arX

iv:1

407.

8269

v4 [

cs.M

A]

12 S

ep 2

016

Justified Representation inApproval-Based Committee Voting

Haris Aziz · Markus Brill · Vincent Conitzer ·Edith Elkind · Rupert Freeman · Toby Walsh

Received: date/ Accepted: date

Abstract We consider approval-based committee voting, i.e. the setting where eachvoter 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, whichwe calljustified representation (JR). This axiom requires that if a large enough groupof voters exhibits agreement by supporting the same candidate, then at least one voterin this group has an approved candidate in the winning committee. We show that forevery list of ballots it is possible to select a committee that providesJR. However,it turns out that several prominent approval-based voting rules may fail to outputsuch a committee. In particular, while Proportional Approval Voting (PAV) alwaysoutputs a committee that providesJR, Reweighted Approval Voting (RAV), a tractableapproximation toPAV, does not have this property. We then introduce a strongerversion of theJRaxiom, which we callextended justified representation (EJR), andshow thatPAV satisfiesEJR, while other rules we consider do not; indeed,EJRcanbe used to characterizePAVwithin the class of weightedPAV rules. We also considerseveral other questions related toJR and EJR, including the relationship betweenJR/EJRand core stability, and the complexity of the associated algorithmic problems.

Keywords Approval voting· committee selection· representation.

H. Aziz · T. WalshData61, CSIRO and UNSW Australia, Sydney 2052 , AustraliaTel.:+61-2-8306 0490Fax:+61-2-8306 0405E-mail: {haris.aziz,toby.walsh}@data61.csiro.au

M. Brill · E. ElkindUniversity of OxfordOxford OX1 3QD, UKE-mail: {mbrill,elkind}@cs.ox.ac.uk

V. Conitzer· R. FreemanDuke UniversityDurham, NC 27708, USAE-mail: {conitzer, rupert}@cs.duke.edu

2 Haris Aziz et al.

JEL Classification: C70 · D61 · D71

1 Introduction

The aggregation of preferences is a central problem in the field of social choice.While the most-studied scenario is that of selecting a single candidate out of many, itis often the case that one needs to select a fixed-sizeset of winners (committee): thisincludes domains such as parliamentary elections, the hiring of faculty members, or(automated) agents deciding on a set of plans (LeGrand et al.2007; Davis et al. 2014;Elkind et al. 2014, 2015; Skowron et al. 2015a). The study of algorithmic complexityof voting rules that output committees is an active researchdirection (Procaccia et al.2008; Meir et al. 2008; Caragiannis et al. 2010; Lu and Boutilier 2011; Cornaz et al.2012; Betzler et al. 2013; Skowron et al. 2015b,c).

In this paper we consider approval-based rules, where each voter lists the subsetof candidates that she approves of. There is a growing literature on voting rules thatare based on approval ballots: the Handbook on Approval Voting (Laslier and Sanver2010) provides a very useful survey of pre-2010 research on this topic, and after thisseminal book was published, various aspects of approval voting continued to attracta considerable amount of attention (see, e.g., the papers ofCaragiannis et al. 2010;Endriss 2013; Duddy 2014). One of the advantages of approvalballots is their sim-plicity: such ballots reduce the cognitive burden on voters(rather than providing a fullranking of the candidates, a voter only needs to decide whichcandidates to approve)and are also easier to communicate to the election authority. The most straightforwardway to aggregate approvals is to have every approval for a candidate contribute onepoint to that candidate’s score and select the candidates with the highest score. Thisrule is calledApproval Voting (AV). AV has many desirable properties in the single-winner case (Brams et al. 2006; Endriss 2013), including its“simplicity, propensityto elect Condorcet winners (when they exist), its robustness to manipulation and itsmonotonicity” (Brams 2010, p. viii). However, for the case of multiple winners, themerits ofAV are “less clear” (Brams 2010, p. viii). For example,AV may fail pro-portional representation: if the goal is to selectk winners,k > 1, 51% of the votersapprove the samek candidates, and the remaining voters approve a disjoint setof kcandidates, then the voters in minority do not get any of their approved candidatesselected.

As a consequence, over the years, several multi-winner rules based on approvalballots have been proposed (see, e.g., the survey by Kilgour2010); we will nowbriefly describe the rules that will be considered in this paper (see Section 2 for for-mal definitions). UnderProportional Approval Voting (PAV), each voter’s contribu-tion to the committee’s total score depends on how many candidates from the voter’sapproval set have been elected. In the canonical variant of this rule the marginal util-ity of the ℓ-th approved candidate is1

ℓ, i.e. this rule is associated with the weight

vector (1, 12 ,

13 , . . . ); other weight vectors can be used as well, resulting in the family

of weighted PAV rules. A sequential variant ofPAV is known asReweighted Ap-proval Voting (RAV); again, by varying the weight vector, we obtain the family ofweighted RAV rules. Another way to modulate the approvals is through computinga

Justified Representation in Approval-Based Committee Voting 3

satisfaction score for each voter based on the ratio of the number of their approvedcandidates appearing in the committee and their total number of approved candi-dates; this idea leads toSatisfaction Approval Voting (SAV). One could also use adistance-based approach:Minimax Approval Voting (MAV)selects a set ofk candi-dates that minimizes the maximum Hamming distance from the submitted ballots.Finally, one could adapt classic rules that provide fully proportional representation,such as the Chamberlin–Courant rule (Chamberlin and Courant 1983) or the Monroerule (Monroe 1995), to work with approval ballots, by using each voter’s ballot asa scoring vector. All the rules informally described above have a more egalitarianobjective thanAV. For example, Steven Brams, a proponent ofAV in single-winnerelections, has argued thatSAVis more suitable for equitable representation in multi-winner elections (Brams and Kilgour 2014).

The relative merits of approval-based multi-winner rules and the complexityof winner determination under these rules have been examined in great detail inboth economics and computer science in recent years (Brams and Fishburn 2007;LeGrand et al. 2007; Meir et al. 2008; Caragiannis et al. 2010; Aziz et al. 2015;Byrka and Sornat 2014; Misra et al. 2015). On the other hand, there has been lim-ited axiomatic analysis of these rules from the perspectiveof representation (see,however, Section 7).

In this paper, we introduce the notion ofjustified representation(JR) in approval-based voting. Briefly, a committee is said to provide justified representation for agiven set of ballots if every large enough group of voters with shared preferencesis allocated at least one representative. A rule is said to satisfy justified representa-tion if it always outputs a committee that provides justifiedrepresentation. This con-cept is related to theDroop proportionality criterion(Droop 1881) and Dummett’ssolid coalition property(Dummett 1984; Tideman and Richardson 2000; Elkind et al.2014), but is specific to approval-based elections.

We show that every set of ballots admits a committee that provides justified rep-resentation; moreover, such a committee can be computed efficiently, and checkingwhether a given committee providesJRcan be done in polynomial time as well. Thisshows that justified representation is a reasonable requirement. However, it turns outthat many popular multi-winner approval-based rules failJR; in particular, this is thecase forAV, SAV, MAV and the canonical variant ofRAV. On the positive side,JR issatisfied by some of the weightedPAV rules, including the canonicalPAV rule, as wellas by the weightedRAV rule associated with the weight vector (1, 0, . . . ) and by theMonroe rule. Also,MAV satisfiesJR for a restricted domain of voters’ preferences.We then consider a strengthening of theJRaxiom, which we callextended justifiedrepresentation (EJR). This axiom captures the intuition that a very large group ofvoters with similar preferences may deserve not just one, but several representatives.EJRturns out to be a more demanding property thanJR: of all voting rules consideredin this paper, only the canonicalPAV rule satisfiesEJR. Thus, in particular,EJRchar-acterizes the canonicalPAV rule within the class of weightedPAV rules. However,we show that it is computationally hard to check whether a given committee providesEJR.

We also consider other strengthenings ofJR, which we callsemi-strong justifiedrepresentationandstrong justified representation; however, it turns out that for some

4 Haris Aziz et al.

inputs the requirements imposed by these axioms are impossible to satisfy. Finally,we explore the relationship betweenJR/EJRand core stability in a non-transferableutility game that can be associated with a multiwinner approval voting scenario. Weshow that, even thoughEJRmay appear to be similar to core stability, it is, in fact,a strictly weaker condition. Indeed, the core stability condition appears to be too de-manding, as none of the voting rules considered in our work isguaranteed to producea core stable outcome, even when the core is known to be non-empty. We concludethe paper by showing howJR can be used to formulate other attractive approval-based multi-winner rules, discussing related work, and identifying several directionsfor future work.

2 Preliminaries

We consider a social choice setting with a setN = {1, . . . , n} of voters and a setCof candidates. Each voteri ∈ N submits an approval ballotAi ⊆ C, which representsthe subset of candidates that she approves of. We refer to thelist A = (A1, . . . ,An) ofapproval ballots as theballot profile. We will considerapproval-based multi-winnervoting rulesthat take as input a tuple (N,C,A, k), wherek is a positive integer thatsatisfiesk ≤ |C|, and return a subsetW ⊆ C of sizek, which we call thewinning set,or committee(Kilgour and Marshall 2012). We omitN andC from the notation whenthey are clear from the context. Several approval-based multi-winner rules are definedbelow. Whenever the description of the rule does not uniquely specify a winning set,we assume that ties are broken according to some deterministic procedure; however,most of our results do not depend on the tie-breaking rule.

2.1 Approval-Based Multi-Winner Rules

Approval Voting (AV) UnderAV, the winners are thek candidates that receive thelargest number of approvals. Formally, theapproval scoreof a candidatec ∈ C isdefined as|{i | c ∈ Ai}|, andAV outputs a setW of sizek that maximizes

∑

c∈W |{i |c ∈ Ai}|. AV has been adopted by several academic and professional societies suchas the Institute of Electrical and Electronics Engineers (IEEE) and the InternationalJoint Conference on Artificial Intelligence (IJCAI).

Satisfaction Approval Voting (SAV) A voter’s satisfaction scoreis the fraction ofher approved candidates that are elected.SAVmaximizes the sum of voters’ satisfac-tion scores. Formally,SAVoutputs a setW ⊆ C of sizek that maximizes

∑

i∈N|W∩Ai |

|Ai |.

