State dependent expected utility for Savage's state space

This paper generalizes the Debreu/Gorman characterization of additively dec omposable functionals and separable preferences to infinite dimensions. The first novelty concerns the very definition of additively decomposable func tion as for infinite dimensions. For decision under uncertainty, our result provides a state-dependent extension of Savage's expected utility. A chara cterization in terms of preference conditions idensitifies the empirical co ntent of the models; it amounts to Savage's axiom system with P4 (likelihoo d ordering) dropped. Our approach does not require that a (probability) mea sure on the state space be given a priori, or can be derived from extraneou s conditions outside the realm of decision theory, Bayesian updating of new information is still possible, even though no prior probabilities are give n. The finding suggests that the sure- thing principle, rather than prior p robability, is at the heart of Bayesian updating.