Variational representation of preferences under ambiguity.

In the classic Anscombe and Aumann decision setting, we give necessary and sufficient conditions that guarantee the existence of a utility function u on outcomes and an ambiguity index c on the set of all probabilities on the states of the world such that acts are ranked according to the criterion V(f)=min{E(u(f),p)+c(p)} where p ranges over all probability distributions and c is a non-negative convex funcion on the set of all probability distributions. The preferences we characterize include as special cases the multiple priors preferences of Gilboa and Schmeidler, the multiplier preferences of Hansen and Sargent, and the mean-variance preferences of Markowitz and Tobin. In this way we are able to provide a rigorous ambiguity perspective on the latter two models, which have been widely used in macroeconomics and finance.