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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 (Econometrica 33: 520, 1965) and that in a team by Radner (Ann. Math. Stat. 33: 857, 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.

Suggested Citation

  • UI, Takashi, 2015. "Bayesian Nash Equilibrium and Variational Inequalities," Discussion Papers 2015-08, Graduate School of Economics, Hitotsubashi University.
  • Handle: RePEc:hit:econdp:2015-08
    Note: First Version: November 2004, This Version: October 2015
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    File URL: http://hermes-ir.lib.hit-u.ac.jp/rs/bitstream/10086/27485/1/070econDP15-08.pdf
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    References listed on IDEAS

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    More about this item

    Keywords

    Bayesian game; network game; potential game; team; variational inequality; payoff gradient; strict monotonicity;

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D82 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Asymmetric and Private Information; Mechanism Design

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