Bayesian Nash equilibrium and variational inequalities

By Takashi Ui. 2016

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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.

Subfields: Economics and Econometrics, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/b748e2209f81ebe34c4c02a0ec5b7e00301c8ff4