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 Z u (f) dp+ c (p) min p Z 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 themultiple 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 …nance. An extended version of this paper was previously circulated with the title “Variational Representation of Preferences under Ambiguity” ICER Working Paper 5/2004, March 2004. We thank Erio Castagnoli, Rose-Anne Dana, Larry Epstein, Peter Klibano¤, Bart Lipman, Mark Machina, Jianjun Miao, Sujoy Mukerji, Emre Ozdenoren, and, especially, Andy Postlewaite and four ananymous referees for helpful discussions and suggestions. We also thank several seminar audiences. Part of this research was done while the …rst two authors were visiting the Department of Economics of Boston University, which they thank for its hospitality. They also gratefully acknowledge the …nancial support of the Ministero dell’Istruzione, dell’Università e della Ricerca. Rustichini gratefully acknowledges the …nancial support of the National Science Foundation (grant # SES-0452477).