Van Zuylen et al. introduced the notion of a popular ranking in a voting context, where each voter submits a strictly-ordered list of all candidates. A popular ranking pi of the candidates is at least as good as any other ranking sigma in the following sense: if we compare pi to sigma, at least half of all voters will always weakly prefer pi. Whether a voter prefers one ranking to another is calculated based on the Kendall distance.



A more traditional definition of popularity---as applied to popular matchings, a well-established topic in computational social choice---is stricter, because it requires at least half of the voters who are not indifferent between pi and sigma to prefer pi. In this paper, we derive structural and algorithmic results in both settings, also improving upon the results by van Zylen et al. We also point out connections to the famous open problem of finding a Kemeny consensus with 3 voters

Cseh, Agnes

Kraiczy, Sonja

Manlove, David

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     Kraiczy, S., Cseh, A. and Manlove, D. (2021) On Weakly and Strongly Popular Rankings. In: 20th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2021), London, UK (Virtual), 3-7 May 2021, pp. 1563-1565. ISBN 9781450383073 (doi:10.5555/3463952.3464160)  There may be differences between this version and the published version. You are advised to consult the publisher’s version if you wish to cite from it.  © The Authors 2020. This is the author's version of the work. It is posted here for your personal use. Not for redistribution. The definitive Version of Record was published in the Proceedings of the 20th International Conference on Autonomous Agents and Multiagent Systems (AAMS ’21), Virtual Event, United Kingdom, May 2021, 9781450383073, p. 1563–1565, (doi:10.5555/3463952.3464160)    http://eprints.gla.ac.uk/233427/            Deposited on: 05 February 2021           Enlighten – Research publications by members of the University of                  Glasgow http://eprints.gla.ac.uk   OnWeakly and Strongly Popular RankingsExtended AbstractSonja KraiczyMerton CollegeUniversity of OxfordOxford, UKsonja.kraiczy@merton.ox.ac.ukÁgnes CsehHasso-Plattner-InstituteUniversity of PotsdamPotsdam, Germanyagnes.cseh@hpi.deDavid ManloveSchool of Computing ScienceUniversity of GlasgowGlasgow, UKdavid.manlove@glasgow.ac.ukABSTRACTVan Zuylen et al. [26] introduced the notion of a popular ranking ina voting context, where each voter submits a strictly-ordered list ofall candidates. A popular ranking π of the candidates is at least asgood as any other ranking σ in the following sense: if we compare πto σ , at least half of all voters will always weakly prefer π . Whethera voter prefers one ranking to another is calculated based on theKendall distance.Amore traditional definition of popularity—as applied to popularmatchings, a well-established topic in computational social choice—is stricter, because it requires at least half of the voters who are notindifferent between π and σ to prefer π . In this paper, we derivestructural and algorithmic results in both settings, also improvingupon the results in [26].We also point out connections to the famousopen problem of finding a Kemeny consensus with 3 voters.KEYWORDSmajority rule; Kemeny consensus; complexity; preference aggrega-tion; popular matchingACM Reference Format:Sonja Kraiczy, Ágnes Cseh, and David Manlove. 2021. On Weakly andStrongly Popular Rankings: Extended Abstract. In Proc. of the 20th Interna-tional Conference on Autonomous Agents and Multiagent Systems (AAMAS2021), Online, May 3–7, 2021, IFAAMAS, 3 pages.1 INTRODUCTIONA fundamental question in preference aggregation is the follow-ing: given a number of voters who rank candidates, can we con-struct a ranking that expresses the preferences of the entire set ofvoters as a whole? A common way of evaluating how close theconstructed ranking is to a voter’s preferences is the Kendall dis-tance, which measures the pairwise disagreements between tworankings. Among others, a well-known rank aggregation methodis the Kemeny ranking method [20], in which the winning rankingminimises the sum of its Kendall distances to the voters’ rankings.For the preference aggregation problem, van Zuylen et al. [26]introduce a new rank aggregation method called popular ranking,which is also based on the Kendall distance. Each voter can comparetwo given rankings π and σ , and prefers the one that is closer toher submitted ranking in terms of the Kendall distance. Van Zuylenet al. define π to be a winning ranking in an instance if for anyranking σ , at least half of the voters prefer π to σ or are indifferentProc. of the 20th International Conference on Autonomous Agents and Multiagent Systems(AAMAS 2021), U. Endriss, A. Nowé, F. Dignum, A. Lomuscio (eds.), May 3–7, 2021, Online.