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A Game with No Bayesian Approximate Equilibria

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  • Ziv Hellman

Abstract

Simon (2003) presented an example of a 3-player Bayesian games with no Bayesian equilibria but it has been an open question whether or not there are games with no approximate Bayesian equilibria. We present an example of a Bayesian game with two players, two actions and a continuum of states that possesses no approximate Bayesian equilibria, thus resolving the question. As a side benefit we also have for the first time an an example of a 2-player Bayesian game with no Bayesian equilibria and an example of a strategic-form game with no approximate Nash equilibria. The construction makes use of techniques developed in an example by Y. Levy of a discounted stochastic game with no stationary equilibria.

Suggested Citation

  • Ziv Hellman, 2012. "A Game with No Bayesian Approximate Equilibria," Discussion Paper Series dp615, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    • Handle: RePEc:huj:dispap:dp615
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    References listed on IDEAS

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    1. SCHMEIDLER, David, 1973. "Equilibrium points of nonatomic games," LIDAM Reprints CORE 146, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Yehuda (John) Levy, 2012. "A Discounted Stochastic Game with No Stationary Nash Equilibrium," Discussion Paper Series dp596r, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem, revised May 2012.
    3. Radner, Roy, 1980. "Collusive behavior in noncooperative epsilon-equilibria of oligopolies with long but finite lives," Journal of Economic Theory, Elsevier, vol. 22(2), pages 136-154, April.
    4. John C Harsanyi, 1997. "Games with incomplete information played by "bayesian" players," Levine's Working Paper Archive 1175, David K. Levine.
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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. 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.
    3. Bergemann, Dirk & Morris, Stephen & Takahashi, Satoru, 2017. "Interdependent preferences and strategic distinguishability," Journal of Economic Theory, Elsevier, vol. 168(C), pages 329-371.
    4. Wei He & Xiang Sun & Yeneng Sun & Yishu Zeng, 2021. "Characterization of equilibrium existence and purification in general Bayesian games," Papers 2106.08563, arXiv.org.
    5. 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.
    6. He, Wei & Sun, Yeneng, 2019. "Pure-strategy equilibria in Bayesian games," Journal of Economic Theory, Elsevier, vol. 180(C), pages 11-49.
    7. , & ,, 2017. "Bayesian games with a continuum of states," Theoretical Economics, Econometric Society, vol. 12(3), September.
    8. 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.
    9. 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.

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