Hierarchies of Beliefs and Common Knowledge

A type of an individual is an infinite hierarchy of beliefs—over some state space S, over other individuals’ beliefs over S, and so on. We show that a coherent type determines a belief over S and other individuals’ types, and that common knowledge of coherency is needed to “close” such a model of beliefs. Belief-closed sets always satisfy common knowledge of coherency and the largest belieft-closed set (which satisfies only this restriction) is equivalent to the universal type space constructed in Mertens and Zamir [9]. In Aumann’s definition of common knowledge [1], the information structure is itself assumed to be common knowledge. We show that this assumption is without loss of generality if the underlying state space is expanded to include the type spaces, and common knowledge of coherency is satisfied.