Bayes and Hurwicz without Bernoulli

We provide a theory of decision under ambiguity that does not require expected utility maximization under risk. Instead, we require only that a decision maker be probabilistically sophisticated in evaluating a subcollection of acts. Three components determine the decision maker's ranking of acts: a prior, a map from ambiguous acts to equivalent risky lotteries, and a generalized notion of certainty equivalent. The prior is Bayesian, defined over the inverse image of acts for which the decision maker is probabilistically sophisticated. Ambiguity preferences are similar to Hurwicz, depending on an act's best- and worst-case interpretations. The generalized certainty equivalent may, but need not, come from a Bernoulli utility. The ability to combine appealing theories of risk and ambiguity at will has been sought after but missing from the literature, and our decomposition provides a promising way forward.