From Type Spaces to Probability Frames and Back, via Language

TL;DR

This paper explores the deep connection between Harsanyi type spaces and probability frames, showing they are essentially equivalent when considering the underlying logical language, thus unifying two major frameworks for modeling interactive beliefs.

Contribution

It reveals that universal type spaces and canonical models are fundamentally the same when the background logical language is explicitly considered, providing a new perspective on their relationship.

Findings

Translation from type spaces to probability frames is straightforward.

Reverse translation depends on the background logical language.

Universal type spaces and canonical models are essentially the same construct.

Abstract

We investigate the connection between the two major mathematical frameworks for modeling interactive beliefs: Harsanyi type spaces and possible-worlds style probability frames. While translating the former into the latter is straightforward, we demonstrate that the reverse translation relies implicitly on a background logical language. Once this "language parameter" is made explicit, it reveals a close relationship between universal type spaces and canonical models: namely, that they are essentially the same construct. As the nature of a canonical model depends heavily on the background logic used to generate it, this work suggests a new view into a corresponding landscape of universal type spaces.