Isotone Equilibrium in Multi-Unit Auctions

The main result is that a Bayesian Nash equilibrium in isotone pure strategies exists in a class of multi-unit auctions that includes the uniform-price, discriminatory, and Vickrey auctions, given riskneutrality and independent atomless types. We begin by considering all games of incomplete information in which types are atomless, each player’s action set is a finite lattice, and payoff is quasisupermodular in his action and satisfies a strict single-crossing property in his action and type. Then there exists an equilibrium in isotone pure strategies. The key observation that allows us to apply this result to multi-unit auctions is that each bidder’s surplus is modular in his own bid whenever (i) his quantity is determined by market-clearing and (ii) his payment is modular in his bid. If bidders are risk-neutral, have independent atomless types, and tremble with positive probability in an auction satisfying property (i), furthermore, then each bidder’s payoff satisfies strict increasing differences in his bid and type. Thus, under these conditions an equilibrium in isotone pure strategies exists. Finally, a limit of these equilibria, as the probability of trembling goes to zero, is an isotone pure strategy equilibrium without trembling.