State Dependent Expected Utility for Savage's State Space

Abstract

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