A savage-style utility theory for belief functions

Abstract

In this paper, we provide an axiomatic justification for decision making with belief functions by studying the belief-function counterpart of Savage's Theorem where the state space is finite and the consequence set is a continuum [l, M](l < M). We propose six axioms for a preference relation over acts, and then show that this axiomatization admits a definition of qualitative belief functions comparing preferences over events that guarantees the existence of a belief function on the state space. The key axioms are uniformity and an analogue of the independence axiom. The uniformity axiom is used to ensure that all acts with the same maximal and minimal consequences must be equivalent. And our independence axiom shows the existence of a utility function and implies the uniqueness of the belief function on the state space. Moreover, we prove without the independence axiom the neutrality theorem that two acts are indifferent whenever they generate the same belief functions over consequences. At the end of the paper, we compare our approach with other related decision theories for belief functions.