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Bayesian Nash equilibrium and variational inequalities

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  • Ui, Takashi

Abstract

This paper provides a sufficient condition for the existence and uniqueness of a Bayesian Nash equilibrium by regarding it as a solution of a variational inequality. The payoff gradient of a game is defined as a vector whose component is a partial derivative of each player’s payoff function with respect to the player’s own action. If the Jacobian matrix of the payoff gradient is negative definite for each state, then a Bayesian Nash equilibrium is unique. This result unifies and generalizes the uniqueness of an equilibrium in a complete information game by Rosen (1965) and that in a team by Radner (1962). In a Bayesian game played on a network, the Jacobian matrix of the payoff gradient coincides with the weighted adjacency matrix of the underlying graph.

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  • Ui, Takashi, 2016. "Bayesian Nash equilibrium and variational inequalities," Journal of Mathematical Economics, Elsevier, vol. 63(C), pages 139-146.
  • Handle: RePEc:eee:mateco:v:63:y:2016:i:c:p:139-146
    DOI: 10.1016/j.jmateco.2016.02.004
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    Cited by:

    1. Xu, Jin & Zenou, Yves & Zhou, Junjie, 2019. "Networks in Conflict: A Variational Inequality Approach," CEPR Discussion Papers 13647, C.E.P.R. Discussion Papers.
    2. Christian Ewerhart & Federico Quartieri, 2020. "Unique equilibrium in contests with incomplete information," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 70(1), pages 243-271, July.
    3. Ui, Takashi, 2019. "The Lucas Imperfect Information Model with Imperfect Common Knowledge," Discussion Papers 2019-04, Graduate School of Economics, Hitotsubashi University.
    4. Leister, C. Matthew, 2020. "Information acquisition and welfare in network games," Games and Economic Behavior, Elsevier, vol. 122(C), pages 453-475.
    5. Ceparano, Maria Carmela & Quartieri, Federico, 2017. "Nash equilibrium uniqueness in nice games with isotone best replies," Journal of Mathematical Economics, Elsevier, vol. 70(C), pages 154-165.
    6. Parise, Francesca & Ozdaglar, Asuman, 2019. "A variational inequality framework for network games: Existence, uniqueness, convergence and sensitivity analysis," Games and Economic Behavior, Elsevier, vol. 114(C), pages 47-82.
    7. Takashi Ui, 2020. "LQG Information Design," Working Papers on Central Bank Communication 018, University of Tokyo, Graduate School of Economics.
    8. Emerson Melo, 2018. "A Variational Approach to Network Games," Working Papers 2018.05, Fondazione Eni Enrico Mattei.
    9. Ori Haimanko, 2021. "Bayesian Nash equilibrium existence in (almost continuous) contests," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 71(3), pages 1231-1258, April.
    10. Takashi Ui, 2019. "The Lucas Imperfect Information Model with Imperfect Common Knowledge," Working Papers on Central Bank Communication 007, University of Tokyo, Graduate School of Economics.
    11. Takashi Ui, 2020. "The Lucas imperfect information model with imperfect common knowledge," The Japanese Economic Review, Springer, vol. 71(1), pages 85-100, January.
    12. P'eter Bayer & Gyorgy Kozics & N'ora Gabriella SzH{o}ke, 2021. "Best-response dynamics in directed network games," Papers 2101.03863, arXiv.org.

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