Cycles in synchronous iterative voting: general robustness and examples in Approval Voting
Benoît Kloeckner
备注：
v2: added a numerical study of rarity of bad cycles and equilibriums, and a case of chaotic Continuous Polling Dynamics. Many other improvements throughout the text. v3: reorganization and change of order. The robustness result is generalized to all voting systems. v4 Added a simple example to the stability Theorem, various other small modifications
We consider synchronous iterative voting, where voters are given the opportunity to strategically choose their ballots depending on the outcome deduced from the previous collective choices.We propose two settings for synchronous iterative voting, one of classical flavor with a discrete space of states, and a more general continuous-space setting extending the first one. We give a general robustness result for cycles not relying on a tie-breaking rule, showing that they persist under small enough perturbations of the behavior of voters. Then we give examples in Approval Voting of electorates applying simple, sincere and consistent heuristics (namely Laslier's Leader Rule or a modification of it) leading to cycles with bad outcomes, either not electing an existing Condorcet winner, or possibly electing a candidate ranked last by a majority of voters. Using the robustness result, it follows that those "bad cycles" persist even if only a (large enough) fraction of the electorate updates its choice of ballot at each iteration.We complete these results with examples in other voting methods, including ranking methods satisfying the Condorcet criterion; an in silico experimental study of the rarity of preference profiles exhibiting bad cycles; and an example exhibiting chaotic behavior.
Several multi-winner systems that use approval voting have been developed but they each suffer from various problems. Six of these methods are discussed in this paper. They are Satisfaction Approval Voting, Minimax Approval Voting, Proportional Approval Voting, Monroe's Fully Proportional Representation, Chamberlin-Courant's Rule, and Ebert's method. They all fail at least one of Proportional Representation (PR), strong PR, monotonicity or positive support. However, the new method described in this paper - Proportional Approval Method using Squared loads, Approval removal and Coin-flip approva…
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…
We introduce several electoral systems for multi-winner elections with approval ballots, generalizing the classical methods of Sainte-Laguë and D'Hondt. Our approach is based on the works of Phragmén and Thiele. In the last section we discuss possible generalizations to score voting.
Social choice is replete with various settings including single-winner voting, multi-winner voting, probabilistic voting, multiple referenda, and public decision making. We study a general model of social choice called Sub-Committee Voting (SCV) that simultaneously generalizes these settings. We then focus on sub-committee voting with approvals and propose extensions of the justified representation axioms that have been considered for proportional representation in approval-based committee voting. We study the properties and relations of these axioms. For each of the axioms, we analyse whether…
Multi-winner voting rules based on approval ballots have received increased attention in recent years. In particular Satisfaction Approval Voting (SAV) and its variants have been proposed. In this note, we show that the winning set can be determined in polynomial time for two prominent and natural variants of SAV. We thank Arkadii Slinko for suggesting these problems in a talk at the Workshop on Challenges in Algorithmic Social Choice, Bad Belzig, October 11, 2014.
NICTA（国家信息与通信技术研究院）；UNSW（新南威尔士大学）；University of Sydney（悉尼大学）
We study computational aspects of three prominent voting rules that use approval ballots to elect multiple winners. These rules are satisfaction approval voting, proportional approval voting, and reweighted approval voting. We first show that computing the winner for proportional approval voting is NP-hard, closing a long standing open problem. As none of the rules are strategyproof, even for dichotomous preferences, we study various strategic aspects of the rules. In particular, we examine the computational complexity of computing a best response for both a single agent and a group of agents.…
Proportional representation (PR) is one of the central principles in voting. Elegant rules with compelling PR axiomatic properties have the potential to be adopted for several important collective decision making settings. I survey some recent ideas and results on axioms and rules for proportional representation in committee voting.
