Finitely additive beliefs and universal type spaces

Details Finitely additive beliefs and universal type spaces Finitely additive beliefs and universal type spaces

2006-02-28

arxiv.org/abs/math/0602656v1

Annals of Probability 2006, Vol. 34, No. 1, 386-422

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117905000000576 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117905000000576

The probabilistic type spaces in the sense of Harsanyi [Management Sci. 14 (1967/68) 159--182, 320--334, 486--502] are the prevalent models used to describe interactive uncertainty. In this paper we examine the existence of a universal 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 ($\kappa$-measurability, for some fixed regular cardinal $\kappa$), there is a universal type space (i.e., a terminal object) to which every type space can be mapped in a unique beliefs-preserving way. However, by a probabilistic adaption of the elegant sober-drunk example of Heifetz and Samet [Games Econom. Behav. 22 (1998) 260--273] we show that if all subsets of the spaces are required to be measurable, then there is no universal type space.

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