On Some k -scoring Rules for Committee Elections: Agreement and Condorcet Principle

Abstract

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

Scoring rules, ballot truncation, and the truncation paradox

“…As we only deal with large electorates, we follow a procedure that was developed in Cervone et al (2005). This technique which is based on the computation of polytope volumes has recently been used in many research papers; see among others Diss andGehrlein (2015, 2012), Diss et al (2020Diss et al ( , 2018, El Ouafdi et al (2020), Gehrlein et al (2015), Kamwa (2019), Kamwa and Moyouwou (2021), Kamwa and Valognes (2017), Moyouwou and Tchantcho (2015). In the online supplementary material we report Propositions 8 to 10 which give the probability representations that we obtain from this technique.…”

Computations of volumes in five candidates elections

“…But the volume computation was already substantially improved by the descent algorithm described in 5 . Examples of Normaliz being used for voting theory computations by independent authors can be found in 6 , 7 and 8 . The purpose of this paper is to present precise probability computations in five candidates elections under the assumption of the Impartial Anonymous Culture (IAC).…”

Computations of volumes and Ehrhart series in four candidates elections

“…But the volume computation was already substantially improved by the descent algorithm described in [7]. Examples of Normaliz being used for voting theory computations by independent authors can be found in [3], [4] and [14].…”