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

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.