Multi-Winner Scoring Election Methods: Condorcet Consistency and Paradoxes

Abstract

Abstract:The goal of this paper is to propose a comparison of four multi-winner voting rules, k-Plurality, k-Negative Plurality, k-Borda, and Bloc, which can be considered as generalisations of well-known single-winner scoring rules. The first comparison is based on the Condorcet committee efficiency which is defined as the conditional probability that a given voting rule picks out the Condorcet committee, given that such a committee exists. The second comparison is based on the likelihood of two paradoxes of committee elections: The Prior Successor Paradox and the Leaving Member Paradox which occur when a member of an elected committee leaves. In doing so, using the well-known Impartial Anonymous Culture condition, we extend the results of Kamwa and Merlin (2015) in two directions. First, our paper is concerned with the probability of the paradoxes no matter the ranking of the leaving candidate. Second, we do not only focus on the occurrence of these paradoxes when one wishes to select a committee of size k = 2 out of m = 4 candidates but we consider more values of k and m. Keywords:Multi-winner voting rules, committee, Condorcet committee efficiency, paradoxes, probability. JEL codes:D71, D72 AbstractThe goal of this paper is to propose a comparison of four multi-winner voting rules, k−Plurality, k−Negative Plurality, k−Borda, and Bloc, which can be considered as generalisations of well-known single-winner scoring rules. The first comparison is based on the Condorcet committee efficiency which is defined as the conditional probability that a given voting rule picks out the Condorcet committee, given that such a committee exists. The second comparison is based on the likelihood of two paradoxes of committee elections: The Prior Successor Paradox and the Leaving Member Paradox which occur when a member of an elected committee leaves. In doing so, using the well-known Impartial Anonymous Culture condition, we extend the results of Kamwa and Merlin (2015) in two directions. First, our paper is concerned with the probability of the paradoxes no matter the ranking of the leaving candidate. Second, we do not only focus on the occurrence of these paradoxes when one wishes to select a committee of size k = 2 out of m = 4 candidates but we consider more values of k and m.

The Chamberlin-Courant Rule and the k-Scoring Rules: Agreement and Condorcet Committee Consistency

“…We present and dene some well-known k-scoring rules in Section 2. Without being exhaustive, the following papers deal with this approach: Bock et al (1998), Debord (1993), Diss and Doghmi (2016), Dummett (1984), Kamwa and Merlin (2015). It is important to mention that Elkind et al (2017), Skowron et al (2016) recently examined the properties of some voting rules in multi-winner context and proposed a set of natural properties (axioms) by which these voting rules can be examined.…”

“…A member of a committee is said to be dominated according to the Condorcet principle if there is an external candidate that is majority preferred to this member by more than half of the voters. In a recent paper, Diss and Doghmi (2016) have focused on the conditional probability for the k-Plurality rule, the k-Borda rule, the k-Negative Plurality rule and the Bloc rule to select the Condorcet committee à la Gehrlein when one exists: The Condorcet Committee Eciency (CCE). The Condorcet committee à la Gehrlein is a subset of k candidates such that each of its members defeats every outside candidate in pairwise majority comparisons (Gehrlein, 1985).…”

“…For further details on the Condorcet eciency in the single-winner framework, the reader may refer to the works of Diss andGehrlein (2015, 2012) and the recent books of Gehrlein andLepelley (2017, 2011), among others. The second contribution of this paper is to compute the Condorcet committee eciency of the CCR according to the same values of the pair (k, m) that have been considered in Diss and Doghmi (2016), i.e., m ∈ {4, 5, 6} and 1 ≤ k ≤ m − 1. Notice that we focus here on large electorates as in Diss and Doghmi (2016).…”

“…The second contribution of this paper is to compute the Condorcet committee eciency of the CCR according to the same values of the pair (k, m) that have been considered in Diss and Doghmi (2016), i.e., m ∈ {4, 5, 6} and 1 ≤ k ≤ m − 1. Notice that we focus here on large electorates as in Diss and Doghmi (2016). Again, the probability model on which our results are based is the Impartial Anonymous Culture assumption.…”

“…For this, our volumes are found with the use of the algorithm Convex which is a MAPLE package for convex geometry written by Franz (2016). This package works with the same general procedure that was implemented in Cervone et al (2005) and recently used in other studies, e.g., Diss and Doghmi (2016), Diss andGehrlein (2015, 2012), Gehrlein et al (2015) and Moyouwou and Tchantcho (2017).…”