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…
Approval-like voting rules, such as Sincere-Strategy Preference-Based Approval voting (SP-AV), the Bucklin rule (an adaptive variant of $k$-Approval voting), and the Fallback rule (an adaptive variant of SP-AV) have many desirable properties: for example, they are easy to understand and encourage the candidates to choose electoral platforms that have a broad appeal. In this paper, we investigate both classic and parameterized computational complexity of electoral campaign management under such rules. We focus on two methods that can be used to promote a given candidate: asking voters to move t…
We extend Approval voting to the settings where voters may have intransitive preferences. The major obstacle to applying Approval voting in these settings is that voters are not able to clearly determine who they should approve or disapprove, due to the intransitivity of their preferences. An approach to address this issue is to apply tournament solutions to help voters make the decision. We study a class of voting systems where first each voter casts a vote defined as a tournament, then a well-defined tournament solution is applied to select the candidates who are assumed to be approved by th…
In this paper we study several monotonicity axioms in approval-based multi-winner voting rules. We consider monotonicity with respect to the support received by the winners and also monotonicity in the size of the committee. Monotonicity with respect to the support is studied when the set of voters does not change and when new voters enter the election. For each of these two cases we consider a strong and a weak version of the axiom. We observe certain incompatibilities between the monotonicity axioms and well-known representation axioms (extended/proportional justified representation) for the…
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 implies the existence of voters, called manipulators, who can change the election outcome in their favour by voting strategically. When a given preference profile admits several such manipulators, voting becomes a game played by these voters, who have to reason strategically about each others' actions. To complicate the game even further, counter-manipulators may then try to counteract the actions of manipulators. Our voters are boundedly rational and do not think beyond manipulating or countermanipulating. We call these games Gibbard--Satterthwaite Games. In …
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 …
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 p…
University of Warsaw（华沙大学）；University of Wrocław（弗罗茨瓦夫大学）；Warsaw Center of Mathematics and Computer Science（华沙数学与计算机科学中心）
We present three results on the complexity of Minimax Approval Voting. First, we study Minimax Approval Voting parameterized by the Hamming distance $d$ from the solution to the votes. We show Minimax Approval Voting admits no algorithm running in time $\mathcal{O}^\star(2^{o(d\log d)})$, unless the Exponential Time Hypothesis (ETH) fails. This means that the $\mathcal{O}^\star(d^{2d})$ algorithm of Misra et al. [AAMAS 2015] is essentially optimal. Motivated by this, we then show a parameterized approximation scheme, running in time $\mathcal{O}^\star(\left({3}/ε\right)^{2d})$, which is essent…
This paper is an axiomatic study of consistent approval-based multi-winner rules, i.e., voting rules that select a fixed-size group of candidates based on approval ballots. We introduce the class of counting rules and provide an axiomatic characterization of this class based on the consistency axiom. Building upon this result, we axiomatically characterize three important consistent multi-winner rules: Proportional Approval Voting, Multi-Winner Approval Voting and the Approval Chamberlin--Courant rule. Our results demonstrate the variety of multi-winner rules and illustrate three different, or…
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…
Epistemic voting interprets votes as noisy signals about a ground truth. We consider contexts where the truth consists of a set of objective winners, knowing a lower and upper bound on its cardinality. A prototypical problem for this setting is the aggre-gation of multi-label annotations with prior knowledge on the size of the ground truth. We posit noisemodels, for which we define rules that output an optimal set of winners. We report on experiments on multi-label annotations (which we collected).
We study the computational complexity of committee selection problem for several approval-based voting rules in the presence of outliers. Our first result shows that outlier consideration makes committee selection problem intractable for approval, net approval, and minisum approval voting rules. We then study parameterized complexity of this problem with five natural parameters, namely the target score, the size of the committee (and its dual parameter, the number of candidates outside the committee), the number of outliers (and its dual parameter, the number of non-outliers). For net approval…
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…
We study sincere-strategy preference-based approval voting (SP-AV), a system proposed by Brams and Sanver [Electoral Studies, 25(2):287-305, 2006], and here adjusted so as to coerce admissibility of the votes (rather than excluding inadmissible votes a priori), with respect to procedural control. In such control scenarios, an external agent seeks to change the outcome of an election via actions such as adding/deleting/partitioning either candidates or voters. SP-AV combines the voters' preference rankings with their approvals of candidates, where in elections with at least two candidates the v…
In many real world situations, collective decisions are made using voting. Moreover, scenarios such as committee or board elections require voting rules that return multiple winners. In multi-winner approval voting (AV), an agent may vote for as many candidates as they wish. Winners are chosen by tallying up the votes and choosing the top-$k$ candidates receiving the most votes. An agent may manipulate the vote to achieve a better outcome by voting in a way that does not reflect their true preferences. In complex and uncertain situations, agents may use heuristics to strategize, instead of inc…
We consider Approval Voting systems where each voter decides on a subset to candidates he/she approves. We focus on the optimization problem of finding the committee of fixed size k minimizing the maximal Hamming distance from a vote. In this paper we give a PTAS for this problem and hence resolve the open question raised by Carragianis et al. [AAAI'10]. The result is obtained by adapting the techniques developed by Li et al. [JACM'02] originally used for the less constrained Closest String problem. The technique relies on extracting information and structural properties of constant size subse…
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…
The growing need for labeled training data has made crowdsourcing an important part of machine learning. The quality of crowdsourced labels is, however, adversely affected by three factors: (1) the workers are not experts; (2) the incentives of the workers are not aligned with those of the requesters; and (3) the interface does not allow workers to convey their knowledge accurately, by forcing them to make a single choice among a set of options. In this paper, we address these issues by introducing approval voting to utilize the expertise of workers who have partial knowledge of the true answe…
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…
In approval voting, individuals vote for all platforms that they find acceptable. In this situation it is natural to ask: When is agreement possible? What conditions guarantee that some fraction of the voters agree on even a single platform? Berg et. al. found such conditions when voters are asked to make a decision on a single issue that can be represented on a linear spectrum. In particular, they showed that if two out of every three voters agree on a platform, there is a platform that is acceptable to a majority of the voters. Hardin developed an analogous result when the issue can be repre…
Multi-winner approval elections are seen in a variety of settings ranging from academic societies and associations to public elections. In such elections, it is often the case that ballot-length restrictions are enforced; that is, where voters have a limit on the number of candidates which they can vote for. Despite this common feature, there does not seem to be any theoretical justification for ballot-length restrictions (Laslier and Van der Straeten, 2016). This work endogenously derives the set of voter best-response ballot lengths under complete information and with general assumptions on …
