Committee scoring rules form a rich class of aggregators of voters' preferences for the purpose of selecting subsets of objects with desired properties, e.g., a shortlist of candidates for an interview, a representative collective body such as a parliament, or a set of locations for a set of public facilities. In the spirit of celebrated Young's characterization result that axiomatizes single-winner scoring rules, we provide an axiomatic characterization of multiwinner committee scoring rules. We show that committee scoring rules---despite forming a remarkably general class of rules---are characterized by the set of four standard axioms, anonymity, neutrality, consistency and continuity, and by one axiom specific to multiwinner rules which we call committee dominance. In the course of our proof, we develop several new notions and techniques. In particular, we introduce and axiomatically characterize multiwinner decision scoring rules, a class of rules that broadly generalizes the well-known majority relation.
The state of research on axiomatic characterizations of multiwinner voting rules is far less advanced. Indeed, we are aware of only one unconditional characterization of a multiwinner rule: Debord has characterized the @math -Borda rule as the only rule that satisfies neutrality, faithfulness, consistency, and the cancellation property @cite_19 . Yet, there exists an interesting line of research, where the properties of multiwinner election rules are studied. A large bulk of this literature focuses on the principle of Condorcet consistency @cite_58 @cite_40 @cite_23 @cite_64 , and on approval-based multiwinner rules @cite_0 @cite_59 @cite_65 @cite_44 . Properties of other types of multiwinner election rules have been studied by Felsenthal and Maoz @cite_49 , Elkind et al. , and---in a somewhat different context---Skowron @cite_26 .
@cite_64 @cite_26 @cite_65 @cite_0 @cite_19 @cite_40 @cite_44 @cite_23 @cite_59 @cite_49 @cite_58
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This paper is an axiomatic study of consistent approval-based multi-winner rules, i.e., voting rules that select a fixed-size group of candidates based on approval ballots. We introduce the class of counting rules, provide an axiomatic characterization of this class and, in particular, show that counting rules are consistent. Building upon this result, we axiomatically characterize three important consistent multi-winner rules: Proportional Approval Voting, Multi-Winner Approval Voting and the Approval Chamberlin--Courant rule. Our results demonstrate the variety of multi-winner rules and illustrate three different, orthogonal principles that multi-winner voting rules may represent: individual excellence, diversity, and proportionality. The goal of this paper is to propose and study properties of multiwinner voting rules which can be consider as generalisations of single-winner scoring voting rules. We consider SNTV, Bloc, k-Borda, STV, and several variants of Chamberlin--Courant's and Monroe's rules and their approximations. We identify two broad natural classes of multiwinner score-based rules, and show that many of the existing rules can be captured by one or both of these approaches. We then formulate a number of desirable properties of multiwinner rules, and evaluate the rules we consider with respect to these properties. In computational social choice, one important problem is to take the votes of a subelectorate (subset of the voters), and summarize them using a small number of bits. This needs to be done in such a way that, if all that we know is the summary, as well as the votes of voters outside the subelectorate, we can conclude which of the m alternatives wins. This corresponds to the notion of compilation complexity, the minimum number of bits required to summarize the votes for a particular rule, which was introduced by [IJCAI-09]. We study three different types of compilation complexity. The first, studied by , depends on the size of the subelectorate but not on the size of the complement (the voters outside the subelectorate). The second depends on the size of the complement but not on the size of the subelectorate. The third depends on both. We first investigate the relations among the three types of compilation complexity. Then, we give upper and lower bounds on all three types of compilation complexity for the most prominent voting rules. We show that for l-approval (when l ≤ m 2), Borda, and Bucklin, the bounds for all three types are asymptotically tight, up to a multiplicative constant; for l-approval (when l > m 2), plurality with runoff, all Condorcet consistent rules that are based on unweighted majority graphs (including Copeland and voting trees), and all Condorcet consistent rules that are based on the order of pairwise elections (including ranked pairs and maximin), the bounds for all three types are asymptotically tight up to a multiplicative constant when the sizes of the subelectorate and its complement are both larger than m1+∊ for some ∊ > 0. In this paper we study several monotonicity axioms in approval-based multi-winner voting rules. We consider monotonicity with respect to the support received by the winners and also monotonicity in the size of the committee. Monotonicity with respect to the support is studied when the set of voters does not change and when new voters enter the election. For each of these two cases we consider a strong and a weak version of the axiom. We observe certain incompatibilities between the monotonicity axioms and well-known representation axioms (extended proportional justified representation) for the voting rules that we analyze and provide formal proofs of incompatibility between some of these axioms and perfect representation.