### Extensions of the Simpson voting rule to the committee selection setting

Daniela Bubboloni, Mostapha Diss, Michele Gori, Working paper GATE 2018-13

Committee selection rules are procedures selecting sets of candidates of a given size on the basis of the preferences of the voters. There are in the literature two natural extensions of the well-known single-winner Simpson voting rule to the multiwinner setting. The first method gives a ranking of candidates according to their minimum number of wins against the other candidates. Then, if a fixed number k of candidates are to be elected, the k best ranked candidates are chosen as the overall winners. The second method gives a ranking of committees according to the minimum number of wins of committee members against committee nonmembers. Accordingly, the committee of size k with the highest score is chosen as the winner. We propose an in-depth analysis of those committee selection rules, assessing and comparing them with respect to several desirable properties among which unanimity, fixed majority, non-imposition, stability, local stability, Condorcet consistency, some kinds of monotonicity, resolvability and consensus committee. We also investigate the probability that the two methods are resolute and suffer the reversal bias, the Condorcet loser paradox and the leaving member paradox. We compare the results obtained with the ones related to further well-known committee selection rules. The probability assumption on which our results are based is the widely used Impartial Anonymous Culture.

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

Mostapha Diss, Eric Kamwa, Abdelmonaim Tlidi, Working paper GATE 2018-12

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.