Statistical Decisions under Ambiguity

Consider a decision maker who faces a number of possible models of the world. Every model generates objective probabilities, but no probabilities of models are given. This is the classic setting of statistical decision theory; recent and less standard applications include decision making with model uncertainty, e.g. due to concerns for misspecification, treatment choice with partial identification, and robust Bayesian analysis. I characterize a number of decision rules including Bayesianism, maximin utility, the Hurwicz criterion, and especially several variations of minimax regret. The main contributions are the unified axiomatization of these rules in a framework tailored to statistical decision making, an axiomatic system that relaxes transitivity as well as menu-independence of preferences, and the introduction of new, regret-based decision criteria. Interestingly, the axiom that picks regret-based rules over maximin utility is independence.