Decision making in a case of mixed uncertainty: A normative model

Abstract

We consider a case of uncertainty which is frequently met in various fields, e.g., in parametric statistics: Events {θ}, θ ∈ ∵, are members of family E on which the decision maker possesses no information at all; however, conditionally on the realization of {θ}, he is able to affix probabilities to all members of another family of events, F . We assume that the decision maker: (1) has a rational behavior under complete ignorance, for decisions whose results only depend on events of E ; (2) with {θ} known, maximizes his conditional expected utility for decisions whose results only depend on events of F ; (3) has (unconditional) preferences which are consistent with his conditional ones. These assumptions are shown to be sufficient to ensure an approximate representation of the decision maker's preference by a real-valued function W which has the form W(f) = v[ Inf θ∈∵ E θ(u∘f), Sup θ∈∵ E θ(u∘f)] , where u and v, respectively, characterize the decision maker's attitudes toward risk and toward complete ignorance.