© 2021 International Foundation for Autonomous Agents and Multiagent Systems(www.ifaamas.org). All rights reserved.between them. This implies that there is no ranking σ such thatswitching to σ from π would benefit a majority of all voters.According to the definition of popularity in [26], even in a sit-uation where exactly half of the voters are indifferent betweenrankings π and σ , whilst the other half of the voters prefer σ to π ,σ is not more popular than π . This example demonstrates how hardit is for the dissatisfied voters to find a ranking that overrules π—the definition requires them to find a profiting set of voters whobuild an absolute majority, that is, a majority of all voters for thisendeavour.A straightforward option would be to require only a simple ma-jority, this is, a majority of the non-abstaining voters, to profit fromswitching to σ from π . Excluding the abstaining voters in a pairwisemajority voting rule is common practice [12]. It is also analogous tothe classical popularity notion in the matching literature [1, 9, 24].In this paper, we propose an alternative definition of a popularranking. We define π to be a strongly popular ranking if for everyranking σ , at least half of the non-abstaining voters prefer π to σ .This means that switching from π to σ would harm at least as manyvoters as it would benefit.Related literature. The most common approach to aggregatingvoters’ preferences is to search for a ranking that minimises thesum of the distances to the voters’ rankings. If the Kendall dis-tance [21] is used as the metric on rankings, then this optimalityconcept corresponds to the Kemeny consensus [20, 23]. Decidingwhether a given ranking is a Kemeny consensus is NP-complete,and calculating a Kemeny consensus is NP-hard [6] even if thereare only 4 voters [7, 13], or at least 6 of them [2]. Interestingly, thecomplexity of the problem is open for 3 and 5 voters [2, 7].Majority voting rules offer another natural way of aggregatingvoters’ preferences. The earliest reference for this might be fromCondorcet [8], who uses pairwise comparisons to calculate thewinning candidate, establishing his famous paradox on the small-est voting instance not admitting any majority winner. The abso-lute and simple majority voting rules have both been extensivelydiscussed in the setting where the goal is to choose the winningcandidate [4, 5].The concept of majority voting readily translates to other scenar-ios, where voters submit preference lists. One such field is the area ofmatchings under preferences, where popular matchings [17, 1, 24,Chapter 7, 9] serve as a voting-based alternative concept to thewell-known notion of stable matchings [3, 16] in two-sided mar-kets. Besides two-sided matchings, majority voting has also beendefined for the house allocation problem [1, 25], the roommatesproblem [14, 18], spanning trees [10], permutations [26], the or-dinal group activity selection problem [11], and very recently, tobranchings [19]. The notion of popularity is aligned with simplemajority in all these papers, with one exception, namely [26], whichdefines popularity based on the absolute majority rule.A part of this work revisits the paper from van Zuylen et al. [26].They show that a popular ranking—according to their definition ofpopularity—need not necessarily exist. More precisely, they showthat the acyclicity of a structure known as the majority graph is anecessary, but not sufficient condition for the existence of a popularranking. They also prove that if the majority graph is acyclic, thenone can efficiently compute a ranking, which may or may not bepopular, but for which the voters have to solve anNP-hard problemto compute a ranking that a majority of them prefer.Our contribution. We study both the weaker notion of popularityfrom [26] and the stronger notion of popularity analogous to theone in the matching literature, which excludes abstaining voters. Inthe full version of the paper [22], we present the following results.(1) For at most 5 voters, the two notions are equivalent, but with6 voters this does not hold anymore.(2) We give a sufficient condition for the two notions to beequivalent for a given ranking π .(3) In the case of 2 or 3 voters, one can find a popular rankingof either kind and verify the weak or strong popularity of agiven ranking in polynomial time.(4) The problem of verifying the weak or strong popularity ofa given ranking for 4 voters is polynomial-time solvable ifand only if it is polynomial-time solvable for 5 voters.