On some k-scoring rules for committee elections: agreement and Condorcet Principle

Given a collection of individual preferences on a set of candidates and a desired number of winners, a multi-winner voting rule outputs groups of winners, which we call committees. In this paper, we consider five multi-winner voting rules widely studied in the literature of social choice theory: the k-Plurality rule, the k-Borda rule, the k-Negative Plurality rule, the Bloc rule, and the Chamberlin-Courant rule. The objective of this paper is to provide a comparison of these multi-winner voting rules according to many principles by taking into account a probabilistic approach using the well-known Impartial Anonymous Culture (IAC) assumption. We first evaluate the probability that each pair of the considered voting rules selects the same unique committee in order to identify which multi-winner rules are the most likely to agree for a given number of candidates and a fixed target size of the committee. In this matter, our results show that the Chamberlin-Courant rule and the k-Plurality rule on one side, and the k-Borda rule and the Bloc rule on the other side, are the pairs of rules that most agree in comparison to other pairs. Furthermore, we evaluate the probability of every multi-winner voting rule selecting the Condorcet committee à la Gehrlein when it exists. The Condorcet committee à la Gehrlein is a fixed-size committee such that every member defeats every non-member in pairwise comparisons. In addition, we compare the considered multi-winner voting rules according to their ability (susceptibility) to select a committee containing the Condorcet winner (loser) when one exists. Here, our results tell us that in general, the k-Borda rule has the highest performance amongst all the considered voting rules. Finally, we highlight that this paper is one of the very rare contributions in the literature giving exact results under the Impartial Anonymous Culture (IAC) condition for the case of four candidates.