A Non-utilitarian Discrete Choice Model for Preference Aggregation

Martin Durand, Fanny Pascual, Olivier Spanjaard

Abstract: We study in this paper a non-utilitarian discrete choice model for preference aggregation. Unlike the Plackett-Luce model, this model is not based on the assignment of utility values to alternatives, but on probabilities \(p_i\) to choose the best alternative (according to a ground truth ranking \(r^*\)) in a subset of i alternatives. We consider \(k\!-\!1\) parameters \(p_i\) (for \(i\!=\!2\) to k) in the model, where k is bounded by the number m of alternatives. We study the application of this model to voting, where we assume that the input is a set of choice functions provided by voters. If \(k\!=\!2\), our model amounts to the model used by Young [25] in his statistical analysis of Condorcet’s voting method, and a maximum likelihood ranking is a consensus ranking for the Kemeny rule [12]. If \(k\!>\!2\), we show that, under some restrictive assumptions about probabilities \(p_i\), the maximum likelihood ranking is a consensus ranking for the k-wise Kemeny rule [10]. In the general case, we provide a characterization result for the maximum likelihood ranking r and probabilities \(p_i\). We propose an exact and a heuristic algorithm to compute both ranking r and probabilities \(p_i\). Numerical tests are presented to assess the efficiency of these algorithms, and measure the model fitness on synthetic and real data.