This rule was proposed with the aim of “representing more diverse interests” thanAV(Brams and Kilgour 2014).

Proportional Approval Voting (PAV) Under PAV, a voter is assumed to derivea utility of 1 + 1

2 +13 + · · · +

1j from a committee that contains exactlyj of her

approved candidates, and the goal is to maximize the sum of the voters’ utilities.Formally, thePAV-score of a setW ⊆ C is defined as

∑

i∈N r(|W∩ Ai |), wherer(p) =∑p

j=11j , andPAV outputs a setW ⊆ C of sizek with the highestPAV-score. Though

sometimes attributed to Forest Simmons,PAV was already proposed by the Danish

Justified Representation in Approval-Based Committee Voting 5

polymath Thorvald N. Thiele in the 19th century (Thiele 1895).1 PAV captures theidea of diminishing returns: an individual voter’s preferences should count less themore she is satisfied.

We can generalize the definition ofPAVby using an arbitrary score vector in placeof (1, 1

2 ,13 , . . . ). Specifically, for every vector2 w = (w1,w2, . . . ), wherew1,w2, . . . are

non-negative reals, we define a voting rulew-PAV that operates as follows. Given aballot profile (A1, . . . ,An) and a target number of winnersk, w-PAV returns a setWof sizek with the highestw-PAV score, defined by

∑

i∈N rw(|W∩ Ai |), whererw(p) =∑p

j=1 w j . Usually, it is required thatw1 = 1 andw1 ≥ w2 ≥ . . . . The latter constraint isappropriate in the context of representative democracy: itis motivated by the intuitionthat once an agent already has one or more representatives inthe committee, thatagent should have less priority for further representation. It what follows, we willalways impose the constraintw1 = 1 (as we can always rescale the weight vector, thisis equivalent to requiring thatw1 > 0; while the casew1 = 0 may be of interest insome applications, we omit it in order to keep the length of the paper manageable3)and explicitly indicate which of our results require thatw1 ≥ w2 ≥ . . . ; in particular,for our characterization ofPAV in Theorem 11 this is not the case.

Reweighted Approval Voting (RAV) RAVconvertsPAV into a multi-round rule, byselecting a candidate in each round and then reweighing the approvals for the subse-quent rounds. Specifically,RAVstarts by settingW = ∅. Then in roundj, j = 1, . . . , k,it computes theapproval weightof each candidatec as

∑

i:c∈Ai

11+|W∩Ai |

, selects acandidate with the highest approval weight, and adds him toW. After k rounds, itoutputs the setW. RAV has also been referred to as “sequential proportional AV”(Brams and Kilgour 2014), and was used briefly in Sweden during the early 1900s.

Thiele (1895) proposedRAVas a tractable approximation toPAV (see Section 2.2for a discussion of the computational complexity of these rules and the relation-ship between them). We note that there are several other examples of voting rulesthat were conceived as approximate versions of other rules,yet became viewed aslegitimate voting rules in and of themselves; two representative examples are theSimplified Dodgson rule of Tideman (2006), which was designed as an approxi-mate version of the Dodgson rule (see the discussion by Caragiannis et al. 2014), andthe Greedy Monroe rule of Skowron et al. (2015b), which approximates the Monroerule (Monroe 1995).

Just as forPAV, we can extend the definition ofRAV to score vectors other than(1, 1

2 ,13 , . . . ): every vectorw = (w1,w2, . . . ) defines a sequential voting rulew-RAV,

which proceeds asRAV, except that it computes the approval weight of a candidatecin round j as

∑

i:c∈Aiw|W∩Ai |+1, whereW is the winning set after the firstj − 1 rounds.

Again, we impose the constraintw1 = 1 (note that ifw1 = 0, thenw-RAVcan pick anarbitrary candidate at the first step, which is obviously undesirable).

1 We are grateful to Xavier Mora and Svante Janson for pointingthis out to us.2 It is convenient to think ofw as an infinite vector; note that for an election withmcandidates only the

first m entries ofw matter. To analyze the complexity ofw-PAV rules, one would have to place additionalrequirements onw; however, we do not consider algorithmic properties of suchrules in this paper.

3 Generalizations ofPAV with w1 = 0 have been considered by Fishburn and Pekec (2004) andSkowron et al. (2015d). We note that such rules do not satisfyjustified representation (as defined in Sec-tion 3).

6 Haris Aziz et al.

A particularly interesting rule in this class is (1, 0, . . . )-RAV: this rule, whichwe will refer to asGreedy Approval Voting (GAV), can be seen as a variant of theSweetSpotGreedy (SSG) algorithm of Lu and Boutilier (2011), and admits a verysimple description: we pick candidates one by one, trying to‘cover’ as many cur-rently ‘uncovered’ voters as possible. In more detail, a winning committee under thisrule can be computed by the following algorithm. We start by settingC′ = C, A′ = A,andW = ∅. As long as|W| < k andA′ is non-empty, we pick a candidatec ∈ C′ thathas the highest approval score with respect toA′, and setW :=W∪{c}, C′ := C′ \{c}.Also, we remove fromA′ all ballotsAi such thatc ∈ Ai . If at some point we have|W| < k andA′ is empty, we add an arbitrary set ofk − |W| candidates fromC′ to Wand returnW; if this does not happen, we terminate after having pickedk candidates.

We will also consider a variant ofGAV, where, at each step, after selecting acandidatec, instead of removing all voters inAc = {i | c ∈ Ai} from A′, we removea subset ofAc of size min

{

⌈ nk⌉, |Ac|

}

. This rule can be seen as an adaptation of the

classic STV rule to approval ballots, and we will refer to it asGAVT (whereT standsfor ‘threshold’).4

Minimax Approval Voting (MAV) MAV returns a committeeW that minimizesthe maximumHamming distancebetweenW and the voters’ ballots; this rule wasproposed by Brams et al. (2007). Formally, letd(Q,T) = |Q \ T | + |T \ Q| and definetheMAV-score of a setW ⊆ C as max(d(W,A1), . . . , d(W,An)). MAV outputs a size-kset with the lowestMAV-score.

Chamberlin–Courant and Monroe Approval Voting (CCAV and Mo nAV) TheChamberlin–Courant rule (Chamberlin and Courant 1983) is usually defined for thesetting where each voter provides a full ranking of the candidates. Each voteri ∈ Nis associated with a scoring vectorui = (ui

1, . . . , uim) whose entries are non-negative

reals; we think ofuij as voteri’s satisfaction from being represented by candidatec j . A

voter’s satisfaction from a committeeW is defined as maxc j∈W uij, and the rule returns

a committee of sizek that maximizes the sum of voters’ satisfactions. For the case ofapproval ballots, it is natural to define the scoring vectorsby settingui

j = 1 if c j ∈ Ai

anduij = 0 otherwise; that is, a voter is satisfied by a committee if this committee

contains one of her approved candidates. Thus, the resulting rule is equivalent to(1, 0, . . . )-PAV (and therefore we will not discuss it separately).

The Monroe rule (Monroe 1995) is a modification of the Chamberlin–Courantrule where each committee member represents roughly the same number of vot-ers. Just as under the Chamberlin–Courant rule, we have a scoring vectorui =

(ui1, . . . , u

im) for each voteri ∈ N. Given a committeeW of size k, we say that a

mappingπ : N → W is valid if it satisfies |π−1(c)| ∈{

⌊ nk⌋, ⌈

nk⌉

}

for eachc ∈ W.The Monroe score of a valid mappingπ is given by

∑

i∈N uiπ(i), and the Monroe score

of a committeeW is the maximum Monroe score of a valid mapping fromN toW. The Monroe rule returns a size-k committee with the maximum Monroe score.For approval ballots, we define the scoring vectors in the same manner as for the

4 For readability, we use the Hare quota⌈ nk ⌉; however, all our proofs go through if we use the Droop

quota⌈ nk+1⌉+1 instead. For a discussion of differences between these two quotas, see the article of Tideman

(1995).

Justified Representation in Approval-Based Committee Voting 7

Chamberlin–Courant Approval Voting rule; we call the resulting rule the MonroeApproval Voting rule (MonAV).

We note that fork = 1,AV, PAV, RAV, GAV, GAVT andMonAVproduce the sameoutput if there is a unique candidate with the highest approval score. However, sucha candidate need not be a winner underSAVor MAV.

2.2 Computational Complexity

The rules listed above differ from an algorithmic perspective. For some of theserules, namely,AV, SAV, RAV, GAV andGAVT , a winning committee can be com-puted in polynomial time; this is also true forw-RAV as long the entries of theweight vector are rational numbers that can be efficiently computed given the num-ber of candidates. In contrast,PAV, MAV, and MonAV are computationally hard(Aziz et al. 2015; Skowron et al. 2015a; LeGrand et al. 2007; Procaccia et al. 2008);for w-PAV, the hardness result holds for most weight vectors, including (1, 0, . . . ),(i.e., it holds forCCAV). However, bothPAV andMAV admit efficient approxima-tion algorithms (i.e., algorithms that output committees which are approximatelyoptimal with respect to the optimization criteria of these rules) and have been an-alyzed from the perspective of parameterized complexity. Specifically,w-PAVadmitsan efficient

(

1− 1e

)

-approximation algorithm as long as the weight vectorw is ef-ficiently computable and non-increasing; in fact, such an algorithm is provided byw-RAV (Skowron et al. 2015a). ForMAV, LeGrand et al. (2007) propose a simple 3-approximation algorithm; Caragiannis et al. (2010) improve the approximation ratioto 2 and Byrka and Sornat (2014) develop a polynomial-time approximation scheme.Misra et al. (2015) show thatMAV is fixed-parameter tractable for a number of natu-ral parameters; Elkind and Lackner (2015) obtain fixed parameter tractability resultsfor PAV when voters’ preferences are, in some sense, single-dimensional. There isalso a number of tractability results forCCAV, and, to a lesser extent, forMonAV; werefer the reader to the work of Skowron et al. (2015a) and references therein.

3 Justified Representation

We will now define one of the main concepts of this paper.

