Axiomatization of the Choquet integral for 2-dimensional heterogeneous product sets

TL;DR

This paper establishes an axiomatic foundation for the Choquet integral on a two-dimensional heterogeneous product set, enabling preference modeling without requiring comparability or commensurateness of criteria.

Contribution

It provides the first axiomatization of the Choquet integral for heterogeneous product sets without assuming criteria comparability or special structure.

Findings

Representation theorem for the Choquet integral on heterogeneous sets

Analysis of uniqueness properties of the model

Applications in decision analysis, utility theory, and social choice

Abstract

We prove a representation theorem for the Choquet integral model. The preference relation is defined on a two-dimensional heterogeneous product set $X = X_1 \times X_2$ where elements of $X_1$ and $X_2$ are not necessarily comparable with each other. However, making such comparisons in a meaningful way is necessary for the construction of the Choquet integral (and any rank-dependent model). We construct the representation, study its uniqueness properties, and look at applications in multicriteria decision analysis, state-dependent utility theory, and social choice. Previous axiomatizations of this model, developed for decision making under uncertainty, relied heavily on the notion of comonotocity and that of a "constant act". However, that requires $X$ to have a special structure, namely, all factors of this set must be identical. Our characterization does not assume commensurateness of criteria a priori, so defining comonotonicity becomes impossible.