Terminal Type Structures

Say a type structure is terminal if, for every hierarchy of beliefs satisfying a certain coherency condition, there is a type in that structure associated with that hierarchy of beliefs. When is a type structure terminal? This paper attempts to characterize terminal structures. In doing so, it relates terminal structures to other concepts of “large type structures” found in the literature. It goes on to apply these results to game theory. In particular, the characterization sheds light on the epistemic conditions for the iteratively undominated strategies. Many game theoretic analyses require specifying players’ hierarchies of beliefs. This is true of both the epistemic program and so-called games of incomplete information. Harsanyi [12, 1967-1968] introduced the concept of a type structure and argued that it is a model for hierarchies of beliefs. For a type structure to be such a model, it should be able to represent all hierarchies of beliefs. This raises the question: Does there exist a terminal type structure—a type structure that “contains all hierarchies of beliefs”? The question goes back to Böge-Eisele [4, 1979], who also used the phrase “terminal type structure” and constructed one such structure. (Section 6.a discusses the relationship between this question and the question of the existence of a “universal type structure.”) This paper asks: Which type structures are terminal? That is, can we characterize the class of terminal type structures? The paper also introduces the concept of a finitely terminal structure and asks about the characterization of these structures. In addressing these questions, the paper treats the relationship between terminal (resp. finitely terminal) structures and other notions of “large type structures” found in the literature. This paper owes much to joint research and many conversations with Adam Brandenburger. Indeed, I believe the paper would not have existed without Adam’s important input. I also thank the Olin School of Business for financial support. t e rm in a l 1 0 0 9 0 6 . r a p Olin School of Business, Washington University, 1 Brookings Drive, Campus Box 1133, St. Louis, MO 63130, [email protected].