Properties of multiwinner voting rules

arXiv (Cornell University), 2016

We characterize the class of committee scoring rules that satisfy the fixed-majority criterion. In some sense, the committee scoring rules in this class are multiwinner analogues of the singlewinner Plurality rule, which is uniquely characterized as the only single-winner scoring rule that satisfies the simple majority criterion. We define top-k-counting committee scoring rules and show that the fixed majority consistent rules are a subclass of the top-k-counting rules. We give necessary and sufficient conditions for a top-k-counting rule to satisfy the fixed-majority criterion. We find that, for most of the rules in our new class, the complexity of winner determination is high (that is, the problem of computing the winners is NP-hard), but we also show examples of rules with polynomial-time winner determination procedures. For some of the computationally hard rules, we provide either exact FPT algorithms or approximate polynomial-time algorithms.

2016

We present and advertise the class of committee scoring rules, recently introduced as multiwinner analogues of single-winner scoring rules. We present a hierarchy of committee scoring rules (which includes all previously studied subclasses of committee scoring rules, as well as two new subclasses) and discuss their axiomatic properties (while focusing on fixed-majority consistency and monotonicity) as well as their computational properties (with a special focus on positive results).

arXiv (Cornell University), 2017

We consider the approval-based model of elections, and undertake a computational study of voting rules which select committees whose size is not predetermined. While voting rules that output committees with a predetermined number of winning candidates are quite well studied, the study of elections with variable number of winners has only recently been initiated by Kilgour [18]. This paper aims at achieving a better understanding of these rules, their computational complexity, and on scenarios for which they might be applicable.

Social Choice and Welfare, 2018

We characterize the class of committee scoring rules that satisfy the fixedmajority criterion. We argue that rules in this class are multiwinner analogues of the single-winner Plurality rule, which is uniquely characterized as the only single-winner scoring rule that satisfies the simple majority criterion. We define top-k-counting committee scoring rules and show that the fixed-majority consistent rules are a subclass of the top-k-counting rules. We give necessary and sufficient conditions for a top-kcounting rule to satisfy the fixed-majority criterion. We show that, for many top-kcounting rules, the complexity of winner determination is high (formally, we show that the problem of deciding if there exists a committee with at least a given score is NPhard), but we also show examples of rules with polynomial-time winner determination procedures. For some of the computationally hard rules, we provide either exact FPT algorithms or approximate polynomial-time algorithms.

2014

Positional scoring rules in voting compute the score of an alternative by summing the scores for the alternative induced by every vote. This summation principle ensures that all votes contribute equally to the score of an alternative. We relax this assumption and, instead, aggregate scores by taking into account the rank of a score in the ordered list of scores obtained from the votes. This defines a new family of voting rules, rank-dependent scoring rules (RDSRs), based on ordered weighted average (OWA) operators, which, include all scoring rules, and many others, most of which of new. We study some properties of these rules, and show, empirically, that certain RDSRs are less manipulable than Borda voting, across a variety of statistical cultures.

Social Choice and Welfare, 2006

Different scoring rules can result in the selection of any of the k competing candidates, given the same preference profile, Saari (2001). It is also possible that a candidate, and even a Condorcet winning candidate, cannot be selected by any scoring rule, Saari (2000). These findings are balanced by Saari's result (1992) that specifies the necessary and sufficient condition for the selection of the same candidate by all scoring rules. This condition is, however, indirect. We provide a sufficient condition that is stated directly in terms of the preference profile and therefore its testability does not require the verdict of any voting rule.

2020

Suppose you want to design a voting rule that can be used to elect a committee or parliament by asking each voter to approve of a subset of the candidates standing. There are several properties you may want that rule to satisfy. First, voters should enjoy some form of proportional representation. Second, voters should not have an incentive to misrepresent their preferences. Third, outcomes should be Pareto efficient. We show that it is impossible to design a voting rule that satisfies all three properties. We also explore what possibilities there are when we weaken our requirements. Of special interest is the methodology we use, as a significant part of the proof is outsourced to a SAT solver. While prior work has considered similar questions for the special case of resolute voting rules, which do not allow for ties between outcomes, we focus on the fact that, in practice, most voting rules allow for the possibility of such ties.

Public Choice, 2016

The goal of this paper is to propose a comparison of four multi-winner voting rules, k-Plurality, k-Negative Plurality, k-Borda, and Bloc, which can be considered as generalisations of well-known single-winner scoring rules. The first comparison is based on the Condorcet committee efficiency which is defined as the conditional probability that a given voting rule picks out the Condorcet committee, given that such a committee exists. The second comparison is based on the likelihood of two paradoxes of committee elections: The Prior Successor Paradox and the Leaving Member Paradox which occur when a member of an elected committee leaves. In doing so, using the well-known Impartial Anonymous Culture condition, we extend the results of Kamwa and Merlin (2015) in two directions. First, our paper is concerned with the probability of the paradoxes no matter the ranking of the leaving candidate. Second, we do not only focus on the occurrence of these paradoxes when one wishes to select a committee of size k = 2 out of m = 4 candidates but we consider more values of k and m.

Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence, 2018

Multiwinner voting rules are used to select a small representative subset of candidates or items from a larger set given the preferences of voters. However, if candidates have sensitive attributes such as gender or ethnicity (when selecting a committee), or specified types such as political leaning (when selecting a subset of news items), an algorithm that chooses a subset by optimizing a multiwinner voting rule may be unbalanced in its selection -- it may under or over represent a particular gender or political orientation in the examples above. We introduce an algorithmic framework for multiwinner voting problems when there is an additional requirement that the selected subset should be ``fair'' with respect to a given set of attributes. Our framework provides the flexibility to (1) specify fairness with respect to multiple, non-disjoint attributes (e.g., ethnicity and gender) and (2) specify a score function. We study the computational complexity of this constrained multi...

Journal of Economic Theory, 2019

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.