Definition 1 (Justified representation (JR))Given a ballot profileA = (A1, . . . ,An)over a candidate setC and a target committee sizek, we say that a set of candidatesW of size |W| = k provides justified representation for(A, k) if there does not exista set of votersN∗ ⊆ N with |N∗| ≥ n

k such that⋂

i∈N∗ Ai , ∅ andAi ∩W = ∅ forall i ∈ N∗. We say that an approval-based voting rulesatisfies justified representation(JR) if for every profileA = (A1, . . . ,An) and every target committee sizek it outputsa winning set that provides justified representation for (A, k).

The logic behind this definition is that ifk candidates are to be selected, then, intu-itively, each group ofnk voters “deserves” a representative. Therefore, a set ofn

k votersthat have at least one candidate in common should not be completely unrepresented.We refer the reader to Section 6 for a discussion of alternative definitions.

8 Haris Aziz et al.

3.1 Existence and Computational Properties

We start our analysis of justified representation by observing that, for every ballotprofile A and every value ofk, there is a committee that provides justified represen-tation for (A, k), and, moreover, such a committee can be computed efficiently giventhe voters’ ballots. In fact, bothGAV andGAVT output a committee that providesJR.

Theorem 1 GAV and GAVT satisfy JR.

Proof We present a proof that applies to bothGAV andGAVT . Suppose for the sakeof contradiction that for some ballot profileA = (A1, . . . ,An) and somek > 0, GAV(respectively,GAVT) outputs a committee that does not provide justified representa-tion for (A, k). Then there exists a setN∗ ⊆ N with |N∗| ≥ n

k such that⋂

i∈N∗ Ai , ∅

and, whenGAV (respectively,GAVT) terminates, every ballotAi such thati ∈ N∗ isstill in A′. Consider some candidatec ∈

⋂

i∈N∗ Ai . At every point in the execution ofour algorithm,c’s approval score is at least|N∗| ≥ n

k . As c was not elected, at everystage the algorithm selected a candidate whose approval score was at least as high asthat ofc. Thus, at the end of each stage the algorithm removed fromA′ at least⌈ n

k⌉

ballots containing the candidate added toW at that stage, so altogether the algorithmhas removed at leastk · n

k ballots fromA′. This contradicts the assumption thatA′

contains at leastnk ballots when the algorithm terminates. ⊓⊔

Theorem 1 shows that it is easy to find a committee that provides justified repre-sentation for a given ballot profile. It is also not too hard tocheck whether a givencommitteeW providesJR. Indeed, while it may seem that we need to consider everysubset of voters of sizenk , in fact it is sufficient to consider the candidates one by one,and, for each candidatec, computes(c) = |{i ∈ N | c ∈ Ai ,Ai ∩W = ∅}|; the setWfails to provide justified representation for (A, k) if and only if there exists a candidatec with s(c) ≥ n

k . We obtain the following theorem.

Theorem 2 There exists a polynomial-time algorithm that, given a ballot profileAover a candidate set C, and a committee W,|W| = k, decides whether W providesjustified representation for(A, k).

3.2 Justified Representation and Unanimity

A desirable property of single-winner approval-based voting rules isunanimity: avoting rule is unanimous if, given a ballot profile (A1, . . . ,An) with ∩i∈NAi , ∅, itoutputs a candidate in∩i∈NAi . This property is somewhat similar in spirit toJR, sothe reader may expect that fork = 1 it is equivalent toJR. However, it turns out thatthe JR axiom is strictly weaker than unanimity fork = 1: while unanimity impliesJR, the converse is not true, as illustrated by the following example.

Example 1Let N = {1, . . . , n}, C = {a, b1, . . . , bn}, Ai = {a, bi} for i ∈ N. Considera voting rule that fork = 1 outputsb1 on this profile and coincides withGAV inall other cases. Clearly, this rule is not unanimous; however, it satisfiesJR, as it isimpossible to find a group ofnk = n unrepresented voters for (A1, . . . ,An).

Justified Representation in Approval-Based Committee Voting 9

It is not immediately clear how to define unanimity for multi-winner voting rules;however, any reasonable definition would be equivalent to the standard definition ofunanimity whenk = 1, and therefore would be different from justified representation.

We remark that a rule can be unanimous fork = 1 and provideJR for all valuesof k: this is the case, for instance, forGAV.

4 Justified Representation under Approval-Based Rules

We have argued that justified representation is a reasonablecondition: there alwaysexists a committee that provides it, and, moreover, such a committee can be computedefficiently. It is therefore natural to ask whether prominent voting rules satisfyJR. Inthis section, we will answer this question forAV, SAV, MAV, PAV, RAV, andMonAV.We will also identify conditions onw that are sufficient/necessary forw-PAV andw-RAV to satisfyJR.

In what follows, for each rule we will try to identify the range of values ofk forwhich this rule satisfiesJR. Trivially, all rules that we consider satisfyJR for k = 1.It turns out thatAV fails JR for k > 2, and fork = 2 the answer depends on thetie-breaking rule.

Theorem 3 For k = 2, AV satisfies JR if ties are broken in favor of sets that provideJR. For k≥ 3, AV fails JR.

Proof Suppose first thatk = 2. Fix a ballot profileA. If every candidate is approvedby fewer thann

2 voters inA, JR is trivially satisfied. If some candidate is approvedby more thann

2 voters inA, thenAV selects some such candidate, in which case nogroup of⌈ n

2⌉ voters is unrepresented, soJR is satisfied in this case as well. It remainsto consider the case wheren = 2n′, some candidates are approved byn′ voters, andno candidate is approved by more thann′ voters. ThenAV necessarily picks at leastone candidate approved byn′ voters; denote this candidate byc. In this situationJRcan only be violated if then′ voters who do not approvec all approve the samecandidate (say,c′), and this candidate is not elected. But the approval score of c′ isn′, and, by our assumption, the approval score of every candidate is at mostn′, sothis is a contradiction with our tie-breaking rule. This argument also illustrates whythe assumption on the tie-breaking rule is necessary: it canbe the case thatn′ votersapprovec andc′′, and the remainingn′ voters approvec′, in which case the approvalscore of{c, c′′} is the same as that of{c, c′}.

For k ≥ 3, we letC = {c0, c1, . . . , ck}, n = k, and consider the profile where thefirst voter approvesc0, whereas each of the remaining voters approves all ofc1, . . . , ck.JRrequiresc0 to be selected, butAV selects{c1, . . . , ck}. ⊓⊔

On the other hand,SAVandMAV fail JReven fork = 2.

Theorem 4 SAV and MAV do not satisfy JR for k≥ 2.

Proof We first considerSAV. Fix k ≥ 2, let X = {x1, . . . , xk, xk+1}, Y = {y1, . . . , yk},C = X ∪ Y, and consider the profile (A1, . . . ,Ak), whereA1 = X, A2 = {y1, y2},Ai = {yi} for i = 3, . . . , k. JR requires each voter to be represented, butSAV will

10 Haris Aziz et al.

chooseY: theSAV-score ofY is k− 1, whereas theSAV-score of every committeeWwith W ∩ X , ∅ is at mostk − 2 + 1

2 +1

k+1 < k − 1. Therefore, the first voter willremain unrepresented.

For MAV, we use the following construction. Fixk ≥ 2, let X = {x1, . . . , xk},Y = {y1, . . . , yk}, C = X ∪ Y ∪ {z}, and consider the profile (A1, . . . ,A2k), whereAi = {xi , yi} for i = 1, . . . , k, Ai = {z} for i = k + 1, . . . , 2k. Every committee of sizek that providesJR for this profile containsz. However,MAV fails to selectz. Indeed,theMAV-score ofX is k + 1: we haved(X,Ai) = k for i ≤ k andd(X,Ai) = k + 1 fori > k. Now, consider some committeeW with |W| = k, z ∈W. We haveAi∩W = ∅ forsomei ≤ k, sod(W,Ai) = k+ 2. Thus,MAV prefersX to any committee that includesz. ⊓⊔

The constructions used in the proof of Theorem 4 show thatMAV andSAVmay be-have very differently:SAVappears to favor voters who approve very few candidates,whereasMAV appears to favor voters who approve many candidates.

Interestingly, we can show thatMAV satisfiesJR if we assume that each voterapproves exactlyk candidates and ties are broken in favor of sets that provideJR.

Theorem 5 If the target committee size is k,|Ai | = k for all i ∈ N, and ties are brokenin favor of sets that provide JR, then MAV satisfies JR.

Proof Consider a profileA = (A1, . . . ,An) with |Ai | = k for all i ∈ N.Observe that if there exists a set of candidatesW with |W| = k such thatW∩Ai , ∅

for all i ∈ N, thenMAV will necessarily select some such set. Indeed, for any such setW we haved(W,Ai) ≤ 2k− 1 for eachi ∈ N, whereas ifW′ ∩ Ai = ∅ for some setW′

with |W′| = k and somei ∈ N, thend(W′,Ai) = 2k. Further, by definition, every setW such that|W| = k andW∩ Ai , ∅ for all i ∈ N provides justified representation for(A, k).

On the other hand, if there is nok-element set of candidates that intersects eachAi , i ∈ N, then theMAV-score of every set of sizek is 2k, and thereforeMAV can pickan arbitrary size-k subset. Since we assumed that the tie-breaking rule favors sets thatprovideJR, our claim follows. ⊓⊔

While Theorem 5 provides an example of a setting whereMAV satisfiesJR, thisresult is not entirely satisfactory: first, we had to place a strong restriction on voters’preferences, and, second, we used a tie-breaking rule that was tailored toJR.

We will now show thatPAV satisfiesJR, for all ballot profiles and irrespective ofthe tie-breaking rule.

Theorem 6 PAV satisfies JR.

Proof Fix a ballot profileA = (A1, . . . ,An) and ak > 0 and lets = ⌈ nk⌉. Let W be

the output ofPAV on (A, k). Suppose for the sake of contradiction that there exists asetN∗ ⊂ N, |N∗| ≥ s, such that

⋂

i∈N∗ Ai , ∅, butW ∩⋃

i∈N∗ Ai = ∅. Let c be somecandidate approved by all voters inN∗.

For each candidatew ∈ W, define itsmarginal contributionas the differencebetween thePAV-score ofW and that ofW\{w}. Letm(W) denote the sum of marginalcontributions of all candidates inW. Observe that ifc were to be added to the winning

Justified Representation in Approval-Based Committee Voting 11

set, this would increase thePAV-score by at leasts. Therefore, it suffices to argue thatthe marginal contribution of some candidate inW is less thans: this would mean thatswapping this candidate withc increases thePAV-score, a contradiction. To this end,we will prove thatm(W) ≤ s(k − 1); as|W| = k, our claim would then follow by thepigeonhole principle.

