We consider elections where both voters and candidates can be associated with points in a metric space and voters prefer candidates that are closer to those that are farther away. It is often assumed that the optimal candidate is the one that minimizes the total distance to the voters. Yet, the voting rules often do not have access to the metric space $M$ and only see preference rankings induced by $M$.Consequently, they often are incapable of selecting the optimal candidate. The distortion of a voting rule measures the worst-case loss of the quality being the result of having access only to preference rankings. We extend the idea of distortion to approval-based preferences. First, we compute the distortion of Approval Voting. Second, we introduce the concept of acceptability-based distortion---the main idea behind is that the optimal candidate is the one that is acceptable to most voters. We determine acceptability-distortion for a number of rules, including Plurality, Borda, $k$-Approval, Veto, the Copeland's rule, Ranked Pairs, the Schulze's method, and STV.
Proportional Approval Method using Squared loads, Approval removal and Coin-flip approval transformation (PAMSAC) - a new system of proportional representation using approval voting
We study the multifaceted question of how to sample approval elections in a meaningful way. Our analysis aims to discern the properties of various statistical cultures (both established and new ones). Based on the map-of-elections framework by Szufa et al. [2020], we graphically represent statistical cultures; and, by that, provide an intuitive understanding of their differences and properties.
AGH University of Kraków（克拉科夫AGH科技大学）；Department of Computer Science The University of Chicago（芝加哥大学计算机科学系）；CNRS, LAMSADE Université Paris Dauphine – PSL（法国国家科学研究中心，LAMSADE，巴黎第九大学-PSL）
Approval-preferential voting is problematical since it combines two different kinds of information that could by themselves lead to different choices. This article analyses the problem and studies a new proposal to deal with it. The existing methods are overviewed with special attention to several desirable properties that are not always met. Particular emphasis is made on certain rather unknown views of Condorcet about this subject. The proposed procedure definitely aims for a choice in the spirit of approval voting. However, it still makes use of the preferential information so as to improve…
Tulane University（杜兰大学）；Louisiana State University（路易斯安那州立大学）；University of West Florida（西佛罗里达大学）；Institute for Human and Machine Cognition（人机认知研究所）
In many real world situations, collective decisions are made using voting and, in scenarios such as committee or board elections, employing voting rules that return multiple winners. In multi-winner approval voting (AV), an agent submits a ballot consisting of approvals for as many candidates as they wish, and winners are chosen by tallying up the votes and choosing the top-$k$ candidates receiving the most approvals. In many scenarios, an agent may manipulate the ballot they submit in order to achieve a better outcome by voting in a way that does not reflect their true preferences. In complex…
How long is a piece of string? An exploration of multi-winner approval voting and ballot-length restrictions
从学术团体和协会到公共选举，在各种场合都可以看到多赢家批准选举。在这类选举中，往往实行选票长度限制;也就是说，选民对他们可以投票选举的候选人人数有限制。尽管有这一共同特征，但选票长度限制似乎没有任何理论依据（Laslier和Van der Straeten，2016）。这项工作内生地得出了一套选民最佳反应选票长度在完全信息和选民效用和投票规则的一般假设。这些结果为实践中观察到的一些选票长度限制提供了理由，但是当考虑均衡结果时，我们的分析表明这种理由不再有效。均衡分析被认为是懒惰和真理偏见的二阶倾向的选民和均衡解决方案的概念是纯粹的纳什均衡。关键的见解表明，选票长度的限制或制度的特点，使投票成本可能会导致不稳定的选举结果时，选民有不同的偏好，通过不存在的均衡。另一方面，当均衡确实存在时，它们满足理想的性质，而这些性质是在无成本投票和没有投票长度限制的情况下获得的均衡所不能保证的。总之，我们的研究结果突出了稳定和理想的选举结果之间的鲜明权衡。
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…
Multi-winner voting is the process of selecting a fixed-size set of representative candidates based on voters' preferences. It occurs in applications ranging from politics (parliamentary elections) to the design of modern computer applications (collaborative filtering, dynamic Q&A platforms, diversifying search results). All these applications share the problem of identifying a representative subset of alternatives -- and the study of multi-winner voting is the principled analysis of this task. This book provides a thorough and in-depth look at multi-winner voting based on approval preferences…
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 …
Technische Universität Berlin（柏林工业大学）；Paris School of Economics（巴黎经济学院）；University of Warsaw（华沙大学）
We establish a link between multiwinner elections and apportionment problems by showing how approval-based multiwinner election rules can be interpreted as methods of apportionment. We consider several multiwinner rules and observe that they induce apportionment methods that are well-established in the literature on proportional representation. For instance, we show that Proportional Approval Voting induces the D'Hondt method and that Monroe's rule induces the largest reminder method. We also consider properties of apportionment methods and exhibit multiwinner rules that induce apportionment m…
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…
Epistemic social choice aims at unveiling a hidden ground truth given votes, which are interpreted as noisy signals about it. We consider here a simple setting where votes consist of approval ballots: each voter approves a set of alternatives which they believe can possibly be the ground truth. Based on the intuitive idea that more reliable votes contain fewer alternatives, we define several noise models that are approval voting variants of the Mallows model. The likelihood-maximizing alternative is then characterized as the winner of a weighted approval rule, where the weight of a ballot decr…
The Gibbard-Satterthwaite theorem established that no non-trivial voting rule is strategy-proof, but that does not mean that all voting rules are equally susceptible to strategic manipulation. Over the past fifty years numerous approaches have been proposed to compare the manipulability of voting rules in terms of the probability of manipulation, the domains on which manipulation is possible, the complexity of finding such a manipulation, and others. In the closely related field of matching, Pathak and Sonmez pioneered a notion of manipulability based on case-by-case comparison of manipulable …
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…
In the apportionment problem, a fixed number of seats must be distributed among parties in proportion to the number of voters supporting each party. We study a generalization of this setting, in which voters can support multiple parties by casting approval ballots. This approval-based apportionment setting generalizes traditional apportionment and is a natural restriction of approval-based multiwinner elections, where approval ballots range over individual candidates instead of parties. Using techniques from both apportionment and multiwinner elections, we identify rules that generalize the D'…
