Discussion Paper, Center for Mathematical Studies in Economics and Management Science 1584
Many important economic situations can be modelled as dynamic games of incomplete information with strategic complementarities of actions and types. Of special interest is the question of the existence of a perfect Bayesian equilibrium in which actions are monotonic in types. In this paper, we extend the results of Athey (2001) and Reny (2011) from static Bayesian games to dynamic environments, providing conditions that guarantee the existence of monotone equilibria. Specifically, we define a belief mapping which pins down beliefs over types at any subgame, thereby allowing for the translation of the dynamic game into a static one and an extension of previous results. Difficulties arise when attempting to extend to a continuum of actions due to belief entanglement, which does not occur in the static environment, making extensions to a continuum of actions possible only under stronger conditions. We also provide conditions which guarantee that there will exist monotone best-replies to monotone strategies of one's opponents in a dynamic environment. Applications are given to signalling games and stopping games such as auctions.