Consider the setN \ N∗; we haven ≤ sk, so |N \ N∗| ≤ n − s ≤ s(k − 1). Picka voteri ∈ N \ N∗, and let j = |Ai ∩W|. If j > 0, this voter contributes exactly1j tothe marginal contribution of each candidate inAi ∩W, and hence her contribution tom(W) is exactly 1. If j = 0, this voter does not contribute tom(W) at all. Therefore,we havem(W) ≤ |N \ N∗| ≤ s(k− 1), which is what we wanted to prove. ⊓⊔

The reader may observe that the proof of Theorem 6 applies to all voting rules ofthe formw-PAV where the weight vector satisfiesw1 = 1, w j ≤

1j for all j ≥ 1. In

Section 5 we will see that this condition onw is also necessary forw-PAV to satisfyJR.

Next, we considerRAV. As this voting rule can be viewed as a tractable approx-imation ofPAV (recall thatPAV is NP-hard to compute), one could expect thatRAVsatisfiesJR as well. However, this turns out not to be the case, at least ifk is suffi-ciently large.

Theorem 7 RAV satisfies JR for k= 2, but fails it for k≥ 10.

Proof For k = 2, we can use essentially the same argument as forAV; however, wedo not need to assume anything about the tie-breaking rule. This is because if thereare three candidates,c, c′, andc′′, such thatc andc′′ are approved by the samen2voters, whereasc′ is approved by the remainingn2 voters, andRAV selectsc in thefirst round, then in the second roundRAV favorsc′ overc′′.

Now, suppose thatk = 10. Consider a profile over a candidate setC = {c1, . . . , c11}

with 1199 voters who submit the following ballots:

81×{c1, c2}, 81×{c1, c3}, 80×{c2}, 80×{c3},

81×{c4, c5}, 81×{c4, c6}, 80×{c5}, 80×{c6},

49×{c7, c8}, 49×{c7, c9}, 49×{c7, c10},

96×{c8}, 96×{c9}, 96×{c10}, 120×{c11}.

Candidatesc1 andc4 are each approved by 162 voters, the most of any candidate,and these blocks of 162 voters do not overlap, soRAV selectsc1 andc4 first. Thisreduces theRAVscores ofc2, c3, c5 andc6 from 80+ 81= 161 to 80+ 40.5 = 120.5,soc7, whoseRAVscore is 147, is selected next. Now, theRAVscores ofc8, c9 andc10

become 96+ 24.5 = 120.5. The selection of any ofc2, c3, c5, c6, c8, c9 or c10 does notaffect theRAV score of the others, so all seven of these candidates will be selectedbeforec11, who has 120 approvals. Thus, after the selection of 10 candidates, thereare 120> 1199

10 =nk unrepresented voters who jointly approvec11.

To extend this construction tok > 10, we createk− 10 additional candidates and120(k − 10) additional voters such that for each new candidate, there are 120 newvoters who approve that candidate only. Note that we still have 120> n

k . RAV willproceed to selectc1, . . . , c10, followed byk− 10 additional candidates, andc11 or oneof the new candidates will remain unselected. ⊓⊔

12 Haris Aziz et al.

While RAV itself does not satisfyJR, one could hope that this can be fixed by tweak-ing the weights, i.e. thatw-RAVsatisfiesJRfor a suitable weight vectorw. However,it turns out that (1, 0, . . . ) is essentially the only weight vector for which this is thecase: Theorem 7 extends tow-RAV for everyweight vectorw with w1 = 1, w2 > 0.

Theorem 8 For every vectorw = (w1,w2, . . . ) with w1 = 1, w2 > 0, there exists avalue of k0 > 0 such thatw-RAV does not satisfy JR for k> k0.

Proof Pick a positive integers≥ 8 such thatw2 ≥1s. LetC = C1∪C2∪ {x, y}, where

C1 = {ci, j | i = 1, . . . , 2s+ 3, j = 1, . . . , 2s+ 1}, C2 = {ci | i = 1, . . . , 2s+ 3}.

For eachi = 1, . . . , 2s+ 3 and eachj = 1, . . . , 2s+ 1 we construct 2s3− svoters whoapproveci, j only ands2 voters who approveci, j andci only. Finally, we construct2s3 − 1 voters who approvex only ands2 − 7s− 5 voters who approvey only (notethat the number of voters who approvey is positive by our choice ofs).

Setk0 = (2s+2)(2s+3) = |C1∪C2|. Note that the number of votersn is given by

(2s+ 3)(2s+ 1)(2s3 + s2 − s) + (2s3 − 1)+ (s2 − 7s− 5)

= (2s+ 2)(2s+ 3)(2s3 − 1) = (2s3 − 1)k0,

and hencenk0= 2s3 − 1.

Underw-RAV initially the score of each candidate inC2 is s2(2s+ 1) = 2s3 + s2,the score of each candidate inC1 is 2s3+ s2− s, the score ofx is 2s3−1, and the scoreof y is s2−7s−5, so in the first 2s+3 rounds the candidates fromC2 get elected. Afterthat, the score of every candidate inC1 becomes 2s3 − s+ w2s2 ≥ 2s3 − s+ s= 2s3,while the scores ofx andy remains unchanged. Therefore, in the next (2s+3)(2s+1)rounds the candidates fromC1 get elected. At this point,k candidates are elected, andx is not elected, even though the 2s3−1 = n

k0voters who approve him do not approve

of any of the candidates in the winning set.To extend this argument to larger values ofk, we proceed as in the proof of The-

orem 7: fork > k0, we addk − k0 new candidates, and for each new candidate weconstruct 2s3− 1 new voters who approve that candidate only. Let the resulting num-ber of voters ben′; we haven′

k = 2s3 − 1, sow-RAVwill first select the candidates inC2, followed by the candidates inC1, and then it will choosek−k0 winners among thenew candidates andx. As a result, eitherx or one of the new candidates will remainunselected. ⊓⊔

Remark 1Theorem 8 partially subsumes Theorem 7: it implies thatRAVfails JR, butthe proof only shows that this is the case fork ≥ 18·19= 342, while Theorem 7 statesthat RAV fails JR for k ≥ 10 already. We chose to include the proof of Theorem 7because we feel that it is useful to know what happens for relatively small values ofk. Note, however, that Theorem 7 leaves open the question of whetherRAVsatisfiesJR for k = 3, . . . , 9. Very recently, Sanchez-Fernandez et al. (2016) have answeredthis question by showing thatRAVsatisfiesJR for k ≤ 5 and fails it fork ≥ 6.

Justified Representation in Approval-Based Committee Voting 13

If we allow the entries of the weight vector to depend on the number of votersn, wecan obtain another class of rules that provide justified representation: the argumentused to show thatGAV satisfiesJR extends tow-RAV where the weight vectorwsatisfiesw1 = 1, w j ≤

1n for j > 1. In particular, the rule (1, 1

n ,1n2 , . . . , )-RAV is

somewhat more appealing thanGAV: for instance, if⋂

i∈N Ai = {c} andk > 1, GAVwill pick c, and then behave arbitrarily, whereas (1, 1

n ,1n2 , . . . , )-RAV will also pick

c, but then it will continue to look for candidates approved byas many voters aspossible.

We conclude this section by showing thatMonAVsatisfiesJR.

Theorem 9 MonAV satisfies JR.

Proof Fix a ballot profileA = (A1, . . . ,An) and ak > 0. Let W be an output ofMonAVon (A, k). If Ai ∩W , ∅ for all i ∈ N, thenW provides justified representationfor (A, k). Thus, assume that this is not the case, i.e. there exists some voteri withAi ∩W = ∅. Consider a valid mappingπ : N → W whose Monroe score equals theMonroe score ofW, let c = π(i), and sets= |π−1(c)|; note thats ∈ {⌊ n

k⌋, ⌈nk⌉}.

Suppose for the sake of contradiction thatW does not provide justified represen-tation for (A, k). Then by our choice ofs there exists a setN∗ ⊂ N, |N∗| = s, suchthat

⋂

i∈N∗ Ai , ∅, butW ∩ (⋃

i∈N∗ Ai) = ∅. Let c′ be some candidate approved by allvoters inN∗, and setW′ = (W \ {c}) ∪ {c′}. To obtain a contradiction, we will arguethatW′ has a higher Monroe score thanW.

To this end, we will modifyπ by first swapping the voters inN∗ with votersin π−1(c) and then assigning the voters inN∗ to c′. Formally, letσ : π−1(c) \ N∗ →N∗ \π−1(c) be a bijection betweenπ−1(c)\N∗ andN∗ \π−1(c). We construct a mappingπ : N→W′ by setting

π(i) =

π(σ(i)) for i ∈ π−1(c) \ N∗,

c′ for i ∈ N∗,

π(i) for i < π−1(c) ∪ N∗.

Note thatπ is a valid mapping: we have|π−1(c′)| = s and |π−1(c′′)| = |π−1(c′′)| foreachc′′ ∈ W′ \ {c′}. Now, let us consider the impact of this modification on theMonroe score. Thes voters inN∗ contributed nothing to the Monroe score ofπ,and they contributes to the Monroe score of ˆπ. By our choice ofc, the voters inπ−1(c) contributed at mosts− 1 to the Monroe score ofπ, and their contribution tothe Monroe score of ˆπ is non-negative. For all other voters their contribution totheMonroe score ofπ is equal to their contribution to the Monroe score ofπ′. Thus, thetotal Monroe score of ˆπ is higher than that ofπ. Since the Monroe score ofW is equalto the Monroe score ofπ, and, by definition, the Monroe score ofW′ is at least theMonroe score of ˆπ, we obtain a contradiction. ⊓⊔

5 Extended Justified Representation

We have identified four (families of) voting rules that satisfy JR for arbitrary ballotprofiles:w-PAV with w1 = 1, w j ≤

1j for j > 1 (this class includesPAV), w-RAV

14 Haris Aziz et al.

with w1 = 1, w j ≤1n for j > 1 (this class includesGAV), GAVT andMonAV. The

obvious advantage ofGAVandGAVT is that their output can be computed efficiently,whereas computing the outputs ofPAV or MonAV is NP-hard. However,GAV putsconsiderable emphasis on representingeveryvoter, at the expense of ensuring thatlarge sets of voters with shared preferences are allocated an adequate number of rep-resentatives. This approach may be problematic in a varietyof applications, such asselecting a representative assembly, or choosing movies tobe shown on an airplane,or foods to be provided at a banquet (see the discussion by Skowron et al. 2015a).In particular, it may be desirable to have several assembly members that represent awidely held political position, both to reflect the popularity of this position, and tohighlight specific aspects of it, as articulated by different candidates. Consider, forinstance, the following example.

