UTA-poly and UTA-splines: additive value functions with polynomial marginals

Additive utility function models are widely used in multiple criteria decision analysis. In such models, a numerical value is associated to each alternative involved in the decision problem. It is computed by aggregating the scores of the alternative on the different criteria of the decision problem. The score of an alternative is determined by a marginal value function that evolves monotonically as a function of the performance of the alternative on this criterion. Determining the shape of the marginals is not easy for a decision maker. It is easier for him/her to make statements such as "alternativea is preferred tob". In order to help the decision maker, UTA disaggregation procedures use linear programming to approximate the marginals by piecewise linear functions based only on such statements. In this paper, we propose to infer polynomials and splines instead of piecewise linear functions for the marginals. In this aim, we use semidefinite programming instead of linear programming. We illustrate this new elicitation method and present some experimental results. Introduction The theory of value functions aims at assigning a number to each alternative in such a way that the decision maker's preference order on the alternatives is the same as the order on the numbers associated with the alternatives. The number or value associated to an alternative is a monotone function of its evaluations on the various relevant criteria. For preferences satisfying some additional properties (includingpreferential independence), the value of an alternative can be obtained as the sum of marginal value functions each depending only on a single criterion [20, Chapter 6]. These functions usually are monotone, i.e., marginal value functions either increase or decrease with the assessment of the alternative on the associated criterion. Many questioning protocols have been proposed aiming to elicit an additive value function [20, 9] through interactions with the decision maker (DM). These direct elicitation methods are time-consuming and require a substantial cognitive effort from the DM. Therefore, in certain cases, an indirect approach may prove fruitful. The latter consists inlearning an additive value model (or a set of such models) from a set of declared or observed preferences. Learning approaches have been proposed not only for inferring an additive value function that is used to rank all other alternatives.