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

For committee or multiwinner elections, the Chamberlin-Courant rule (CCR), which combines the Borda rule and the proportional representation, aims to pick the most representative committee (Chamberlin and Courant, 1983). Chamberlin and Courant (1983) have shown that if the size of the committee to be elected is k = 1 among m ≥ 3 candidates, the CCR is equivalent to the Borda rule ; Kamwa and Merlin (2014) claimed that if k = m − 1, the CCR is equivalent to the k-Plurality rule. In this paper, we explore what happens for 1 < k < m − 1 by computing the probability of agreement between the CCR and four k-scoring rules : k-Plurality, k-Borda, k-Negative Plurality and Bloc. Our results show that for committees of at least two members, the CCR usually leads to a committee recommended by the k-Plurality rule. Furthermore, we evaluate the probability of the CCR to select the Condorcet committee à la Gehrlein when it exists. The Condorcet committee à la Gehrlein is a fixed size subset of candidates such that every meber defeats every non-member in pairwise comparisons. In this matter, our results indicate that the CCR performs less well than the k-Borda rule and the Bloc rule but better than the k-Plurality and the k-Negative Plurality rules