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.
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Notes
- 1.
Note that this assumption allows the preferences to be cyclic.
- 2.
When the votes are viewed as noisy perceptions of a ground truth ranking \(r^*\), a noise model is the mathematical description of the probabilities of the votes based on \(r^*\).
- 3.
From now on, we use indifferently \(\overrightarrow{p}\) or \(\overrightarrow{\alpha }\), because one vector can be inferred from the other.
- 4.
All algorithms have been implemented in C++, and the tests have been carried out on an Intel Core I5-8250 1.6 GHz processor with 8 GB of RAM.
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We acknowledge a financial support from the project THEMIS ANR20-CE23-0018 of the French National Research Agency (ANR).
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Durand, M., Pascual, F., Spanjaard, O. (2022). A Non-utilitarian Discrete Choice Model for Preference Aggregation. In: Dupin de Saint-Cyr, F., Öztürk-Escoffier, M., Potyka, N. (eds) Scalable Uncertainty Management. SUM 2022. Lecture Notes in Computer Science(), vol 13562. Springer, Cham. https://doi.org/10.1007/978-3-031-18843-5_11
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