Construction of a Choquet integral and the value functions without any commensurateness assumption in multi-criteria decision making

We consider a multi-criteria evaluation function U defined over a Cartesian product of attributes. We assume that U is written as the combination of an aggregation function and one value function over each attribute. The aggregation function is assumed to be a Choquet integral w.r.t. an unknown capacity. The problem we wish to address in this paper is the following one: if U is known, can we construct both the value functions and the capacity? The approaches that have been developed so far in the literature to answer this question in an analytical way assume some commensurateness hypothesis. We propose in this paper a method to construct the value functions and the capacity without any commensurateness assumption. Moreover, we show that the construction of the value functions is unique up to an affine transformation.