Multi-utility representations of incomplete preferences induced by set-valued risk measures

TL;DR

This paper develops a unified framework for representing incomplete preferences induced by set-valued risk measures, emphasizing parsimonious and well-behaved multi-utility representations with a duality-based approach.

Contribution

It introduces a general dual representation that unifies existing results and connects set-valued and scalar risk measure dualities.

Findings

Unified dual representation for set-valued risk measures

Characterization of multi-utility representations for incomplete preferences

Enhanced understanding of the structure of preferences induced by risk measures

Abstract

We establish a variety of numerical representations of preference relations induced by set-valued risk measures. Because of the general incompleteness of such preferences, we have to deal with multi-utility representations. We look for representations that are both parsimonious (the family of representing functionals is indexed by a tractable set of parameters) and well behaved (the representing functionals satisfy nice regularity properties with respect to the structure of the underlying space of alternatives). The key to our results is a general dual representation of set-valued risk measures that unifies the existing dual representations in the literature and highlights their link with duality results for scalar risk measures.