(5) If finding a ranking that is more popular in either sense thana given ranking in an instance with 4 (or 5) voters werepolynomial-time solvable, then the famously open Kemenyconsensus problem for 3 voters would also be polynomial-time solvable.2 DEFINITIONS AND EXAMPLEWe are given a set of candidates and a set of voters, each of whomsubmits a ranking. A ranking π is a permutation of all candidates.The rank of candidate a in ranking π is the position it appears atin π , and it is denoted by rankπ [a]. The Kendall distance K(π ,σ )between two rankings π and σ is defined as the number of pairwisedisagreements between π and σ , or, formally asK (π , σ ) = | {(a, b) : rankπ [a] > rankπ [b] and rankσ [a] < rankσ [b]} |+ | {(a, b) : rankπ [a] < rankπ [b] and rankσ [a] > rankσ [b]} |.The Kendall distance is equivalent to the number of swaps thatthe bubble sort algorithm [15] executes when converting rankingπ to ranking σ . Voters prefer rankings that are similar to theirsubmitted ranking. More precisely, voter v prefers ranking σ toranking π if K(σ ,πv ) < K(π ,πv ). Analogously, voter v abstainsbetween π and σ if K(σ ,πv ) = K(π ,πv ).In the instance depicted by Figure 1, let σ1 = [1, 2, 3], [5, 6, 4],[9, 7, 8] and σ2 = [1, 2, 3], [4, 5, 6], [7, 8, 9]. Clearly v4 prefers σ2to σ1, since πv4 = σ2 and πv4 , σ1, that is, K(πv4 ,σ2) = 0 <K(πv4 ,σ1). In the same instance, v1 is an abstaining voter sinceK(πv1 ,σ2) = 4 = K(πv1 ,σ1).We now define the two different notions of popularity. The firstnotion of an weakly popular ranking corresponds to the popularranking as defined in [26].πv1 = [1, 2, 3], [6, 4, 5], [8, 9, 7]πv2 = [2, 3, 1], [4, 5, 6], [9, 7, 8]πv3 = [3, 1, 2], [5, 6, 4], [7, 8, 9]πv4 = [1, 2, 3], [4, 5, 6], [7, 8, 9]πv5 = [1, 2, 3], [5, 4, 6], [9, 7, 8]πv6 = [1, 2, 3], [5, 6, 4], [7, 9, 8]Figure 1: A voting instance with 6 voters v1,v2, . . .v6 and 9candidates 1, 2, . . . , 9.Definition 2.1. Ranking σ is more popular than ranking π inthe weak sense ifK(σ ,πv ) < K(π ,πv ) for an absolute majority of allvoters. Ranking π is weakly popular if no ranking σ is more popularthan π in the weak sense.If we consider σ3 = [2, 1, 3], [4, 5, 6], [7, 8, 9], then in the instancein Figure 1, σ2 = [1, 2, 3], [4, 5, 6], [7, 8, 9] is more popular than σ3 inthe weak sense. Notice that σ3 and σ2 only differ in their orderingof the pair of candidates {1, 2}. So since five out of six voters prefercandidate 1 to candidate 2, they form an absolute majority of allvoters who prefer σ2 to σ3.This definition requires more than half of the n voters to preferσ to π in order to declare σ to be more popular than π . Abstain-ing voters make it hard to beat π in such a pairwise comparison.However, if σ only needs to receive more votes than π amongthe voters not abstaining between these two rankings, then it canbeat π . This leads to the notion of strong popularity. Let Vabs(π ,σ )be the set of voters who abstain in the vote between π and σ , thatis, v ∈ Vabs(π ,σ ) if and only if K(πv ,π ) = K(πv ,σ ). In Figure 1,Vabs(σ1,σ2) = {v1,v2,v3} with σ1 and σ2 as before.Definition 2.2. Ranking σ is more popular than ranking π inthe strong sense if K(σ ,πv ) < K(π ,πv ) for a majority of the non-abstaining voters V \Vabs(π ,σ ). Ranking π is strongly popular if noranking σ is more popular than π in the strong sense.It follows directly from the two definitions above that stronglypopular rankings are weakly popular as well, but weakly popularrankings are not necessarily strongly popular. In the instance inFigure 1, σ1 = [1, 2, 3], [5, 6, 4], [9, 7, 8] is more popular than σ2 =[1, 2, 3], [4, 5, 6], [7, 8, 9] in the strong sense, since v5 and v6 preferσ1 to σ2, while v1,v2, and v3 abstain. Notice that σ1 is not morepopular than σ2 in the weak sense, because two voters do notconstitute an absolute majority of all 6 voters, only a majority ofthe non-abstaining 3 voters.Acknowledgement. We thank Markus Brill for fruitful discus-sions. Sonja Kraiczy was supported by Undergraduate ResearchBursary 19-20-66 from the LondonMathematical Society, and by theSchool of Computing Science, University of Glasgow. Ágnes Csehwas supported by the Federal Ministry of Education and Researchof Germany in the framework of KI-LAB-ITSE (project number01IS19066). David Manlove was supported by grant EP/P028306/1from the Engineering and Physical Sciences Research Council.REFERENCES[1] David J. Abraham, Robert W. Irving, Telikepalli Kavitha, and Kurt Mehlhorn.2007. Popular Matchings. SIAM J. Comput. 37 (2007), 1030–1045.[2] Georg Bachmeier, Felix Brandt, Christian Geist, Paul Harrenstein, Keyvan Kardel,Dominik Peters, and Hans Georg Seedig. 2019. k -Majority Digraphs and theHardness of Voting with a Constant Number of Voters. J. Comput. System Sci.105 (2019), 130–157.[3] Mourad Baïou and Michel Balinski. 2002. The stable allocation (or ordinal trans-portation) problem. 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On Weakly and Strongly Popular Rankings

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On Weakly and Strongly Popular Rankings

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