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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We present a new model that describes the process of electing a group of representatives (e.g., a parliament) for a group of voters. In this model, called the voting committee model, the elected group of representatives runs a number of ballots to make final decisions regarding various issues. The satisfaction of voters comes from the final decisions made by the elected committee. Our results suggest that depending on a decision system used by the committee to make these final decisions, different multi-winner election rules are most suitable for electing the committee. Furthermore, we show that if we allow not only a committee, but also an election rule used to make final decisions, to depend on the voters' preferences, we can obtain an even better representation of the voters. Approval voting is a well-known voting procedure for single-winner elections. Voters approve of as many candidates as they like, and the candidate with the most approvals wins (Brams and Fishburn 1978, 1983, 2005). But Merrill and Nagel (1987) point out that there are many ways to aggregate approval votes to determine a winner, justifying a distinction between approval balloting, in which each voter submits a ballot that identifies the candidates the voter approves of, and approval voting, the procedure of ranking the candidates according to their total numbers of approvals. This paper presents an extension of Borda's choice function to k-choice function (the k-choice set may be understood as the set of equally best “elites” of k alternatives). Young's axiomatic conditions are then generalized to k-choice functions and a simple combinatorial proof is exposed for the axiomatic characterization of Borda's k-choice function. This article focuses on decision-making by voting in systems at the levels of the organization, the community, the society, and the international system. It examines the compatibility of four voting procedures (plurality, approval voting, Borda's count, and Hare's single transferable vote) with ten normative properties commonly used to evaluate the desirability of voting procedures by social choice theorists. The analysis alternately assumes that one candidate, as well as more than one candidate, must be elected under each of the procedures. It thus extends previous analyses of the same issues which were mainly restricted to situations where only a single candidate must be elected. We show that: (i) under all four procedures, if a property is violated when only a single candidate must be elected it is also violated when more than one candidate must be elected; (ii) all properties that are satisfied under the AV procedure when a single candidate must be elected are also satisfied when more than one candidate must be elected; (iii) at least one property that is satisfied under each of the remaining three procedures when only a single candidate must be elected, may be violated when the number of available slots is larger than one. Theoretical issues and practical implications of these findings are discussed. Barbera and Coelho (WP 264, CREA-Barcelona Economics, 2007) documented six screening rules associated with the rule of k names that are used by diferent institutions around the world. Here, we study whether these screening rules satisfy stability. A set is said to be a weak Condorcet set a la Gehrlein (Math Soc Sci 10:199–209) if no candidate in this set can be defeated by any candidate from outside the set on the basis of simple majority rule. We say that a screening rule is stable if it always selects a weak Condorcet set whenever such set exists. We show that all of the six procedures which are used in reality do violate stability if the voters do not act strategically. We then show that there are screening rules which satisfy stability. Finally, we provide two results that can explain the widespread use of unstable screening rules.