Model and Predictive Uncertainty: A Foundation for Smooth Ambiguity Preferences

Abstract

Smooth ambiguity preferences (Klibanoff, Marinacci, and Mukerji (2005)) describe a decision maker who evaluates each act f according to the twofold expectation $V(f)={\int }_{\mathcal{P}}\varphi ({\int }_{\mathrm{\Omega }}u(f)\mathrm{d}p)\mathrm{d}\mu (p)$ defined by a utility function u, an ambiguity index ϕ, and a belief μ over a set $\mathcal{P}$ of probabilities. We provide an axiomatic foundation for the representation, taking as a primitive a preference over Anscombe–Aumann acts. We study a special case where $\mathcal{P}$ is a subjective statistical model that is point identified, that is, the decision maker believes that the true law $p\in \mathcal{P}$ can be recovered empirically. Our main axiom is a joint weakening of Savage's sure‐thing principle and Anscombe–Aumann's mixture independence. In addition, we show that the parameters of the representation can be uniquely recovered from preferences, thereby making operational the separation between ambiguity attitude and perception, a hallmark feature of the smooth ambiguity representation.