Example 2Let k = 3, C = {a, b, c, d}, andn = 100. One voter approvesc, one voterapprovesd, and 98 voters approvea andb. GAV would include bothc andd in thewinning set, whereas in many settings it would be more reasonable to choose bothaandb (and one ofc andd); indeed, this is exactly whatGAVT would do.

This issue is not addressed by theJRaxiom, as this axiom does not care if a givenvoter is represented by one or more candidates. Thus, if we want to capture the intu-ition that large cohesive groups of voters should be allocated several representatives,we need a stronger condition. Recall thatJRsays that each group ofn

k voters that allapprove the same candidate “deserves” at least one representative. It seems reason-able to scale this idea and say that, for everyℓ > 0, each group ofℓ · n

k voters thatall approve the sameℓ candidates “deserves” at leastℓ representatives. This approachcan be formalized as follows.

Definition 2 (Extended justified representation (EJR)) Given a ballot profile(A1, . . . ,An) over a candidate setC, a target committee sizek, k ≤ |C|, and a posi-tive integerℓ, ℓ ≤ k, we say that a set of candidatesW, |W| = k, providesℓ-justifiedrepresentation for(A, k) if there does not exist a set of votersN∗ ⊆ N with |N∗| ≥ ℓ · nksuch that|

⋂

i∈N∗ Ai | ≥ ℓ, but |Ai ∩W| < ℓ for eachi ∈ N∗; we say thatW providesextended justified representation (EJR) for(A, k) if it providesℓ-JR for (A, k) for allℓ, 1 ≤ ℓ ≤ k. We say that an approval-based voting rulesatisfiesℓ-justified represen-tation (ℓ-JR)if for every profileA = (A1, . . . ,An) and every target committee sizek itoutputs a committee that providesℓ-JRfor (A, k). Finally, we say that a rulesatisfiesextended justified representation (EJR)if it satisfiesℓ-JRfor all ℓ, 1≤ ℓ ≤ k.

Observe thatEJRimpliesJR, because the latter coincides with 1-JR.The definition ofEJRinterprets “a groupN∗ deserves at leastℓ representatives”

as “at least one voter inN∗ getsℓ representatives”. Of course, other interpretationsare also possible: for instance, we can require that each voter in N∗ is represented byℓ candidates in the winning committee or, alternatively, that the winning committeecontains at leastℓ candidates each of which is approved by some member ofN∗.However, the former requirement is too strong: it differs fromJR for ℓ = 1 andin Section 6 we show that there are ballot profiles for which commitees with thisproperty do not exist even forℓ = 1. The latter approach, which was very recently

Justified Representation in Approval-Based Committee Voting 15

proposed by Sanchez-Fernandez et al. (2016) (see the discussion in Section 7), is notunreasonable; in particular, it coincides withJR for ℓ = 1. However, it is strictlyless demanding than the approach we take: clearly, every rule that satisfiesEJRalsosatisfies this condition. As it turns out (Theorem 10) that every ballot profile admits acommittee that providesEJR, theEJRaxiom offers more guidance in choosing a goodwinning committee than its weaker cousin, while still leaving us with a non-empty setof candidate committees to choose from. Finally, theEJRaxiom in its present form isvery similar to a core stability condition for a natural NTU game associated with theinput profile (see Section 5.2); it is not clear if the axiom ofSanchez-Fernandez et al.(2016) admits a similar interpretation.

5.1 Extended Justified Representation under Approval-Based Rules

It is natural to ask which of the voting rules that satisfyJRalso satisfyEJR. Example 2immediately shows that forGAV the answer is negative. Consequently, now-RAVrulesuch that the entries ofw do not depend onn satisfiesEJR: if w2 = 0, this rule isGAVand ifw2 > 0, our claim follows from Theorem 8. Moreover, Example 2 alsoimpliesthatw-RAVrules withw j ≤

1n for j > 1 also failEJR.

The next example shows thatMAV fails EJReven if each voter approves exactlyk candidates (recall that under this assumptionMAV satisfiesJR).

Example 3Let k = 4, C = C1 ∪ C2 ∪ C3 ∪ C4, where|C1| = |C2| = |C3| = |C4| = 4and the setsC1,C2,C3,C4 are pairwise disjoint. LetA = (A1, . . . ,A8), whereAi = Ci

for i = 1, 2, 3, andAi = C4 for i = 4, 5, 6, 7, 8.MAV will select exactly one candidatefrom each of the setsC1,C2,C3 andC4, butEJRdictates that at least two candidatesfrom C4 are chosen.

Further,MonAV fails EJRas well.

Example 4Let k = 4, C = {c1, c2, c3, c4, a, b}, N = {1, . . . , 8}, Ai = {ci} fori = 1, . . . , 4, Ai = {ci−4, a, b} for i = 5, . . . , 8. MonAV outputs{c1, c2, c3, c4} on thisprofile, as this is the unique set of candidates with the maximum Monroe score. Thus,every voter is represented by a single candidate, though thevoters inN∗ = {5, 6, 7, 8}“deserve” two candidates.

Example 4 illustrates the conflict between theEJRaxiom and the requirement to rep-resent all voters whenever possible. We discuss this issue in more detail in Section 7.

For GAVT , it is not hard to construct an example where this rule failsEJR forsome way of breaking intermediate ties.

Example 5Let N = {1, . . . , 8}, C = {a, b, c, d, e, f }, A1 = A2 = {a}, A3 = A4 =

{a, b, c}, A5 = A6 = {d, b, c}, A7 = {d, e}, A8 = {d, f }. Suppose thatk = 4. Note thatall voters inN∗ = {3, 4, 5, 6} approveb andc, and|N∗| = 2 · n

k . UnderGAVT , at thefirst step candidatesa, b, c andd are tied, so we can selecta and remove voters 3 and4. Next, we have to selectd; we can then remove voters 5 and 6. In the remainingtwo steps, we adde and f to the committee. The resulting committee violatesEJR,as each voter inN∗ = {3, 4, 5, 6} is only represented by a single candidate.

16 Haris Aziz et al.

We note that in Example 5 we can remove voters 1 and 2 after selecting a, whichenables us to selectb or c in the second step and thereby obtain a committee thatprovidesEJR. In fact, we were unable to construct an example whereGAVT failsEJRfor all ways of breaking intermediate ties; we now conjecture that it is alwayspossible to break intermediate ties inGAVT so as to satisfyEJR. However, it is notclear if a tie-breaking rule with this property can be formulated in a succinct manner.Thus,GAVT does not seem particularly useful if we want to find a committee thatprovidesEJR: even if our conjecture is true, we may have to explore all ways ofbreaking intermediate ties.

In contrast, we will now show thatPAV satisfiesEJR irrespective of the tie-breaking rule.

Theorem 10 PAV satisfies EJR.

Proof Suppose thatPAV violatesEJR for some value ofk, and consider a ballotprofile A1, . . . ,An, a value ofℓ > 0 and a set of votersN∗, |N∗| = s ≥ ℓ · n

k , thatwitness this. LetW, |W| = k, be the winning set. We know that at least one of theℓcandidates approved by all voters inN∗ is not elected; letc be some such candidate.Each voter inN∗ has at mostℓ − 1 representatives inW, so the marginal contributionof c (if it were to be added toW) would be at leasts · 1

ℓ≥ n

k . On the other hand, theargument in the proof of Theorem 6 can be modified to show that the sum of marginalcontributions of candidates inW is at mostn.

Now, consider some candidatew ∈ W with the smallest marginal contribution;clearly, his marginal contribution is at mostn

k . If it is strictly less thannk , we are

done, as we can improve the totalPAV-score by swappingw andc, a contradiction.Therefore suppose it is exactlynk , and therefore the marginal contribution of eachcandidate inW is exactly n

k . SincePAV satisfiesJR, we know thatAi ∩ W , ∅ forsomei ∈ N∗. Pick some candidatew′ ∈ W ∩ Ai , and setW′ = (W \ {w′}) ∪ {c}.Observe that afterw′ is removed, addingc increases the totalPAV-score by at least(s− 1) · 1

ℓ+ 1ℓ−1 >

nk . Indeed,i approves at mostℓ − 2 candidates inW \ {w′} and

therefore addingc to W \ {w′} contributes at least1ℓ−1 to her satisfaction. Thus, the

PAV-score ofW′ is higher than that ofW, a contradiction again. ⊓⊔

Interestingly, Theorem 10 does not extend to weight vectorsother than(1, 1

2 ,13 , . . . ): our next theorem shows thatPAV is essentially the uniquew-PAV rule

that satisfiesEJR.

Theorem 11 For every weight vectorw with w1 = 1, w , (1, 12 ,

13 , . . . ), the rule

w-PAV does not satisfy EJR.

Theorem 11 follows immediately from Lemmas 1 and 2, which arestated below.

Lemma 1 Consider a weight vectorw with w1 = 1. If w j >1j for some j> 1, then

w-PAV fails JR.

Proof Suppose thatw j =1j + ε for somej > 1 andε > 0. Pickk > ⌈ 1

ε j ⌉ + 1 so thatj

dividesk; let t = kj . LetC = C0 ∪C1 ∪ · · · ∪Ct, whereC0 = {c}, |C1| = · · · = |Ct| = j,

and the setsC0,C1, . . . ,Ct are pairwise disjoint. Note that|C| = t j + 1 = k+ 1. Also,

Justified Representation in Approval-Based Committee Voting 17

constructt+1 pairwise disjoint groups of votersN0,N1, . . . ,Nt so that|N0| = k, |N1| =

· · · = |Nt| = j(k−1), and for eachi = 0, 1, . . . , t the voters inNi approve the candidatesin Ci only. Observe that the total number of voters is given byn = k+ t j(k− 1) = k2.

