On the existence of monotone pure-strategy perfect Bayesian equilibrium in games with complementarities

Many important economic situations, such as auctions and signaling games, can be modeled as dynamic games of incomplete information with strategic complementarities of actions and types. In this paper, we extend the results of Athey (2001) and Reny (2011) from static Bayesian games to dynamic environments with observable actions, providing conditions that guarantee the existence of monotone equilibria in types in such games. A feature that distinguishes this environment from those of previous results is the endogeneity of beliefs, which can complicate continuity of payoffs, needed to find a fixed point. To address this, we perturb the strategies of the game, which pins down beliefs while preserving continuity of payoffs. We then provide conditions which guarantee that there will exist monotone best-replies to monotone strategies of one's opponents in a dynamic environment, enabling verification of existence by merely looking at the primitives of the model.