Committee scoring voting rules are multiwinner analogues of positional scoring rules which constitute an important subclass of single-winner voting rules. We identify several natural subclasses of committee scoring rules, namely, weakly separable, representation-focused, top- @math -counting, OWA-based, and decomposable rules. We characterize SNTV, Bloc, and @math -Approval Chamberlin--Courant as the only nontrivial rules in pairwise intersections of these classes. We provide some axiomatic characterizations for these classes, where monotonicity properties appear to be especially useful. The class of decomposable rules is new to the literature. We show that it strictly contains the class of OWA-based rules and describe some of the applications of decomposable rules.
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@cite_82: The election methods introduced in 1894--1895 by Phragmen and Thiele, and their somewhat later versions for ordered (ranked) ballots, are discussed in detail. The paper includes definitions and examples and discussion of whether the methods satisfy some properties, including monotonicity, consistency and various proportionality criteria. The relation with STV is also discussed. The paper also contains historical information on the methods.
@cite_29: The Single Transferable Vote (STV) is an attractive way of achieving representation that is proportional in terms of whatever characteristics of candidates voters value. Increasingly sophisticated methods of implementing STV have been advanced to overcome identified limitations of earlier methods. But every refinement comes at a cost of increased difficulty of understanding the vote-counting algorithm and increased cost of undertaking the count. This paper uses votes from actual elections to provide evidence about the frequency with which the choice of a particular STV method affects the outcome, and about the type of difference that different methods make. The most sophisticated STV method is CPO-STV, the comparison of pairs of outcomes by STV. This method avoids sequential exclusions and therefore overcomes the limitations of previous methods, that a paucity of votes in the early stages of a count can lead to the exclusion of a candidate who is the consensus choice of voters whose preferred candidates will be excluded at later stages of the count.
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@cite_50: We generalize the idea of a Condorcet winner to committee elections to select a Condorcet committee of size m. As in the case of a Condorcet winner, the Condorcet committee need not exist. We adapt two methods to measure how far a set of m candidates is from being the Condorcet committee. In particular, we generalize a procedure proposed by Lewis Carroll for selecting the candidate that is closest to being the Condorcet winner to allow the selection of a committee. We also generalize Kemeny’s method, which gives a complete transitive ranking, to the selection of committees and show that it is closely related to the first method.We show that these methods lead to some surprising inconsistencies. For example, the committee of size k may be disjoint from the committee of size j or they may overlap in any manner, the committee arising from Carroll’s method may appear at any locations in the Kemeny ranking, and except for two highly restrictive cases, the members of the committee arising from Kemeny’s method may appear at any location in the Kemeny ranking.
@cite_71: This outstanding volume brings together major arguments on what constitutes the best electoral system, particulary on the relative merits of plurality and proportional representation.
@cite_15: This paper analyzes nonranked voting systems for electing an m-member committee from a slate of @math nominees. The analysis generalizes an approach used previously for the election of a single candidate. Each voting system allows a voter to vote for k nominees, where k is any integer in a subset k of @math . The rn nominees with the most votes comprise the elected committee.Independence assumptions for voters’ preferences between committees lead to a basic dominance theorem for voting strategies. This theorem generates several corollaries that permit comparisons among the nonranked voting systems. These comparisons suggest that system @math may be the most satisfactory system of the type considered.
@cite_58: An alternative is said to be a Condorcet winner of an election if it is preferred to any other alternative by a majority of voters. While this is a very attractive solution concept, many elections do not have a Condorcet winner. In this paper, we propose a set-valued relaxation of this concept, which we call a Condorcet winning set: such sets consist of alternatives that collectively dominate any other alternative. We also consider a more general version of this concept, where instead of domination by a majority of voters we require domination by a given fraction @math θ of voters; we refer to such sets as @math θ -winning sets. We explore social choice-theoretic and algorithmic aspects of these solution concepts, both theoretically and empirically. Copyright Springer-Verlag Berlin Heidelberg 2015
@cite_51: Abstract Recent studies have evaluated election procedures on their propensity to select committees that meet a Condorcet criterion. The Condorcet criterion has been defined to use majority agreement from voters' preferences to compare the selected committee to all other committees. This study uses a different definition of the Condorcet criterion as defined on committees. The focus of the new definition is on candidates. That is, we consider majority agreement on each candidate in the selected committee as compared to each candidate not in the selected committee. This new definition of the Condorcet criterion allows for the existence of majority cycles on candidates within the selected committee. However, no candidate in the non-selected group is able to defeat any candidate in the selected committee by majority rule. Of particular interest is the likelihood that a committee meeting this Condorcet criterion exists. Attention is also given to the likelihood that various simple voting procedures will select a committee meeting this Condorcet criterion when one does exist.
@cite_11: This paper establishes the computational complexity status for a problem of deciding on the quality of a committee. Starting with individual preferences over alternatives, we analyse when it can be determined efficiently if a given committee C satisfies a weak (resp. strong) Condorcet criterion–i.e., if C is at least as good as (resp. better than) every other committee in a pairwise majority comparison. Scoring functions used in classic voting rules are adapted for these comparisons. In particular, we draw the sharp separation line between computationally tractable and intractable instances with respect to different voting rules. Finally, we show that deciding if there exists a committee which satisfies the weak (resp. strong) Condorcet criterion is computationally hard.
Naturally, there exist many interesting multiwinner rules beyond the class of committee scoring rules. These include, for example, Single Transferable Vote (see, e.g., the work of Tideman and Richardson @cite_29 ), a number of rules based on the Condorcet principle @cite_63 @cite_60 @cite_11 @cite_58 @cite_15 @cite_56 @cite_51 @cite_50 @cite_73 , Monroe's rule @cite_76 , and different variants of the rule invented by Phragm 'en @cite_32 @cite_87 @cite_40 @cite_82 @cite_4 . For an overview of electoral systems used to select committees of representatives in practice, we refer the reader to the book of Lijphart and Grofman @cite_71 .