We have|N0| = k = nk , so every committee that provides justified representation

for this profile must electc. However, we claim thatw-PAV elects all candidates inC \ {c} instead. Indeed, if we replace an arbitrary candidate inC \ {c} with c, thenunderw-PAV the total score of our committee changes by

k− j(k− 1) ·

(

1j+ ε

)

= 1− j(k− 1)ε < 1− jε

⌈

1ε j

⌉

≤ 0,

i.e.C \ {c} has a strictly higher score than any committee that includesc. ⊓⊔

Lemma 2 Consider a weight vectorw with w1 = 1. If w j <1j for some j> 1, then

w-PAV fails j-JR.

Proof Suppose thatw j =1j − ε for somej > 1 andε > 0. Pickk > j + ⌈ 1

ε⌉. Let C =

C0 ∪C1, where|C0| = j, C1 = {c1, . . . , ck− j+1} andC0 ∩C1 = ∅. Note that|C| = k+ 1.Also, constructk− j + 2 pairwise disjoint groups of votersN0,N1, . . . ,Nk− j+1 so that|N0| = j(k− j+1), |N1| = · · · = |Nk− j+1| = k− j, the voters inN0 approve the candidatesin C0 only, and for eachi = 1, . . . , k− j + 1 the voters inNi approveci only. Note thatthe number of voters is given byn = j(k− j + 1)+ (k− j + 1)(k− j) = k(k− j + 1).

We havenk = k− j+1 and|N0| = j · n

k , so every committee that providesEJRmustselect all candidates inC0. However, we claim thatw-PAV elects all candidates fromC1 and j − 1 candidates fromC0 instead. Indeed, letc be some candidate inC0, let c′

be some candidate inC1, and letW = C \ {c}, W′ = C \ {c′}. The difference betweenthe total score ofW and that ofW′ is

j(k− j + 1)

(

1j− ε

)

− (k− j) < 1− j ·1ε· ε < 1− j < 0,

i.e. w-PAV assigns a higher score toW. As this argument does not depend on thechoice ofc in C0 andc′ in C1, the proof is complete. ⊓⊔

5.2 JR, EJRand Core Stability

One can view (extended) justified representation as a stability condition, by associat-ing committees that provideJR/EJRwith outcomes of a certain NTU game that areresistant to certain types of deviations.

Specifically, given a pair (A, k), whereA = (A1, . . . ,An), we define an NTU gameG(A, k) with the set of playersN as follows. We assume that each coalition of sizex,ℓ n

k ≤ x < (ℓ+1)nk , whereℓ ∈ {1, . . . , k}, can “purchase”ℓ alternatives. Moreover, each

player evaluates a committee of sizeℓ, ℓ ∈ {1, . . . , k}, using thePAV utility function,i.e. i derives a utility of 1+ 1

2 + · · · +1j from a committee that contains exactlyj of

her approved alternatives (the argument goes through forw-PAV utilities, as long asw1 ≥ · · · ≥ wk > 0). Thus, for each coalitionS with ℓ n

k ≤ |S| < (ℓ + 1)nk a payoff

vectorx ∈ Rn is considered to be feasible forS if and only if there exists a committee

18 Haris Aziz et al.

W ⊆ A with |W| ≤ ℓ such thatxi = ui(W) for eachi ∈ S, whereui(W) = 1+· · ·+ 1|Ai∩W| .

We denote the set of all payoff vectors that are feasible for a coalitionS ⊆ N by V(S).We say that a coalitionS ⊆ N has aprofitable deviationfrom a payoff vector

x ∈ V(N) if there exists a payoff vectory ∈ V(S) such thatyi > xi for all i ∈ S. Apayoff vectorx is stableif it is feasible forN and no coalitionS ⊆ N has a profitabledeviation from it; the set of all stable payoff vectors is thecoreof G(A, k).

The following theorem describes the relationship betweenJR, EJR, and outcomesof G(A, k).

Theorem 12 A committee W,|W| = k, provides justified representation for(A, k) ifand only if no coalition of size⌈ n

k⌉ or less has a profitable deviation from the payoffvectorx associated with W. Moreover, W provides extended justified representationfor (A, k) if and only if for everyℓ ≥ 0 no coalition N∗ with ℓ · n

k ≤ |N∗| < (ℓ + 1) · n

k ,| ∩i∈N∗ Ai | ≥ ℓ has a profitable deviation fromx.

Proof Suppose thatW fails to provide justified representation for (A, k), i.e. thereexists a set of votersN∗, |N∗| = ⌈ n

k⌉, who all approve some candidatec <W, but noneof them approves any of the candidates inW. Then we havexi = 0 for eachi ∈ N∗,and players inN∗ can successfully deviate: the payoff vectory that is associated withthe committee{c} is feasible forN∗ and satisfiesyi = 1 for eachi ∈ N∗.

Conversely, suppose thatW provides justified representation for (A, k), and con-sider a coalitionN∗. If |N∗| < ⌈ n

k⌉, then for everyy ∈ V(N∗) we haveyi = 0 for alli ∈ N∗, soN∗ cannot profitably deviate. On the other hand, if|N∗| = ⌈ n

k⌉, then everypayoff vectory ∈ V(N∗) is associated with a committee of size 1. Hence, for everyy ∈ V(N∗) we haveyi ≤ 1 for all i ∈ N∗, and ifyi = 1 for all i ∈ N∗, then∩i∈N∗Ai , ∅,and therefore, sinceW providesJR, we havexi ≥ 1 for somei ∈ N∗.

ForEJRthe argument is similar. IfW fails to provide extended justified represen-tation for (A, k), there exists anℓ > 0 and a set of votersN∗, |N∗| ≥ ℓ · n

k , such that|⋂

i∈N∗ Ai | ≥ ℓ, but |Ai∩W| < ℓ for eachi ∈ N∗. Then we havexi < 1+ · · ·+ 1ℓ

for eachi ∈ N∗, and players inN∗ can successfully deviate: ifS is a committee that consistsof someℓ candidates in

⋂

i∈N∗ Ai , then the payoff vectory that is associated withS isfeasible forN∗ and satisfiesyi = 1+ · · · + 1

ℓfor eachi ∈ N∗.

Conversely, suppose thatW provides extended justified representation for (A, k),and consider someℓ ≥ 0 and some coalitionN∗ with ℓ · n

k ≤ |N∗| < (ℓ+ 1)n

k . We haveargued above that ifℓ = 0, thenN∗ cannot profitably deviate. Thus, assumeℓ > 0.Every payoff vectory ∈ V(N∗) is associated with a committee of sizeℓ. Hence, foreveryy ∈ V(N∗) we haveyi ≤ 1+ · · · + 1

ℓfor all i ∈ S, and ifyi = 1+ · · · + 1

ℓfor all

i ∈ N∗, then| ∩i∈N∗ Ai | ≥ ℓ. SinceW providesEJR, we havexi ≥ 1+ · · · + 1ℓ

for somei ∈ N∗. ⊓⊔

The second part of Theorem 12 considers deviations by cohesive coalitions.The reader may wonder if it can be strengthened toarbitrary coalitional deviations,i.e. whether a committee providesEJR if and only ifthe associated payoff vector is inthe core ofG(A, k). The following example shows that this is not the case.

Justified Representation in Approval-Based Committee Voting 19

Example 6Let k = 10,C = {x1, x2, . . . , x10, y, z}, N = {1, 2, . . . , 20}, and

A1 = A2 = A3 = {x1, y},

A4 = A5 = A6 = {x1, z},

A7 = . . . = A20 = {x2, . . . , x10}.

ThenPAV outputs the committeeW = {x1, x2, . . . , x10} for (A, k); in particular,WprovidesEJRfor (A, k). However, the associated payoff vectorx is not in the core, asthe players in{1, 2, 3, 4, 5, 6}, a coalition of size 3nk , can successfully deviate: the pay-off vector associated with{x1, y, z} is feasible for{1, 2, 3, 4, 5, 6}and provides a higherpayoff thanx to each of the first six players. We remark that the core ofG(A, k) is notempty: in particular, it contains the payoff vector associated with{x1, . . . , x8, y, z}.

It remains an open question whether the core ofG(A, k) is non-empty for everypair (A, k). Further, while it would be desirable to have a voting rule that outputs acommittee whose associated payoff vector is in the core whenever the core is notempty, we are not aware of any such rule: every voting rule that fails EJRalso failsthis more demanding criterion, and Example 6 illustrates that PAV fails this criterionas well.

5.3 Computational Issues

In Section 3 we have argued that it is easy to find a committee that providesJR for agiven ballot profile, and to check whether a specific committee providesJR. In con-trast, forEJRthese questions appear to be computationally difficult. Specifically, wewere unable to design an efficient algorithm for computing a committee that providesEJR; while PAV is guaranteed to find such a committee, computing its output is NP-hard. We remark, however, that whenℓ is bounded by a constant, we can efficientlycompute a committee that providesℓ-JR, i.e. the challenge is in handling large valuesof ℓ.

Theorem 13 A committee satisfyingℓ-JR can be computed in time polynomial in nand |C|ℓ.

Proof Consider the following greedy algorithm, which we will refer to asℓ-GAV. Westart by settingC′ = C, A′ = A, andW = ∅. As long as|W| ≤ k − ℓ, we check ifthere exists a set of candidates{c1, . . . , cℓ} ⊂ C′ that is unanimously approved by atleastℓ n

k voters inA′ (this can be done in timen · |C|ℓ+1). If such a set exists, we setW := W ∪ {c1, . . . , cℓ} and we remove fromA′ all ballotsAi such that|Ai ∩W| ≥ ℓ(note that this includes all ballotsAi with {c1, . . . cℓ} ⊆ Ai). If at some point we have|W| ≤ k− ℓ and there is no{c1, . . . , cℓ} that satisfy our criterion or|W| > k− ℓ, we addan arbitraryk − |W| candidates fromC′ to W and returnW; if this does not happen,we terminate after having pickedk candidates.

