State Dependent Expected Utility for Savage's State Space

This paper provides a state-dependent extension of Savage's expected utility when outcomes are real-valued (money, distance, etc.) and utility is increasing (or, equivalently, the "loss function" is decreasing). The first novelty concerns the very definition of the functional, which is not an integral. The existing results in the literature always invoke restrictive assumptions to reduce the functional to an integral, mostly by adding empirical primitives outside the realm of decision theory to allow for the identification of probability. A characterization in terms of preference conditions identifies the empirical content of our model; it amounts to a characterization of Savage's axiom system when the likelihood ordering axiom P4 is dropped. Bayesian updating of new information is still possible even while no prior probabilities are specified, suggesting that the sure-thing principle is at the heart of Bayesian updating. Prior probabilities simplify Bayesian updating, but are not essential.