Monotone Equilibria in Bayesian Games of Strategic Complementarities

For Bayesian games of strategic complementarities, we provide a constructive proof of the existence of a greatest and a least Bayes-Nash equilibrium - each one in strategies monotone in type - if the payoff to a player displays increasing differences in own action and the profile of types, and if the posteriors are increasing in type with respect to first-order stochastic dominance (e.g. if types are affiliated). The result holds for multidimensional action and type spaces and also for continuous and discrete type distributions. It uses an intermediate result on monotone comparative statics under uncertainty, which implies that the extremal equilibria increase when there is a first-order stochastic dominant shift in beliefs. We provide an application to strategic information revelation in games of voluntary disclosure.