Suppose for the sake of contradiction that for some profileA = (A1, . . . ,An)and somek > 0, ℓ-GAV outputs a committee that does not provideℓ-JR for (A, k).Then there exists a setN∗ ⊆ N with |N∗| ≥ ℓ n

k such that|⋂

i∈N∗ Ai | ≥ ℓ and, when

20 Haris Aziz et al.

ℓ-GAV terminates, every ballotAi such thati ∈ N∗ is still in A′. Consider somesubset of candidates{c1, . . . , cℓ} ⊆

⋂

i∈N∗ Ai . At every point in the execution ofℓ-GAVthis subset is unanimously approved by at least|N∗| ≥ ℓ n

k ballots inA′. As at leastone of {c1, . . . , cℓ} was not elected, at every stage the algorithm selected a set of ℓcandidates that was approved by at leastℓ n

k ballots (until more thank − ℓ candidateswere selected). Since at the end of each stage the algorithm removed fromA′ allballots containing the candidates that had been added toW at that stage, it followsthat altogether the algorithm has removed at least⌊ k

ℓ⌋ · ℓ n

k > ( kℓ− 1) · ℓ n

k = n − ℓ nk

ballots fromA′. This is a contradiction, since we assumed that, when the algorithmterminates, theℓ n

k ballots (Ai)i∈N∗ are still inA′.

For the problem of checking whether a given committee providesEJRfor a giveninput, we can establish a formal hardness result.

Theorem 14 Given a ballot profileA, a target committee size k, and a committee W,|W| = k, it is coNP-complete to check whether W provides EJR for(A, k).

Proof It is easy to see that this problem is in coNP: to show thatW does not provideEJRfor (A, k), it suffices to guess an integerℓ and a set of votersN∗ of size at leastℓ · n

k such that|⋂

i∈N∗ Ai | ≥ ℓ, but |Ai ∩W| < ℓ for all i ∈ N∗.To prove coNP-completeness, we reduce the classic Balanced Biclique problem

(Garey and Johnson 1979, [GT24]) to the complement of our problem. An instanceof Balanced Biclique is given by a bipartite graph (L,R,E) with partsL andR andedge setE, and an integerℓ; it is a “yes”-instance if we can pick subsets of verticesL′ ⊆ L andR′ ⊆ R so that|L′| = |R′| = ℓ and (u, v) ∈ E for eachu ∈ L′, v ∈ R′;otherwise, it is a “no”-instance.

Given an instance〈(L,R,E), ℓ〉 of Balanced Biclique with R = {v1, . . . , vs}, wecreate an instance of our problem as follows. Assume withoutloss of generality thats ≥ 3, ℓ ≥ 3. We construct 4 pairwise disjoint sets of candidatesC0, C1, C′1, C2, sothatC0 = L, |C1| = |C′1| = ℓ − 1, |C2| = sℓ + ℓ − 3s, and setC = C0 ∪C1 ∪C′1 ∪C2.We then construct 3 sets of votersN0, N1, N2, so thatN0 = {1, . . . , s}, |N1| = ℓ(s− 1),|N2| = sℓ+ ℓ− 3s (note that|N2| > 0 as we assume thatℓ ≥ 3). For eachi ∈ N0 we setAi = {u j | (u j, vi) ∈ E} ∪C1, and for eachi ∈ N1 we setAi = C0 ∪C′1. The candidatesin C2 are matched to voters inN2: each voter inN2 approves exactly one candidatein C2, and each candidate inC2 is approved by exactly one voter inN2. Denote theresulting list of ballots byA. Finally, we setk = 2ℓ − 2, and letW = C1 ∪C′1. Notethat the number of votersn is given bys+ ℓ(s− 1)+ sℓ+ ℓ− 3s= 2s(ℓ− 1), son

k = s.Suppose first that we started with a “yes”-instance of Balanced Biclique, and

let (L′,R′) be the respectiveℓ-by-ℓ biclique. LetC∗ = L′, N∗ = N0 ∪ N1. Then|N∗| = ℓs, all voters inN∗ approve all candidates inC∗, |C∗| = ℓ, but each voter inN∗ is only represented byℓ− 1 candidates inW. Hence,W fails to provideℓ-justifiedrepresentation for (A, k).

Conversely, suppose thatW fails to provideEJRfor (A, k). That is, there exists avalue j > 0, a setN∗ of js voters and a setC∗ of j candidates so that all voters inN∗

approve of all candidates inC∗, but for each voter inN∗ at most j of her approvedcandidates are inW. Note that, sinces> 1, we haveN∗ ∩N2 = ∅. Further, each voterin N \ N2 is represented byℓ − 1 candidates inW, so j ≥ ℓ. As N∗ = js ≥ ℓs ≥ s, it

Justified Representation in Approval-Based Committee Voting 21

follows that|N∗ ∩ N0| ≥ ℓ, |N∗ ∩ N1| > 0. SinceN∗ contains voters from bothN0 andN1, it follows thatC∗ ⊆ C0. Thus, there are at leastℓ voters inN∗ ∩ N0 who approvethe samej ≥ ℓ candidates inC0; any set ofℓ such voters andℓ such candidatescorresponds to anℓ-by-ℓ biclique in the input graph. ⊓⊔

6 Variants of Justified Representation

The definition ofJRrequires that if there is a group of⌈ nk⌉ voters who jointly approve

some candidate, then the elected committee has to contain atleast one candidate ap-proved bysomemember of this group. This condition may appear to be too weak;it may seem more natural to require thateverygroup member approves some can-didate in the committee, or—stronger yet—that the committee contains at least onecandidate approved byall group members. This intuition is captured by the followingdefinitions.

Definition 3 Given a ballot profile (A1, . . . ,An) and a target committee sizek, we saythat a committeeW of sizek provides

– semi-strong justified representation for(A, k) if for each groupN∗ ⊆ N with|N∗| ≥ n

k and⋂

i∈N∗ Ai , ∅ it holds thatW∩ Ai , ∅ for all i ∈ N∗.– strong justified representation for(A, k) if for each groupN∗ ⊆ N with |N∗| ≥ n

kand

⋂

i∈N∗ Ai , ∅ it holds thatW∩ (∩i∈N∗Ai) , ∅.

By definition, a committee providing strong justified representation also providessemi-strong justified representation, and a committee providing semi-strong justifiedrepresentation also provides (standard) justified representation.

However, it turns out that satisfying these stronger requirements is not always fea-sible: there are ballot profiles for which no committee provides semi-strong justifiedrepresentation.

Example 7Let k = 3 and consider the following profile withn = 9 andC =

{a, b, c, d}.

A1 = A2 = {a} A3 = {a, b} A4 = {b} A5 = {b, c}

A6 = {c} A7 = {c, d} A8 = A9 = {d}

For each candidatex ∈ C, there arenk = 3 voters such that∩iAi = {x}, and at least

one of those voters hasAi = {x}. Thus, a committee that satisfies semi-strong justifiedrepresentation would have to contain all four candidates, which is impossible.

While Example 7 shows that no approval-based voting rule canalways find a commit-tee that provides strong or semi-strong justified representation, it may be interestingto identify voting rules that output such committees whenever they exist.

Finally, we remark that strong justified representation does not implyEJR.

22 Haris Aziz et al.

Example 8Let C = {a, b, c, d, e}, n = 4, k = 4, and consider the following ballotprofile.

A1 : {a, b} A2 : {a, b} A3 : {c} A4 : {d, e}

EJR requires that we choose botha andb, but {a, c, d, e} provides strong justifiedrepresentation.

7 Related Work

It is instructive to compareJRandEJRto alternative approaches towards fair repre-sentation, such asrepresentativeness(Duddy 2014) andproportional justified repre-sentation(Sanchez-Fernandez et al. 2016).

Duddy (2014) proposes the notion ofrepresentativeness, which applies to prob-abilistic voting rules. The property Duddy proposes is incomparable withJR: in sit-uations he considers (k = 2, n voters approvex, n + 1 voters approvey andz), JRrequires that one ofy andz should be selected, whereas Duddy requiresx to be se-lected with positive probability. Both are reasonable requirements, but they addressdifferent concerns. Duddy’s axiom say nothing about situationswhere voters are splitequally (say,n voters approve{x, y}, n voters approve{z, t}), whereasJRrequires thateach voter is represented. Another obvious difference is that he allows for randomizedrules.

Very recently (after the conference version of our paper waspublished),Sanchez-Fernandez et al. (2016) came up with the notion ofproportional justifiedrepresentation (PJR), which can be seen as an alternative toEJR. A committee issaid to providePJRfor a ballot profile (A1, . . . ,An) over a candidate setC and a tar-get committee sizek if, for every positive integerℓ, ℓ ≤ k, there does not exist a set ofvotersN∗ ⊆ N with |N∗| ≥ ℓ · n

k such that|⋂

i∈N∗ Ai | ≥ ℓ, but |(⋃

i∈N∗ Ai) ∩W| < ℓ. Incontrast toEJR, thePJRcondition does not require one of the voters inN∗ to haveℓrepresentatives. Rather, a committee providesPJRas long as it containsℓ candidatesthat are approved by (possibly different) voters inN∗, for every groupN∗ satisfyingthe size and cohesiveness constraints. An attractive feature ofPJRis that it is compat-ible with the idea ofperfect representation: a committeeW provides perfect represen-tation for a group ofn voters and a target committee sizek if n = ksfor some positiveintegersand the voters can be split intok pairwise disjoint groupsN1, . . . ,Nk of sizes each in such a way that there is a one-to-one mappingµ : W → {N1, . . . ,Nk} suchthat for each candidatea ∈ W all voters inµ(a) approvea. Sanchez-Fernandez et al.(2016) prove that every committee that provides perfect representation also providesPJR; in contrast,EJRmay rule out all committees that provide perfect representation,as illustrated by Example 4. It is easily seen thatPJR is a weaker requirement thanEJR, and a stronger one thanJR. Interestingly, Sanchez-Fernandez et al. (2016) showthat many results that we have established forEJRalso hold forPJR: in particular,w-RAVviolatesPJRfor every weight vectorw, andw-PAV satisfiesPJRif and onlyif w = (1, 1

2 ,13 , . . . ).

