State-dependent utility and decision theory

This survey paper prepared for the Handbook of Utility Theory covers the axiomatic foundations of decision making under uncertainty when conditional preferences are allowed to be state dependent, leading to an expected state-dependent utility representation. In the context of games against nature, subjective probabilities are not identified from observable choices among games, suggesting recourse to hypothetical preferences. In the context of one-person games with moral hazard, a generalised representation theorem (proved in the Appendix) asserts the existence of a closed convex set O of probabilities in the state space. Choices among games, with associated unobservable strategies, correspond to expected state-dependent utility maximisation, with probabilities chosen in the set O. When that set is full-dimensional, it is uniquely identified. Otherwise, identification is partial, suggesting again recourse to hypothetical preferences, as per a revised version of "conditional expected utility theory". The extent to which the state-dependent utility is identified is the same as that prevailing for the probabilities. The paper includes a discussion of risk aversion with state-dependent preferences; it concludes with a brief survey of applications to the demand for life insurance (a context of games against nature) and for safety (a context of one-person games with moral hazard).