An Alternative Subjective Expected Utility Representation Theorem

A decision maker (DM) is asked to make choices from a set of acts, which entail both risk and uncertainty in the sense of knight (1921). Extending Raiffa's (1961) argument I show that, provided the DM can choose acts objectively randomly (by flipping her own fair coin, for instance), provided the DM's preferences over objective lotteries satisfy the von Neumann and Morgenstern (1944) axioms, and provided the DM's preferences over the space of all random acts satisfy Anscombe and Aumann's (1963) axioms of reversal of order and dominance, then any undominated random act (that is any potential choice the DM can make) can also be rationalized by the DM maximizing her subjective expected utility for some subjective belief