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On the Existence of approximative Equilibria and Sharing Rule Solutions in Discontinuous Games

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  • Philippe Bich

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique, PSE - Paris School of Economics - UP1 - Université Paris 1 Panthéon-Sorbonne - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris Sciences et Lettres - EHESS - École des hautes études en sciences sociales - ENPC - École des Ponts ParisTech - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement)

  • Rida Laraki

    (LAMSADE - Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CNRS - Centre National de la Recherche Scientifique, X - École polytechnique - IP Paris - Institut Polytechnique de Paris)

Abstract

This paper studies the existence of equilibrium solution concepts in a large class of economic models with discontinuous payoff functions. The issue is well understood for Nash equilibria, thanks to Reny's better-reply security condition (Reny (1999)), and its recent improvements (Barelli and Meneghel (2013); McLennan et al. (2011); Reny (2009, 2011)). We propose new approaches, related to Reny's work, and obtain tight conditions for the existence of approximate equilibria and of sharing rule solutions in pure and mixed strategies (Simon and Zame (1990)). As byproducts, we prove that many auction games with correlated types admit an approximate equilibrium, and that many competition models have a sharing rule solution.

Suggested Citation

  • Philippe Bich & Rida Laraki, 2017. "On the Existence of approximative Equilibria and Sharing Rule Solutions in Discontinuous Games," PSE-Ecole d'économie de Paris (Postprint) hal-01396183, HAL.
    • Handle: RePEc:hal:pseptp:hal-01396183
    DOI: 10.3982/TE2081
    Note: View the original document on HAL open archive server: https://hal.science/hal-01396183
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    References listed on IDEAS

    as
    1. Carmona, Guilherme, 2009. "An existence result for discontinuous games," Journal of Economic Theory, Elsevier, vol. 144(3), pages 1333-1340, May.
    2. Laraki, Rida & Solan, Eilon & Vieille, Nicolas, 2005. "Continuous-time games of timing," Journal of Economic Theory, Elsevier, vol. 120(2), pages 206-238, February.
    3. 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.
    4. Guilherme Carmona, 2011. "Understanding some recent existence results for discontinuous games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 48(1), pages 31-45, September.
    5. 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, April.
    6. Andreoni, James & Che, Yeon-Koo & Kim, Jinwoo, 2007. "Asymmetric information about rivals' types in standard auctions: An experiment," Games and Economic Behavior, Elsevier, vol. 59(2), pages 240-259, May.
    7. Ziad, Abderrahmane, 1997. "Pure-Strategy [epsiv]-Nash Equilibrium inn-Person Nonzero-Sum Discontinuous Games," Games and Economic Behavior, Elsevier, vol. 20(2), pages 238-249, August.
    8. Carmona, Guilherme, 2010. "Polytopes and the existence of approximate equilibria in discontinuous games," Games and Economic Behavior, Elsevier, vol. 68(1), pages 381-388, January.
    9. Kim, Jinwoo & Che, Yeon-Koo, 2004. "Asymmetric information about rivals' types in standard auctions," Games and Economic Behavior, Elsevier, vol. 46(2), pages 383-397, February.
    10. Erik J. Balder, 1988. "Generalized Equilibrium Results for Games with Incomplete Information," Mathematics of Operations Research, INFORMS, vol. 13(2), pages 265-276, May.
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    Cited by:

    1. Rabia Nessah, 2022. "Weakly continuous security and nash equilibrium," Theory and Decision, Springer, vol. 93(4), pages 725-745, November.
    2. Barelli, Paulo & Govindan, Srihari, 2024. "Existence of monotone equilibria in large double auctions," Theoretical Economics, Econometric Society, vol. 19(2), May.
    3. 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.
    4. János Flesch & Dries Vermeulen & Anna Zseleva, 2021. "Legitimate equilibrium," International Journal of Game Theory, Springer;Game Theory Society, vol. 50(4), pages 787-800, December.
    5. Duvocelle, Benoit & Mourmans, Niels, 2022. "Contests with discontinuous payoffs," Journal of Mathematical Economics, Elsevier, vol. 98(C).
    6. Olszewski, Wojciech & Siegel, Ron, 2023. "Equilibrium existence in games with ties," Theoretical Economics, Econometric Society, vol. 18(2), May.

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