Learning preference representations based on Choquet integrals for multicriteria decision making

This paper concerns preference elicitation and learning of decision models in the context of multicriteria decision making. We propose an approach to learn a representation of preferences by a non-additive multiattribute utility function, namely a Choquet or bi-Choquet integral. This preference model is parameterized by one-dimensional utility functions measuring the attractiveness of consequences w.r.t. various point of views and one or two set functions (capacities) used to weight the coalitions and control the intensity of interactions among criteria, on the positive and possibly the negative sides of the utility scale. Our aim is to show how we can successively learn marginal utilities from properly chosen preference examples and then learn where the interactions matter in the overall model. We first present a preference elicitation method to learn spline representations of marginal utilities on every component of the model. Then we propose a sparse learning approach based on adaptive \(L_1\)-regularization for determining a compact Möbius representation fitted to the observed preferences. We present numerical tests to compare different regularization methods. We also show the advantages of our approach compared to basic methods that do not seek sparsity or that force sparsity a priori by requiring k-additivity.