Bayesian Learning in Normal Form Games

Abstract

This paper studies the asymptotic behavior of Bayesian learning processes for general finite-player, finite-strategy normal form games. Initially, each player is presumed to know his own payoff function but not the payoff functions of the other players. Strategies are initially determined as a Bayesian Nash equilibrium of the incomplete information game in which each player's private characteristic is his payoff function. These strategies are then observed by all players, causing a revision of beliefs. The new beliefs determine a new Bayesian Nash equilibrium and so on. Assuming that the common prior distribution of payoff functions satisfies independence across players, it is proved that the conditional distributions on strategies converge to the set of Nash equilibria with probability one. Under a further assumption that the prior distributions are sufficiently uniform, convergence to the set of Nash equilibria is proved for every profile of payoff functions, that is, for every normal form game.