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Nash and limit equilibria of games with a continuum of players

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  • Guilherme Carmona

Abstract

We show that a strategy is a Nash equilibrium in a game with a continuum of players if and only if there exists a sequence of finite games such that its restriction is an "n-equilibria, with "n converging to zero. In our characterization, the sequence of finite games approaches the continuum game in the sense that the set of players and the distribution of characteristics and actions in the finite games converge to those of the continuum game. These results render approximate equilibria of large finite economies as an alternative way of obtaining strategic insignificance. Also, they suggest defining a refinement of Nash equilibria for games with a continuum of agents as limit points of equilibria of finite games. This allows us to discard those Nash equilibria that are artifacts of the continuum model, making limit equilibrium a natural equilibrium concept for games with a continuum of players.

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  • Guilherme Carmona, 2004. "Nash and limit equilibria of games with a continuum of players," Nova SBE Working Paper Series wp442, Universidade Nova de Lisboa, Nova School of Business and Economics.
    • Handle: RePEc:unl:unlfep:wp442
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    Cited by:

    1. Guilherme Carmona, 2003. "Symmetric Approximate Equilibrium Distributions with Finite Support," Game Theory and Information 0311006, University Library of Munich, Germany.
    2. Robin Nicole & Peter Sollich, 2018. "Dynamical selection of Nash equilibria using reinforcement learning: Emergence of heterogeneous mixed equilibria," PLOS ONE, Public Library of Science, vol. 13(7), pages 1-37, July.
    3. Robin Nicole & Peter Sollich, 2017. "Dynamical selection of Nash equilibria using Experience Weighted Attraction Learning: emergence of heterogeneous mixed equilibria," Papers 1706.09763, arXiv.org.
    4. Bodoh-Creed, Aaron, 2013. "Efficiency and information aggregation in large uniform-price auctions," Journal of Economic Theory, Elsevier, vol. 148(6), pages 2436-2466.
    5. Hafalir, Isa E. & Hakimov, Rustamdjan & Kübler, Dorothea & Kurino, Morimitsu, 2018. "College admissions with entrance exams: Centralized versus decentralized," Journal of Economic Theory, Elsevier, vol. 176(C), pages 886-934.
    6. Daniel Lacker & Kavita Ramanan, 2019. "Rare Nash Equilibria and the Price of Anarchy in Large Static Games," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 400-422, May.
    7. Aaron Bodoh-Creed & Brent Hickman, 2016. "College Assignment as a Large Contest," Working Papers 2016-27, Becker Friedman Institute for Research In Economics.
    8. Guilherme Carmona, 2004. "On the existence of equilibrium bank runs in a Diamond-Dybvig environment," Nova SBE Working Paper Series wp448, Universidade Nova de Lisboa, Nova School of Business and Economics.
    9. Bodoh-Creed, Aaron L. & Hickman, Brent R., 2018. "College assignment as a large contest," Journal of Economic Theory, Elsevier, vol. 175(C), pages 88-126.
    10. Yang, Jian, 2011. "Asymptotic interpretations for equilibria of nonatomic games," Journal of Mathematical Economics, Elsevier, vol. 47(4-5), pages 491-499.
    11. Carmona, Guilherme, 2007. "Bank failures caused by Large withdrawals: An explanation based purely on liquidity," Journal of Mathematical Economics, Elsevier, vol. 43(7-8), pages 818-841, September.

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    More about this item

    Keywords

    Nash equilibrium; limit equilibrium; noncooperative games; continuum of players;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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