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A game with no Bayesian approximate equilibria

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  • Hellman, Ziv

Abstract

Harsányi [4] showed that Bayesian games over finite games of payoff uncertainty with finite sets of belief types always admit Bayesian equilibria. That still left the question of whether Bayesian games over finite games of payoff uncertainty with infinitely many types are guaranteed to have equilibria. Simon [7] presented an example of a Bayesian game with no measurable Bayesian equilibria, even though the underlying game of payoff uncertainty is finite. We present a new and shorter proof of Simon's result using a simpler Bayesian game that moreover does not even have measurable approximate equilibria. That game in turn is used as the basis for constructing another Bayesian game which has no Bayesian equilibria at all, even in non-measurable strategies, in a construction complementary to one appearing in Friedenberg and Meier [1].

Suggested Citation

  • Hellman, Ziv, 2014. "A game with no Bayesian approximate equilibria," Journal of Economic Theory, Elsevier, vol. 153(C), pages 138-151.
    • Handle: RePEc:eee:jetheo:v:153:y:2014:i:c:p:138-151
    DOI: 10.1016/j.jet.2014.06.005
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    References listed on IDEAS

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    1. , & ,, 2017. "Bayesian games with a continuum of states," Theoretical Economics, Econometric Society, vol. 12(3), September.
    2. Yehuda Levy, 2013. "Discounted Stochastic Games With No Stationary Nash Equilibrium: Two Examples," Econometrica, Econometric Society, vol. 81(5), pages 1973-2007, September.
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    Cited by:

    1. Yehuda John Levy, 2020. "On games without approximate equilibria," International Journal of Game Theory, Springer;Game Theory Society, vol. 49(4), pages 1125-1128, December.
    2. Bergemann, Dirk & Morris, Stephen & Takahashi, Satoru, 2017. "Interdependent preferences and strategic distinguishability," Journal of Economic Theory, Elsevier, vol. 168(C), pages 329-371.
    3. Einy, Ezra & Haimanko, Ori, 2020. "Equilibrium existence in games with a concave Bayesian potential," Games and Economic Behavior, Elsevier, vol. 123(C), pages 288-294.
    4. Dirk Bergemann & Stephen Morris & Rafael Veiel, 2024. "A Strategic Topology on Information Structures," Papers 2411.09149, arXiv.org.
    5. Levy, Yehuda John, 2024. "Bayesian equilibrium: From local to global," Journal of Mathematical Economics, Elsevier, vol. 113(C).
    6. Oriol Carbonell-Nicolau, 2021. "Perfect equilibria in games of incomplete information," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 71(4), pages 1591-1648, June.
    7. Ziv Hellman & Yehuda John Levy, 2020. "Equilibria Existence in Bayesian Games: Climbing the Countable Borel Equivalence Relation Hierarchy," Working Papers 2020-03, Bar-Ilan University, Department of Economics.
    8. Wei He & Xiang Sun & Yeneng Sun & Yishu Zeng, 2021. "Characterization of equilibrium existence and purification in general Bayesian games," Papers 2106.08563, arXiv.org.
    9. He, Wei & Sun, Yeneng, 2019. "Pure-strategy equilibria in Bayesian games," Journal of Economic Theory, Elsevier, vol. 180(C), pages 11-49.
    10. Ziv Hellman & Yehuda John Levy, 2022. "Equilibria Existence in Bayesian Games: Climbing the Countable Borel Equivalence Relation Hierarchy," Mathematics of Operations Research, INFORMS, vol. 47(1), pages 367-383, February.

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    More about this item

    Keywords

    Existence of equilibria; Bayesian games; Incomplete information; Approximate equilibria;
    All these keywords.

    JEL classification:

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

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