What is Bayesian game, Bayes–Nash equilibrium?

A Bayesian game (Harsanyi, 1967–68) adds to a game a type for each player, drawn from a common prior and known only to its owner; a strategy maps types to actions. A Bayes–Nash equilibrium is a strategy profile in which each type’s action maximises its expected payoff given its beliefs about the others’ types and their strategies.