existence theorem in infinite-dimensional spaces," Journal of Global Optimization, Springer, vol. 86(4), pages 989-1003, August.
    3. Mircea Balaj & Dan Florin Serac, 2023. "Generalized Equilibrium Problems," Mathematics, MDPI, vol. 11(9), pages 1-11, May.
    4. Oriol Carbonell-Nicolau & Richard P. McLean, 2018. "On the Existence of Nash Equilibrium in Bayesian Games," Mathematics of Operations Research, INFORMS, vol. 43(2), pages 100-129, February.
    5. Tian, Guoqiang, 2012. "A Full Characterization on Fixed-Point Theorem, Minimax Inequality, Saddle Point, and KKM Theorem," MPRA Paper 57929, University Library of Munich, Germany, revised Jul 2014.
    6. Robert Joliet & Rabia Nessah, 2016. "Euro White and Euro Yolk: Sovereign Debt Structure Stability in the Eurozone," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 18(03), pages 1-15, September.
    7. John Cotrina & Anton Svensson, 2021. "The finite intersection property for equilibrium problems," Journal of Global Optimization, Springer, vol. 79(4), pages 941-957, April.
    8. Rabia Nessah & Raluca Parvulescu, 2017. "On the Existence of Pareto Efficient Nash Equilibria in Discontinuous Games," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 19(03), pages 1-13, September.
    9. Rabia Nessah & Guoqiang Tian, 2016. "On the existence of Nash equilibrium in discontinuous games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 61(3), pages 515-540, March.

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