We study the committee selection problem in the canonical impartial culture model with a large number of voters and an even larger candidate set. Here, each voter independently reports a uniformly random preference order over the candidates. For a fixed committee size $k$, we ask when a committee can collectively beat every candidate outside the committee by a prescribed majority level $α$. We focus on two natural notions of collective dominance, $α$-winning and $α$-dominating sets, and we identify sharp threshold phenomena for both of them using probabilistic methods, duality arguments, and rounding techniques.
We first consider $α$-winning sets. A set $S$ of $k$ candidates is $α$-winning if, for every outside candidate $a \notin S$, at least an $α$-fraction of voters rank some member of $S$ above $a$. We show a sharp threshold at \[ α_{\mathrm{win}}^\star = 1 - \frac{1}{k}. \] Specifically, an $α$-winning set of size $k$ exists with high probability when $α< α_{\mathrm{win}}^\star$, and is unlikely to exist when $α> α_{\mathrm{win}}^\star$.
We then study the stronger notion of $α$-dominating sets. A set $S$ of $k$ candidates is $α$-dominating if, for every outside candidate $a \notin S$, there exists a single committee member $b \in S$ such that at least an $α$-fraction of voters prefer $b$ to $a$. Here we establish an analogous sharp threshold at \[ α_{\mathrm{dom}}^\star = \frac{1}{2} - \frac{1}{2k}. \] As a corollary, our analysis yields an impossibility result for $α$-dominating sets: for every $k$ and every $α> α_{\mathrm{dom}}^\star = 1 / 2 - 1 / (2k)$, there exist preference profiles that admit no $α$-dominating set of size $k$. This corollary improves the best previously known bounds for all $k \geq 2$.
Approval-based committee (ABC) voting rules elect a fixed size subset of the candidates, a so-called committee, based on the voters' approval ballots over the candidates. While these rules have recently attracted significant attention, axiomatic characterizations are largely missing so far. We address this problem by characterizing ABC voting rules within the broad and intuitive class of sequential valuation rules. These rules compute the winning committees by sequentially adding candidates that increase the score of the chosen committee the most. In more detail, we first characterize almost t…
We study the robustness of GreedyCC, GreedyPAV, and Phargmen's sequential rule, using the framework introduced by Bredereck et al. for the case of (multiwinner) ordinal elections and adopted to the approval setting by Gawron and Faliszewski. First, we show that for each of our rules and every committee size $k$, there are elections in which adding or removing a certain approval causes the winning committee to completely change (i.e., the winning committee after the operation is disjoint from the one before the operation). Second, we show that the problem of deciding how many approvals need to …
In the committee selection problem, we are given $m$ candidates, and $n$ voters. Candidates can have different weights. A committee is a subset of candidates, and its weight is the sum of weights of its candidates. Each voter expresses an ordinal ranking over all possible committees. The only assumption we make on preferences is monotonicity: If $S \subseteq S'$ are two committees, then any voter weakly prefers $S'$ to $S$. We study a general notion of group fairness via stability: A committee of given total weight $K$ is stable if no coalition of voters can deviate and choose a committee of p…
In this paper, we study fairness in committee selection problems. We consider a general notion of fairness via stability: A committee is stable if no coalition of voters can deviate and choose a committee of proportional size, so that all these voters strictly prefer the new committee to the existing one. Our main contribution is to extend this definition to stability of a distribution (or lottery) over committees. We consider two canonical voter preference models: the Approval Set setting where each voter approves a set of candidates and prefers committees with larger intersection with this s…
We consider the problem of electing a committee of $k$ candidates, subject to some constraints as to what this committee is supposed to look like. In our framework, the candidates are given labels as an abstraction of a politician's religion, a film's genre, a song's language, or other attribute, and the election outcome is constrained by interval constraints -- of the form "Between 3 and 5 candidates with label X" -- and dominance constraints -- "At least as many candidates with label X as with label Y". The problem is, what shall we do if the committee selected by a given voting rule fails t…
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…
CNRS（法国国家科学研究中心）；Facebook AI Research（脸书人工智能研究院）；University of Warsaw（华沙大学）
Assume $k$ candidates need to be selected. The candidates appear over time. Each time one appears, it must be immediately selected or rejected -- a decision that is made by a group of individuals through voting. Assume the voters use approval ballots, i.e., for each candidate they only specify whether they consider it acceptable or not. This setting can be seen as a voting variant of choosing $k$ secretaries. Our contribution is twofold. (1) We assess to what extent the committees that are computed online can proportionally represent the voters. (2) If a prior probability over candidate approv…
