Archimedean Choice Functions: an Axiomatic Foundation for Imprecise Decision Making

TL;DR

This paper establishes an axiomatic foundation for decision making under imprecise probabilities using choice functions, focusing on E-admissibility and maximality, with a representation theorem based on Archimedean choice functions.

Contribution

It provides necessary and sufficient axioms characterizing E-admissibility and maximality as choice functions, advancing the theoretical understanding of imprecise decision making.

Findings

Axiomatic characterization of E-admissibility and maximality.

Representation theorem for Archimedean choice functions.

Foundation for decision making with sets of probabilities.

Abstract

If uncertainty is modelled by a probability measure, decisions are typically made by choosing the option with the highest expected utility. If an imprecise probability model is used instead, this decision rule can be generalised in several ways. We here focus on two such generalisations that apply to sets of probability measures: E-admissibility and maximality. Both of them can be regarded as special instances of so-called choice functions, a very general mathematical framework for decision making. For each of these two decision rules, we provide a set of necessary and sufficient conditions on choice functions that uniquely characterises this rule, thereby providing an axiomatic foundation for imprecise decision making with sets of probabilities. A representation theorem for Archimedean choice functions in terms of coherent lower previsions lies at the basis of both results.