2015-10 , Graduate School of Economics, Hitotsubashi University
First Version: November 2004, This Version: October 2015 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.