Every hierarchy of beliefs is a type

TL;DR

This paper proves that in a purely measurable framework, every hierarchy of beliefs in game theory can be uniquely represented by an element of the complete universal type space, completing previous foundational results.

Contribution

It establishes that all hierarchies of beliefs are representable by unique types within the complete universal type space in a measurable framework.

Findings

Every hierarchy of beliefs corresponds to a unique element in the type space.

The universal type space is complete and can represent all belief hierarchies.

The results extend the foundational theory of types in game theory.

Abstract

When modeling game situations of incomplete information one usually considers the players' hierarchies of beliefs, a source of all sorts of complications. Hars\'anyi (1967-68)'s idea henceforth referred to as the "Hars\'anyi program" is that hierarchies of beliefs can be replaced by "types". The types constitute the "type space". In the purely measurable framework Heifetz and Samet (1998) formalize the concept of type spaces and prove the existence and the uniqueness of a universal type space. Meier (2001) shows that the purely measurable universal type space is complete, i.e., it is a consistent object. With the aim of adding the finishing touch to these results, we will prove in this paper that in the purely measurable framework every hierarchy of beliefs can be represented by a unique element of the complete universal type space.