Interim Bayesian Nash equilibrium on universal type spaces for supermodular games

We prove the existence of a greatest and a least interim Bayesian Nash equilibrium for supermodular games of incomplete information. There are two main differences from the earlier proofs in Vives (1990) and Milgrom and Roberts (1990): (a) we use the interim formulation of a Bayesian game, in which each player’s beliefs are part of his or her type rather than being derived from a prior; (b) we use the interim formulation of a Bayesian Nash equilibrium, in which each player and every type (rather than almost every type) chooses a best response to the strategy profile of the other players. Given also the mild restrictions on the type spaces, we have a proof of interim Bayesian Nash equilibrium for universal type spaces (for the class of supermodular utilities), as constructed, for example, by Mertens and Zamir (1985). We also weaken restrictions on the set of actions.