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 consider approval-based committee voting, i.e. the setting where each voter approves a subset of candidates, and these votes are then used to select a fixed-size set of winners (committee). We propose a natural axiom for this setting, which we call justified representation (JR). This axiom requires that if a large enough group of voters exhibits agreement by supporting the same candidate, then at least one voter in this group has an approved candidate in the winning committee. We show that for every list of ballots it is possible to select a committee that provides JR. However, it turns out…
We study the complexity of determining a winning committee under the Chamberlin--Courant voting rule when voters' preferences are single-crossing on a line, or, more generally, on a median graph (this class of graphs includes, e.g., trees and grids). For the line, Skowron et al. (2015) describe an $O(n^2mk)$ algorithm (where $n$, $m$, $k$ are the number of voters, the number of candidates and the committee size, respectively); we show that a simple tweak improves the time complexity to $O(nmk)$. We then improve this bound for $k=Ω(\log n)$ by reducing our problem to the $k$-link path problem f…
We study single-candidate voting embedded in a metric space, where both voters and candidates are points in the space, and the distances between voters and candidates specify the voters' preferences over candidates. In the voting, each voter is asked to submit her favorite candidate. Given the collection of favorite candidates, a mechanism for eliminating the least popular candidate finds a committee containing all candidates but the one to be eliminated. Each committee is associated with a social value that is the sum of the costs (utilities) it imposes (provides) to the voters. We design mec…
Central University of Rajasthan（拉贾斯坦中央大学）；University of Hyderabad（海得拉巴大学）
We consider the problem of committee selection from a fixed set of candidates where each candidate has multiple quantifiable attributes. To select the best possible committee, instead of voting for a candidate, a voter is allowed to approve the preferred attributes of a given candidate. Though attribute based preference is addressed in several contexts, committee selection problem with attribute approval of voters has not been attempted earlier. A committee formed on attribute preferences is more likely to be a better representative of the qualities desired by the voters and is less likely to …
Motivated by civic problems such as participatory budgeting and multiwinner elections, we consider the problem of public good allocation: Given a set of indivisible projects (or candidates) of different sizes, and voters with different monotone utility functions over subsets of these candidates, the goal is to choose a budget-constrained subset of these candidates (or a committee) that provides fair utility to the voters. The notion of fairness we adopt is that of core stability from cooperative game theory: No subset of voters should be able to choose another blocking committee of proportiona…
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…
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 char…
The Future Scientists Center–Alpha Program, Davidson Institute, the Educational Arm of Weizmann Institute of Science（魏茨曼科学研究所教育臂大卫森研究所未来科学家中心-阿尔法项目）；Department of Computer Science, Weizmann Institute of Science（魏茨曼科学研究所计算机科学系）
We consider a committee voting setting in which each voter approves of a subset of candidates and based on the approvals, a target number of candidates are to be selected. In particular we focus on the axiomatic property called extended justified representation (EJR). Although a committee satisfying EJR is guaranteed to exist, the computational complexity of finding such a committee has been an open problem and explicitly mentioned in multiple recent papers. We settle the complexity of finding a committee satisfying EJR by presenting a polynomial-time algorithm for the problem. Our algorithmic…
Distortion of Multi-Winner Elections on the Line Metric: The Polar Comparison Rule
Institute of Science and Technology Austria (ISTA)（奥地利科学技术研究院）；University of British Columbia（不列颠哥伦比亚大学）；Tehran Institute for Advanced Studies (TeIAS), Khatam University（德黑兰高等研究院，哈塔姆大学）；Max Planck Institut für Informatik, Universität des Saarlandes（马克斯·普朗克计算机科学研究所，萨尔兰大学）
We consider multiwinner elections in Euclidean space using the minimax Chamberlin-Courant rule. In this setting, voters and candidates are embedded in a $d$-dimensional Euclidean space, and the goal is to choose a committee of $k$ candidates so that the rank of any voter's most preferred candidate in the committee is minimized. (The problem is also equivalent to the ordinal version of the classical $k$-center problem.) We show that the problem is NP-hard in any dimension $d \geq 2$, and also provably hard to approximate. Our main results are three polynomial-time approximation schemes, each of…
We study two notions of stability in multiwinner elections that are based on the Condorcet criterion. The first notion was introduced by Gehrlein: A committee is stable if each committee member is preferred to each non-member by a (possibly weak) majority of voters. The second notion is called local stability (introduced in this paper): A size-$k$ committee is locally stable in an election with $n$ voters if there is no candidate $c$ and no group of more than $\frac{n}{k+1}$ voters such that each voter in this group prefers $c$ to each committee member. We argue that Gehrlein-stable committees…
