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

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 $\text{E}$ 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, $\text{F}$. We assume that the decision maker: (1) has a rational behavior under complete ignorance, for decisions whose results only depend on events of $\text{E}$; (2) with {θ} known, maximizes his conditional expected utility for decisions whose results only depend on events of $\text{F}$; (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 $\text{W(f) = v[}\text{Inf}{}_{\mathrm{\theta \in \because }}\text{E}{}_{\mathrm{\theta }}\text{(u°f),}\text{Sup}{}_{\mathrm{\theta \in \because }}\text{E}{}_{\mathrm{\theta }}\text{(u°f)]}$, where u and v, respectively, characterize the decision maker's attitudes toward risk and toward complete ignorance.