A comparison of the GAI model and the Choquet integral with respect to a k-ary capacity

Two utility models are classically used to represent interaction among criteria: the Choquet integral and the Generalized Additive Independence (GAI) model. We propose a comparison of these models. Looking at their mathematical expression, it seems that the second one is much more general than the first one. The GAI model has been mostly studied in the case where attributes are discrete. We propose an extension of the GAI model to continuous attributes, using the multi-linear interpolation. The values that are interpolated can in fact be interpreted as a k-ary capacity, or its extension – called p-ary capacity – where p is a vector and pi is the number of levels attached to criterion i. In order to push the comparison further, the Choquet integral with respect to a p-ary capacity is generalized to preferences that are not necessarily monotonically increasing or decreasing on the attributes. Then the Choquet integral with respect to a p-ary capacity differs from a GAI model only by the type of interpolation model. The Choquet integral is the Lovász extension of a p-ary capacity whereas the GAI model is the multi-linear extension of a p-ary capacity.