Existence of Equilibrium in Auctions and Discontinuous Bayesian Games: Endogenous and Incentive Compatibility Sharing Rules

We consider discontinuous games with incomplete information. Auctions are a leading example. With standard tie breaking rules (or more generally, sharing rules), these games may not have equilibria. We consider sharing rules that depend on the private information of players. We show that there exists an equilibrium of an augmented game with an incentive compatible sharing rule in which players reveal their private information for the purpose of determining sharing. We also show that for a large class of private value auctions, ties never occur in the equilibrium of the augmented game. This establishes existence of equilibria in such auctions with standard tie breaking rules.