Choice theory when agents can randomize

This paper takes choice theory to risk or uncertainty. Well-known decision models are axiomatized under the premise that agents can randomize. Under a reversal of order assumption, this convexifies choice sets, and even after imposing the weak axiom of revealed preference and nonemptiness of choice correspondences, the preferences directly revealed by choice may be incomplete or cyclical.Choice correspondence characterizations of (von Neumann–Morgenstern) expected utility, (Anscombe–Aumann) subjective expected utility, (Gilboa–Schmeidler) maxmin expected utility, (Schmeidler) Choquet expected utility, (Wald–Milnor) maximin utility, and (Bewley) multiple prior preferences can be established nonetheless by assuming nonempty choice, the weak axiom of revealed preference or weakenings thereof (but avoiding the strong axiom throughout), and choice correspondence analogs of relevant preference axioms. Two salient applications are to games, where agents' ability to randomize is usually presumed, and to statistical decision theory, where agents (i.e., statisticians) randomize in reality.