When do type structures contain all hierarchies of beliefs?

Hierarchies of beliefs play a central role in game theory. In practice, they are modeled by type structures. This allows the analyst to describe hierarchies without explicitly specifying an infinite sequence of beliefs. The focus of this paper is type structures that contain all hierarchies of beliefs. Can the analyst identify these structures without explicit reference to hierarchies? That is, does there exist a test, defined on the type structure alone, so that the structure passes this test only if it contains all hierarchies of beliefs? This paper investigates one such test. The test is based on the concept of completeness [Brandenburger, A., 2003. On the existence of a 'complete' possibility structure. In: Basili, M., Dimitri, N., Gilboa, I. (Eds.), Cognitive Processes and Economic Behavior. Routledge, pp. 30-34], a concept that has played an important role in epistemic game theory.