An empirical comparison of the expressiveness of the additive value function and the Choquet integral models for representing rankings

Recent - and less recent - work has been devoted to learning additive value functions or a Choquet capacity to represent the preference of a decision maker on a set of alternatives described by their performance on the relevant attributes. In this work we compare the ability of related models to represent rankings of such alternatives. Our experiments are designed as follows. We generate a number of alternatives by drawing at random a vector of evaluations for each of them. We then draw a random order on these alternatives and we examine whether this order is representable by a simple weighted sum, a Choquet integral with respect to a 2- or 3-additive capacity, an additive value function in general or a piecewise-linear additive value function with 2 or 3 pieces. This analysis is performed using linear programming and the Kappalab R package. The results explore how representability depends on varying the numbers of alternatives and criteria.