Isotone equilibrium in games of incomplete information

Abstract

An isotone pure strategy equilibrium exists in any game of incomplete information in which each player's action set is infinite sublattice of multidimensional Euclidean space, types are multidimensional and atomless, and each player's interim expected payoff function satisfies two "nonprimitive conditions" whenever others adopt isotone pure strategies: (i) single-crossing in own action and type and (ii) quasi-supermodularity in own action. Conditions (i), (ii) are satisfied in supermodular and log-supermodular games given affiliated types, and in games with independent types in which each player's ex post payoff satisfies supermodularity in own action and nondecreasing differences in own action and type. This result is applied to provide the first proof of pure strategy equilibrium existence in the uniform price auction when bidders have multi-unit demand, nonprivate values, and independent types.