Finitely additive beliefs and universal type spaces. (English) Zbl 1161.91009

Summary: The probabilistic type spaces in the sense of J. C. Harsanyi [Manage. Sci., Theory 14, 159–182 (1967; Zbl 0207.51102), 320–334 (1968; Zbl 0177.48402), 486–502 (1968; Zbl 0177.48501)] 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 A. Heifetz and D. Samet [Games Econ. Behav. 22, No. 2, 260–273 (1998; Zbl 0895.90201)] we show that if all subsets of the spaces are required to be measurable, then there is no universal type space.


91A40 Other game-theoretic models
91A35 Decision theory for games
28A10 Real- or complex-valued set functions
60A05 Axioms; other general questions in probability
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