Ambiguity aversion, robustness, and the variational representation of preferences

We characterize, in the Anscombe-Aumann framework, the preferences for which there are a utility function u on outcomes and an ambiguity index c on the set of probabilities on the states of the world such that, for all acts f and g, f ≳ g ⇔ min_p(∫ u(f)dp + c(p) ≥ min _p(∫ u(g)dp + c(p)). The function u represents the decision maker's risk attitudes, while the index c captures his ambiguity attitudes. These preferences include the multiple priors preferences of Gilboa and Schmeidler and the multiplier preferences of Hansen and Sargent. This provides a rigorous decision-theoretic foundation for the latter model, which has been widely used in macroeconomics and finance.