An Interval-Valued Utility Theory for Decision Making with Dempster-Shafer Belief Functions

TL;DR

This paper develops an axiomatic utility theory for decision making with Dempster-Shafer belief functions, resulting in interval-valued utilities and a partial preference order, extending classical probabilistic utility theory.

Contribution

It introduces an axiomatic framework for belief function lotteries that yields interval-valued utilities, generalizing existing theories and incorporating ambiguity attitudes.

Findings

Representation theorem aligns with Jaffray's for Bayesian belief functions.

Illustrated with literature examples demonstrating the theory.

Proposed a simple utility assessment model using an interval-valued pessimism index.

Abstract

The main goal of this paper is to describe an axiomatic utility theory for Dempster-Shafer belief function lotteries. The axiomatic framework used is analogous to von Neumann-Morgenstern's utility theory for probabilistic lotteries as described by Luce and Raiffa. Unlike the probabilistic case, our axiomatic framework leads to interval-valued utilities, and therefore, to a partial (incomplete) preference order on the set of all belief function lotteries. If the belief function reference lotteries we use are Bayesian belief functions, then our representation theorem coincides with Jaffray's representation theorem for his linear utility theory for belief functions. We illustrate our representation theorem using some examples discussed in the literature, and we propose a simple model for assessing utilities based on an interval-valued pessimism index representing a decision-maker's attitude to ambiguity and indeterminacy. Finally, we compare our decision theory with those proposed by Jaffray, Smets, Dubois et al., Giang and Shenoy, and Shafer.