We extend the recently introduced framework of metric distortion to multiwinner voting. In this framework, $n$ agents and $m$ alternatives are located in an underlying metric space. The exact distances between agents and alternatives are unknown. Instead, each agent provides a ranking of the alternatives, ordered from the closest to the farthest. Typically, the goal is to select a single alternative that approximately minimizes the total distance from the agents, and the worst-case approximation ratio is termed distortion. In the case of multiwinner voting, the goal is to select a committee of $k$ alternatives that (approximately) minimizes the total cost to all agents. We consider the scenario where the cost of an agent for a committee is her distance from the $q$-th closest alternative in the committee. We reveal a surprising trichotomy on the distortion of multiwinner voting rules in terms of $k$ and $q$: The distortion is unbounded when $q \leq k/3$, asymptotically linear in the number of agents when $k/3 < q \leq k/2$, and constant when $q > k/2$.
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…
The goal of this paper is twofold. First and foremost, we aim to experimentally and quantitatively show that the choice of a multiwinner voting rule can play a crucial role on the way minorities are represented. We also test the possibility for some of these rules to achieve proportional representation.
We show a prototype of a system that uses multiwinner voting to suggest resources (such as movies) related to a given query set (such as a movie that one enjoys). Depending on the voting rule used, the system can either provide resources very closely related to the query set or a broader spectrum of options. We show how this ability can be interpreted as a way of controlling the diversity of the results. We test our system both on synthetic data and on the real-life collection of movie ratings from the MovieLens dataset. We also present a visual comparison of the search results corresponding t…
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. This paper aims at achieving a better understanding of these rules, their computational complexity, and on scenarios for which they might be applicable.
We visualize aggregate outputs of popular multiwinner voting rules--SNTV, STV, Bloc, k-Borda, Monroe, Chamberlin--Courant, and HarmonicBorda--for elections generated according to the two-dimensional Euclidean model. We consider three applications of multiwinner voting, namely, parliamentary elections, portfolio/movie selection, and shortlisting, and use our results to understand which of our rules seem to be best suited for each application. In particular, we show that STV (one of the few nontrivial rules used in real high-stake elections) exhibits excellent performance, whereas the Bloc rule …
Aarhus University（奥胡斯大学）；Computer Technology Institute “Diophantus”（“丢番图”计算机技术研究所）；University of Patras（帕特雷大学）
Approval-based multiwinner voting rules have recently received much attention in the Computational Social Choice literature. Such rules aggregate approval ballots and determine a winning committee of alternatives. To assess effectiveness, we propose to employ new noise models that are specifically tailored for approval votes and committees. These models take as input a ground truth committee and return random approval votes to be thought of as noisy estimates of the ground truth. A minimum robustness requirement for an approval-based multiwinner voting rule is to return the ground truth when a…
Given a set of agents with approval preferences over each other, we study the task of finding $k$ matchings fairly representing everyone's preferences. We model the problem as an approval-based multiwinner election where the set of candidates consists of all possible matchings and agents' preferences over each other are lifted to preferences over matchings. Due to the exponential number of candidates in such elections, standard algorithms for classical sequential voting rules (such as those proposed by Thiele and Phragmén) are rendered inefficient. We show that the computational tractability o…
Fairness in multiwinner elections, a growing line of research in computational social choice, primarily concerns the use of constraints to ensure fairness. Recent work proposed a model to find a diverse \emph{and} representative committee and studied the model's computational aspects. However, the work gave complexity results under major assumptions on how the candidates and the voters are grouped. Here, we close this gap and classify the complexity of finding a diverse and representative committee using a monotone, separable positional multiwinner voting rule, conditioned \emph{only} on the a…
We investigate how robust the results of committee elections are to small changes in the input preference orders, depending on the voting rules used. We find that for typical rules the effect of making a single swap of adjacent candidates in a single preference order is either that (1) at most one committee member might be replaced, or (2) it is possible that the whole committee will be replaced. We also show that the problem of computing the smallest number of swaps that lead to changing the election outcome is typically NP-hard, but there are natural FPT algorithms. Finally, for a number of …
The Chamberlin-Courant and Monroe rules are fundamental and well-studied rules in the literature of multi-winner elections. The problem of determining if there exists a committee of size k that has a Chamberlin-Courant (respectively, Monroe) score of at most r is known to be NP-complete. We consider the following natural problems in this setting: a) given a committee S of size k as input, is it an optimal k-sized committee, and b) given a candidate c and a committee size k, does there exist an optimal k-sized committee that contains c? In this work, we resolve the complexity of both problems f…
In party-approval multiwinner elections the goal is to allocate the seats of a fixed-size committee to parties based on the approval ballots of the voters over the parties. In particular, each voter can approve multiple parties and each party can be assigned multiple seats. Two central requirements in this setting are proportional representation and strategyproofness. Intuitively, proportional representation requires that every sufficiently large group of voters with similar preferences is represented in the committee. Strategyproofness demands that no voter can benefit by misreporting her tru…
