Finitely Additive Beliefs and Universal Type Spaces

In this paper we examine the existence of a universal (to be precise: terminal) type space when beliefs are described by finitely additive probability measures. We find that in the category of all type spaces that satisfy certain measurability conditions (κ-measurability, for some fixed regular cardinal κ), there is a universal type space (i.e. a terminal object, that is a type space to which every type space can be mapped in a unique beliefs-preserving way (the morphisms of our category, the so-called type morphisms)), while, by an probabilistic adaption of the elegant sober-drunk example of Heifetz and Samet (1998a), we show that if all subsets of the spaces are required to be measurable there is no universal type space. CORE, 34, Voie du Roman Pays, 1348 Louvain-la-Neuve, Belgium. E-mail: [email protected] The affiliation of the second author. 0 Helpful comments of Aviad Heifetz, Jean-Francois Mertens and Dov Samet are gratefully acknowledged. This text presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister’s Office, Science Policy Programming. The scientific responsibility is assumed by the authors.