Conjoint Axiomatization of the Choquet Integral for Heterogeneous Product Sets

Abstract

We propose an axiomatization of the Choquet integral model for the general case of a heterogeneous product set $X = X_1 \times \ldots \times X_n$. In MCDA elements of X are interpreted as alternatives, characterized by criteria taking values from the sets $X_i$. Previous axiomatizations of the Choquet integral have been given for particular cases $X = Y^n$ and $X = \mathbb {R}^n$. However, within multicriteria context such indenticalness, hence commensurateness, of criteria cannot be assumed a priori. This constitutes the major difference of this paper from the earlier axiomatizations. In particular, the notion of “comonotonicity” cannot be used in a heterogeneous structure, as there does not exist a “built-in” order between elements of sets $X_i$ and $X_j$. However, such an order is implied by the representation model. Our approach does not assume commensurateness of criteria. We construct the representation and study its uniqueness properties.