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Projections and functions of Nash equilibria

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  • Yehuda John Levy

    (University of Oxford)

Abstract

We show that any non-empty compact semi-algebraic subset of mixed action profiles on a fixed player set can be represented as the projection of the set of equilibria of a game in which additional binary players have been added. Even stronger, we show that any semi-algebraic continuous function, or even any semi-algebraic upper-semicontinuous correspondence with non-empty convex values, from a bounded semi-algebraic set to the unit cube can be represented as the projection of an equilibrium correspondence of a game with binary players in which payoffs depend on parameters from the domain of the function or correspondence in a multi-affine way. Some extensions are also presented.

Suggested Citation

  • Yehuda John Levy, 2016. "Projections and functions of Nash equilibria," International Journal of Game Theory, Springer;Game Theory Society, vol. 45(1), pages 435-459, March.
    • Handle: RePEc:spr:jogath:v:45:y:2016:i:1:d:10.1007_s00182-015-0517-3
    DOI: 10.1007/s00182-015-0517-3
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    References listed on IDEAS

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    1. repec:dau:papers:123456789/2960 is not listed on IDEAS
    2. Yehuda Levy, 2013. "A Cantor Set of Games with No Shift-Homogeneous Equilibrium Selection," Mathematics of Operations Research, INFORMS, vol. 38(3), pages 492-503, August.
    3. Kohlberg, Elon & Mertens, Jean-Francois, 1986. "On the Strategic Stability of Equilibria," Econometrica, Econometric Society, vol. 54(5), pages 1003-1037, September.
    4. Lehrer, Ehud & Solan, Eilon & Viossat, Yannick, 2011. "Equilibrium payoffs of finite games," Journal of Mathematical Economics, Elsevier, vol. 47(1), pages 48-53, January.
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    Cited by:

    1. Levy, Yehuda John, 2022. "Uniformly supported approximate equilibria in families of games," Journal of Mathematical Economics, Elsevier, vol. 98(C).
    2. Balkenborg, Dieter & Vermeulen, Dries, 2019. "On the topology of the set of Nash equilibria," Games and Economic Behavior, Elsevier, vol. 118(C), pages 1-6.

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    More about this item

    Keywords

    Nash equilibrium; Structure theorem; Semialgebraic geometry;
    All these keywords.

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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