When selecting a subset of candidates (a so-called committee) based on the preferences of voters, proportional representation is often a major desideratum. When going beyond simplistic models such as party-list or district-based elections, it is surprisingly challenging to capture proportionality formally. As a consequence, the literature has produced numerous competing criteria of when a selected committee qualifies as proportional. Two of the most prominent notions are Dummett's proportionality for solid coalitions (PSC) and Aziz et al.'s extended justified representation (EJR). Both guarant…
Shortlisting of candidates--selecting a group of "best" candidates--is a special case of multiwinner elections. We provide the first in-depth study of the computational complexity of strategic voting for shortlisting based on the perhaps most basic voting rule in this scenario, l-Bloc (every voter approves l candidates). In particular, we investigate the influence of several tie-breaking mechanisms (e.g., pessimistic versus optimistic) and group evaluation functions (e.g., egalitarian versus utilitarian). Among other things, conclude that in an egalitarian setting strategic voting may indeed b…
We develop a model of multiwinner elections that combines performance-based measures of the quality of the committee (such as, e.g., Borda scores of the committee members) with diversity constraints. Specifically, we assume that the candidates have certain attributes (such as being a male or a female, being junior or senior, etc.) and the goal is to elect a committee that, on the one hand, has as high a score regarding a given performance measure, but that, on the other hand, meets certain requirements (e.g., of the form "at least $30\%$ of the committee members are junior candidates and at le…
We present an almost optimal algorithm for the classic Chamberlin-Courant multiwinner voting rule (CC) on single-peaked preference profiles. Given $n$ voters and $m$ candidates, it runs in almost linear time in the input size, improving the previous best $O(nm^2)$ time algorithm of Betzler et al. (2013). We also study multiwinner voting rules on nearly single-peaked preference profiles in terms of the candidate-deletion operation. We show a polynomial-time algorithm for CC where a given candidate-deletion set $D$ has logarithmic size. Actually, our algorithm runs in $2^{|D|} \cdot poly(n,m)$ t…
We study multiwinner elections with approval-based preferences. An instance of a multiwinner election consists of a set of alternatives, a population of voters---each voter approves a subset of alternatives, and the desired committee size $k$; the goal is to select a committee (a~subset) of $k$ alternatives according to the preferences of the voters. We investigate a number of election rules and ask whether the committees that they return represent the voters proportionally. In contrast to the classic literature, we employ quantitative techniques that allow to measure the extent to which the c…
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-$k$-counting, OWA-based, and decomposable rules. We characterize SNTV, Bloc, and $k$-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…
We consider the distributed single-winner metric voting problem on a line, where agents and alternative are represented by points on the line of real numbers, the agents are partitioned into disjoint districts, and the goal is to choose a single winning alternative in a decentralized manner. In particular, the choice is done by a distributed voting mechanism which first selects a representative alternative for each district of agents and then chooses one of these representatives as the winner. In this paper, we design simple distributed mechanisms that achieve distortion at most $2+\sqrt{5}$ f…
We consider a setting with agents that have preferences over alternatives and are partitioned into disjoint districts. The goal is to choose one alternative as the winner using a mechanism which first decides a representative alternative for each district based on a local election with the agents therein as participants, and then chooses one of the district representatives as the winner. Previous work showed bounds on the distortion of a specific class of deterministic plurality-based mechanisms depending on the available information about the preferences of the agents in the districts. In thi…
There has been much recent work on multiwinner voting systems. However, sometimes a committee is highly structured, and if we want to vote for such a committee, our voting method should be more structured as well. We consider committees consisting of representatives for disjoint parts of a collective; for instance, n departments of an organization might each have m possible representatives to choose from. However, in our model the entire population still gets to vote on the committee, and we need to respect that symmetry, which we do by building on earlier work of Barcelo et al. using represen…
The study of fairness in multiwinner elections focuses on settings where candidates have attributes. However, voters may also be divided into predefined populations under one or more attributes (e.g., "California" and "Illinois" populations under the "state" attribute), which may be same or different from candidate attributes. The models that focus on candidate attributes alone may systematically under-represent smaller voter populations. Hence, we develop a model, DiRe Committee WinnerDetermination (DRCWD), which delineates candidate and voter attributes to select a committee by specifying di…
We study the performance of voting mechanisms from a utilitarian standpoint, under the recently introduced framework of metric-distortion, offering new insights along three main lines. First, if $d$ represents the doubling dimension of the metric space, we show that the distortion of STV is $O(d \log \log m)$, where $m$ represents the number of candidates. For doubling metrics this implies an exponential improvement over the lower bound for general metrics, and as a special case it effectively answers a question left open by Skowron and Elkind (AAAI '17) regarding the distortion of STV under l…
