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 = X1 × … × Xn. In MCDA elements of X are interpreted as alternatives, characterized by criteria taking values from the sets Xi. Previous axiomatizations of the Choquet integral have been given for particular cases X = Yn and X = ℝn. 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 Xi and Xj. 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.