Justified Representation in Approval-Based Committee Voting 23

8 Conclusions

JR EJR ComplexityRule

AV – – in PSAV – – in PMAV – – NP-hardRAV – – in PGAV X – in PGAVT

X –∗ in PMonAV X – NP-hardPAV X X NP-hard

Table 1 Satisfaction ofJR andEJRand complexity of approval-based voting rules; the superscript ‘∗’indicates that the rule fails the respective axiom for some way of breaking intermediate ties.

We have formulated a desirable property of approval-based committee selection rules,which we called justified representation (JR). While JR is fairly easy to satisfy, itturns out that many well-known approval-based rules fail it. A prominent exceptionis thePAV rule, which also satisfies a stronger version of this property, namely ex-tended justified representation (EJR). Indeed,EJRcharacterizesPAVwithin the classof w-PAV rules, and we are not aware of any other natural voting rule that satisfiesEJR irrespective of the tie-breaking rule (of course, we can construct voting rulesthat differ from PAV, yet satisfyEJR, by modifying the output ofPAV on profileson whichEJRplaces no constraints on the output). Perhaps the most pressing openquestion suggested by our work is whether there is an efficient algorithm for findinga committee that providesEJRfor a given profile. In particular, we would like to un-derstand whether we can break ties in the execution ofGAVT to always produce sucha committee, and whether some tie-breaking rule with this property is polynomial-time computable. Also, it would be interesting to see ifEJR, in combination withother natural axioms, can be used to axiomatizePAV. Concerning (semi-)strong jus-tified representation, an interesting algorithmic problemis whether there are efficientalgorithm for checking the existence of committees satisfying these requirements.

Justified representation can also be used to formulate new approval-based rules.We mention two rules that seem particularly attractive:

The utilitarian (E)JR rule returns a committee that, among all committeesthat satisfy (E)JR, has the highestAV score.The egalitarian (E)JR rule returns a committee that, among all committeesthat satisfy (E)JR, maximizes the number of representatives of the voter whohas the least number of representatives in the winning committee.

The computational complexity of winner determination for these rules is an interest-ing problem.

SincePAV is NP-hard to compute, our study also provides additional motiva-tion for the use of approximation and parameterized algorithms to computePAVout-comes. Finally, analyzing the compatibility ofJR with other important properties,

24 Haris Aziz et al.

such as, e.g., strategyproofness for dichotomous preferences, is another avenue offuture research.

Acknowledgements The authors thank the anonymous reviewers ofMultidisciplinary Workshop on Ad-vances in Preference Handling (MPREF 2014), and theTwenty-Ninth AAAI Conference (AAAI 2015)fortheir helpful feedback on earlier versions of the paper. We further thank Martin Lackner and Piotr Skowronfor valuable discussions.

Brill, Conitzer, and Freeman were supported by NSF and ARO under grants CCF-1101659, IIS-0953756, CCF-1337215, W911NF-12-1-0550, and W911NF-11-1-0332, by a Feodor Lynen research fel-lowship of the Alexander von Humboldt Foundation, and by COST Action IC1205 on ComputationalSocial Choice. Brill and Elkind were partially supported byERC-StG 639945. Walsh also receives sup-port from the Asian Office of Aerospace Research and Development (AOARD 124056) andthe GermanFederal Ministry for Education and Research through the Alexander von Humboldt Foundation.

References

H. Aziz, S. Gaspers, J. Gudmundsson, S. Mackenzie, N. Mattei, and T. Walsh. Computational aspects ofmulti-winner approval voting. InProceedings of the 14th International Conference on AutonomousAgents and Multiagent Systems (AAMAS), pages 107–115, 2015.

N. Betzler, A. Slinko, and J. Uhlmann. On the computation of fully proportional representation.Journalof Artificial Intelligence Research, 47:475–519, 2013.

S. J. Brams. Preface. In J.-F. Laslier and M. R. Sanver, editors,Handbook on Approval Voting, Studies inChoice and Welfare, pages vii–ix. Springer, 2010.

S. J. Brams and P. C. Fishburn.Approval Voting. Springer, 2nd edition, 2007.S. J. Brams and D. M. Kilgour. Satisfaction approval voting.In R. Fara, D. Leech, and M. Salles, editors,

Voting Power and Procedures, Studies in Choice and Welfare, pages 323–346. Springer, 2014.S. J. Brams, D. M. Kilgour, and M. R. Sanver. How to elect a representative committee using approval

balloting. In B. Simeone and F. Pukelsheim, editors,Mathematics and democracy: Recent advancesin voting systems and collective choice, pages 83–96. Springer, 2006.

S. J. Brams, D. M. Kilgour, and R. M. Sanver. A minimax procedure for electing committees.PublicChoice, 132(3-4):401–420, 2007.

J. Byrka and K. Sornat. PTAS for minimax approval voting. InProceedings of the 10th InternationalConference on Web and Internet Economics (WINE), pages 203–217, 2014.

I. Caragiannis, D. Kalaitzis, and E. Markakis. Approximation algorithms and mechanism design for min-imax approval voting. InProceedings of the 24th AAAI Conference on Artificial Intelligence (AAAI),pages 737–742, 2010.

I. Caragiannis, C. Kaklamanis, N. Karanikolas, and A. D. Procaccia. Socially desirable approximationsfor Dodgson’s voting rule.ACM Transactions on Algorithms, 10(2), 2014.

B. Chamberlin and P. Courant. Representative deliberations and representative decisions: Proportionalrepresentation and the Borda rule.American Political Science Review, 77(3):718–733, 1983.

D. Cornaz, L. Galand, and O. Spanjaard. Bounded single-peaked width and proportional representation.In Proceedings of the 20th European Conference on Artificial Intelligence (ECAI), pages 270–275,2012.

M. Davis, M. E. Orrison, and F. E. Su. Voting for committees inagreeable societies.ContemporaryMathematics, 624:147–157, 2014.

H. R. Droop. On methods of electing representatives.Journal of the Statistical Society of London, 44(2):141–196, 1881.

C. Duddy. Electing a representative committee by approval ballot: An impossibility result. EconomicLetters, 124:14–16, 2014.

M. Dummett.Voting Procedures. Oxford University Press, 1984.E. Elkind and M. Lackner. Structure in dichotomous preferences. InProceedings of the 24th International

Joint Conference on Artificial Intelligence (IJCAI), pages 2019–2025, 2015.E. Elkind, P. Faliszewski, P. Skowron, and A. Slinko. Properties of multiwinner voting rules. InProceed-

ings of the 13th International Conference on Autonomous Agents and Multiagent Systems (AAMAS),pages 53–60, 2014.

Justified Representation in Approval-Based Committee Voting 25

E. Elkind, J. Lang, and A. Saffidine. Condorcet winning sets.Social Choice and Welfare, 44(3):493–517,2015.

U. Endriss. Sincerity and manipulation under approval voting. Theory and Decision, 74(4):335–355, 2013.P. C. Fishburn and A. S. Pekec. Approval voting for committees: Threshold approaches. Technical report,

2004.M. Garey and D. Johnson.Computers and Intractability: A Guide to the Theory of NP-Completeness. W.

H. Freeman and Company, 1979.D. M. Kilgour. Approval balloting for multi-winner elections. InHandbook on Approval Voting, chapter 6.

Springer, 2010.D. M. Kilgour and E. Marshall. Approval balloting for fixed-size committees. In D. S. Felsenthal and

M. Machover, editors,Electoral Systems, Studies in Choice and Welfare, pages 305–326. Springer,2012.

J.-F. Laslier and M. R. Sanver, editors.Handbook on Approval Voting. Studies in Choice and Welfare.Springer, 2010.

R. LeGrand, E. Markakis, and A. Mehta. Some results on approximating the minimax solution in approvalvoting. In Proceedings of the 6th International Conference on Autonomous Agents and Multi-AgentSystems (AAMAS), pages 1185–1187, 2007.

T. Lu and C. Boutilier. Budgeted social choice: From consensus to personalized decision making. InProceedings of the 22nd International Joint Conference on Artificial Intelligence (IJCAI), pages 280–286. AAAI Press, 2011.

R. Meir, A. D. Procaccia, J. S. Rosenschein, and A. Zohar. Complexity of strategic behavior in multi-winner elections.Journal of Artificial Intelligence Research, 33:149–178, 2008.

N. Misra, A. Nabeel, and H. Singh. On the parameterized complexity of minimax approval voting. InProceedings of the 14th International Conference on Autonomous Agents and Multiagent Systemsn(AAMAS), pages 97–105, 2015.

B. Monroe. Fully proportional representation.American Political Science Review, 89(4):925–940, 1995.A. Procaccia, J. Rosenschein, and A. Zohar. On the complexity of achieving proportional representation.

Social Choice and Welfare, 30(3):353–362, 2008.L. Sanchez-Fernandez, N. Fernandez, J. A. Fisteus, and P. Basanta-Val. Some notes on justified represen-

tation. InProceedings of the 10th Multidisciplinary Workshop on Advances in Preference Handling(MPREF), 2016.

P. Skowron, P. Faliszewski, and J. Lang. Finding a collective set of items: From proportional multirep-resentation to group recommendation. InProceedings of the 29th AAAI Conference on ArtificialIntelligence (AAAI), pages 2131–2137, 2015a.

P. Skowron, P. Faliszewski, and A. Slinko. Achieving fully proportional representation: Approximabilityresults.Artificial Intelligence, 222:67–103, 2015b.

P. Skowron, L. Yu, P. Faliszewski, and E. Elkind. The complexity of fully proportional representation forsingle-crossing electorates.Theoretical Computer Science, 569:43–57, 2015c.

P. K. Skowron, P. Faliszewski, and J. Lang. Finding a collective set of items: From proportional mul-tirepresentation to group recommendation. InProceedings of the 29th International Conference onAutonomous Agents and Multi-Agent Systems (AAMAS), pages 2131–2137. AAAI Press, 2015d.

T. N. Thiele. Om flerfoldsvalg. InOversigt over det Kongelige Danske Videnskabernes Selskabs Forhan-dlinger, pages 415–441. 1895.

N. Tideman. The single transferable vote.Journal of Economic Perspectives, 9(1):27–38, 1995.N. Tideman.Collective Decisions and Voting. Ashgate, 2006.N. Tideman and D. Richardson. Better voting methods throughtechnology: The refinement-manageability

trade-off in the single transferable vote.Public Choice, 103(1-2):13–34, 2000.