Variational Bewley preferences

This paper characterizes preference relations % not necessarily complete and transitive over Anscombe-Aumann acts f from a state space S to a convex set of consequences X. The main feature of this model are an a¢ ne utility function u on consequences, an ambiguity index on the set of priors over the state space, and a variational Bewley representation asserting that for all acts f and g f % g , Z u(f)dp+ (p) Z u(g)dp; for all p: Intuitively, the ambiguity index captures the degree of plausibility of each probabilistic scenario. Although the decision maker requires unanimity among all priors in order to consider an act better than the other, depending on the degree of plausibility she is willing to accept some loss in expected utility when comparing two acts according to that prior. The Bewley’s Knightian uncertainty model is the special case of our model where is f0;1g-valued in which every plausible probabilistic scenario p is accorded the same degree of con…dence. In general, our model allow more permissive representations than Bewley’s model, e.g., preferences with ambiguity index given by the relative entropic index. Finally, we obtain an interesting relationship between variational Bewley preferences and variational preferences based on the interplay of a pair of preferences satisfying two natural conditions, which can be viewed as a novel foundation for variational preferences. JEL Classi…cation: D81. KEYWORDS: Ambiguity, Knightian uncertainty, incomplete preferences, intransitive preferences, variational representations. I wish to thank Alain Chateauneuf, Fabio Maccheroni, Leandro Nascimento, Bob Nau, Luca Rigotti, and Roee Teper for comments and suggestions. I also thank seminar audiences at IMPA, UFMG, Duke University, Northwestern University, the Second SWET at Université de Paris 1, the First Workshop in Economic Theory at PUC-Rio, the FURXIII at IESE in Barcelona, the 9th SAET Conference at Ischia, the Workshop on Risk, Ambiguity, and Decisions in Honor of Daniel Ellsberg at IHS in Vienna. I also thank the …nancial support from CNPq-Brasil during the period as visiting professor at IMPA, the …nancial support from Cedeplar and Fapemig during my stay in UFMG, and the …nancial support from FrancoBrazilian Scienti…c Cooperation and CERMSEM Paris 1 for the hospitality during June-2009. E-mail address ([email protected]).