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Existence of pure Nash equilibria in discontinuous and non quasiconcave games

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Abstract

In a recent but well known paper, Reny has proved the existence of Nash equilibria for compact and quasiconcave games, with possibly discontinuous payoff functions. In this paper, we prove that the quasiconcavity assumption in Reny's theorem can be weakened: we introduce a measure allowing to localize the lack of quasiconcavity, which allows to refine the analysis of equilibrium existence

Suggested Citation

  • Philippe Bich, 2009. "Existence of pure Nash equilibria in discontinuous and non quasiconcave games," Documents de travail du Centre d'Economie de la Sorbonne 09061, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    • Handle: RePEc:mse:cesdoc:09061
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    Cited by:

    1. Paulo Barelli & Idione Meneghel, 2013. "A Note on the Equilibrium Existence Problem in Discontinuous Games," Econometrica, Econometric Society, vol. 81(2), pages 813-824, March.
    2. Rabia Nessah, 2022. "Weakly continuous security and nash equilibrium," Theory and Decision, Springer, vol. 93(4), pages 725-745, November.
    3. Pavlo Prokopovych, 2016. "Majorized correspondences and equilibrium existence in discontinuous games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 61(3), pages 541-552, March.
    4. Vincenzo Scalzo, 2013. "Essential equilibria of discontinuous games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 54(1), pages 27-44, September.
    5. Tadeusz Radzik, 2014. "Poor convexity and Nash equilibria in games," International Journal of Game Theory, Springer;Game Theory Society, vol. 43(1), pages 169-192, February.
    6. Rabia Nessah, 2013. "Weakly Continuous Security in Discontinuous and Nonquasiconcave Games: Existence and Characterization," Working Papers 2013-ECO-20, IESEG School of Management.
    7. Philip J. Reny, 2020. "Nash Equilibrium in Discontinuous Games," Annual Review of Economics, Annual Reviews, vol. 12(1), pages 439-470, August.
    8. Guilherme Carmona, 2016. "Reducible equilibrium properties: comments on recent existence results," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 61(3), pages 431-455, March.
    9. Harks, Tobias & Klimm, Max, 2015. "Equilibria in a class of aggregative location games," Journal of Mathematical Economics, Elsevier, vol. 61(C), pages 211-220.
    10. 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.
    11. Iskakov, M. & Iskakov, A. & d'Aspremont, C., 2018. "Games for cautious players: The Equilibrium in Secure Strategies," Games and Economic Behavior, Elsevier, vol. 110(C), pages 58-70.
    12. 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.
    13. Zhiwei Liu & Nicholas C. Yannelis, 2013. "On Discontinuous Games with Asymmetric Information," Economics Discussion Paper Series 1318, Economics, The University of Manchester.
    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. Kukushkin, Nikolai S., 2018. "Better response dynamics and Nash equilibrium in discontinuous games," Journal of Mathematical Economics, Elsevier, vol. 74(C), pages 68-78.
    16. Iskakov, M., 2022. "Existence theorems for Nash equilibrium and equilibrium in secure strategies," Journal of the New Economic Association, New Economic Association, vol. 56(4), pages 12-27.
    17. Carmona, Guilherme & Podczeck, Konrad, 2014. "Existence of Nash equilibrium in games with a measure space of players and discontinuous payoff functions," Journal of Economic Theory, Elsevier, vol. 152(C), pages 130-178.
    18. Alejandro Saporiti, 2014. "Power sharing and electoral equilibrium," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 55(3), pages 705-729, April.
    19. Guilherme Carmona & Konrad Podczeck, 2016. "Existence of Nash equilibrium in ordinal games with discontinuous preferences," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 61(3), pages 457-478, March.
    20. He, Wei & Yannelis, Nicholas C., 2015. "Discontinuous games with asymmetric information: An extension of Reny's existence theorem," Games and Economic Behavior, Elsevier, vol. 91(C), pages 26-35.
    21. Kukushkin, Nikolai S., 2016. "Nash equilibrium with discontinuous utility functions: Reny's approach extended," MPRA Paper 75862, University Library of Munich, Germany.
    22. Oriol Carbonell-Nicolau, 2015. "Further results on essential Nash equilibria in normal-form games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 59(2), pages 277-300, June.
    23. Pavlo Prokopovych, 2013. "The single deviation property in games with discontinuous payoffs," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 53(2), pages 383-402, June.

    More about this item

    Keywords

    Nash equilibrium; discontinuity; quasiconcavity;
    All these keywords.

    JEL classification:

    • C70 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - General

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