Ambiguity through confidence functions

Abstract

We characterize preference relations over bounded below Anscombe and Aumann's acts and give necessary and sufficient conditions that guarantee the existence of a utility function u on consequences, a confidence function φ{symbol} on the set of all probabilities over states of nature, and a positive threshold level of confidence α0 such that our preference relation has a functional representation J, where given an act f. J (f) = under(min , p ∈ Lα0 φ{symbol}) frac(1, φ{symbol} (p)) under(∫, S) u (f) d p . The level set Lα0 φ{symbol} : = {p : φ{symbol} (p) ≥ α0} reflects the priors held by the decision maker and the value φ{symbol} (p) captures the relevance of prior p for his decision. The combination of φ{symbol} and α0 may describe the decision maker's subjective assessment of available information. An important feature of our representation is the characterization of the maximal confidence function which allows us to obtain results on comparative ambiguity aversion and on special cases, namely the subjective expected utility, the Choquet expected utility with convex capacity, and the maxmin expected utility. © 2009 Elsevier B.V. All rights reserved.