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Computational Aspects of Multi-Winner Approval Voting

Haris Aziz and Serge GaspersNICTA and UNSWSydney, Australia

Joachim GudmundssonUniversity of Sydney and NICTA

Sydney, Australia

Simon Mackenzie, Nicholas Mattei and Toby WalshNICTA and UNSWSydney, Australia

Abstract

We study computational aspects of three prominent votingrules that use approval ballots to elect multiple winners.These rules aresatisfaction approval voting, proportional ap-proval voting, andreweighted approval voting. We first showthat computing the winner for proportional approval votingisNP-hard, closing a long standing open problem. As none ofthe rules are strategyproof, even for dichotomous preferences,we study various strategic aspects of the rules. In particu-lar, we examine the computational complexity of computinga best response for both a single agent and a group of agents.In many settings, we show that it is NP-hard for an agent oragents to compute how best to vote given a fixed set of ap-proval ballots from the other agents.

IntroductionThe aggregation of possibly conflicting preference is acentral problem in artificial intelligence (Conitzer 2010).Agents express preferences over candidates and a voting ruleselects a winner or winners based on these preferences. Wefocus here on rules that selectk winners wherek is fixedin advance. This covers settings including parliamentaryelections, the hiring of faculty members, and movie rec-ommendation systems (Obraztsova, Zick, and Elkind 2013).Multi-winner rules can also be used to select a com-mittee (Ratliff 2006; LeGrand, Markakis, and Mehta 2007;Elkind, Lang, and Saffidine 2011).

Generally, in approval-based voting rules, an agent ap-proves of (votes for) a subset of the candidates. The moststraightforward way to aggregate these votes is to have ev-ery approval for a candidate contribute one point to that can-didate; yielding the rule known as Approval Voting (AV ).Approval Voting is an obvious type of voting rule to extendfrom the single winner to the multiple winner case. Un-like, say, plurality voting where agents nominate just theirmost preferred candidate, approval ballots permit agents toidentify multiple candidates that they wish to win. Approvalvoting has many desirable properties in the single winnercase (Fishburn 1978; Brams, Kilgour, and Sanver 2006), in-cluding its ‘simplicity, propensity to elect Condorcet win-ners (when they exist), its robustness to manipulation andits monotonicity’ (Laslier and Sanver 2010). However for

Copyright c© 2014, Association for the Advancement of ArtificialIntelligence (www.aaai.org). All rights reserved.

the case of multiple winners, the merits ofAV are ‘lessclear’ (Laslier and Sanver 2010). In particular, for the multi-winner case,AV does address more egalitarian concernssuch as proportional representation.

Over the years, various methods for counting approvalshave been introduced in the literature, each attempt to ad-dress the fairness concerns when usingAV for multiplewinners (Kilgour 2010). One could, for instance, reducethe weight of an approval from a particular agent based onhow many other candidates the agent approves of have beenelected, as inProportional Approval Voting (PAV ). Anotherway to ensure diversity across agents is vote across a set ofrounds. In each round, the candidate with the most approvalswins. However, in each subsequent round we decrease theweight of agents who have already had a candidate electedin earlier rounds; this method is implemented inReweightedApproval Voting (RAV ). Finally,Satisfaction Approval Vot-ing (SAV ) modulates the weight of approvals with a satis-faction score for each agent, based on the ratio of approvedcandidates appearing in the committee to the agent’s totalnumber of approved candidates.

These approaches to generalizing approval voting to thecase of multiple winners each have their own benefits anddrawbacks. Studying the positive or negative properties ofthese multi-winner rules can help us make informed, objec-tive decisions about which generalization is better depend-ing on the situations to which we are applying a particu-lar multi-winner rule (Elkind et al. 2014). ThoughAV isthe most widely known of these rules,RAV has been used,for example, in elections in Sweden. Rules other thanAV

may have better axiomatic properties in the multi-winnersetting and thus, motivate our study. For example, eachof PAV , SAV , andRAV have a more egalitarian objec-tive thanAV . Steven Brams, the main proponent of AV insingle winner elections, has argued thatSAV is more suit-able for equitable representation in multiple winner elections(Brams and Kilgour 2010).

We undertake a detailed study of computational aspects ofSAV , PAV , andRAV . We first consider the computationalcomplexity of computing the winner, a necessary result ifany voting rule is expected to be used in practice. AlthoughPAV was introduced over a decade ago, a standing openquestion has been the computational complexity of deter-mining the winners, having only been referred to as “com-

putationally demanding” before (Kilgour 2010). We closethis standing open problem, showing that winner determina-tion forPAV is NP-hard. Our reduction applies to a host ofapproval based, multi-winner rules in which the scores con-tributed to an approved candidate by an agent diminish asadditional candidates approved by the agent are elected tothe committee.

We then consider strategic voting for these rules. Weshow that, even with dichotomous preferences,SAV , PAVandRAV are not strategyproof. That is, it may be beneficialfor agents to mis-report their true preferences. We thereforeconsider computational aspects of manipulation. We provethat finding the best response given the preferences of otheragents is NP-hard under a number of conditions forPAV ,RAV , andSAV . In particular, we examine the complexityof checking whether an agent or a set of agents can makea given candidate or a set of candidates win. These resultsoffer support forRAV overPAV or SAV as it is the onlyrule for which winner determination is computationally easybut manipulation is hard.

Related WorkAn important branch of social choice concerns determin-ing how and when agents can benefit by misreportingtheir preferences. In computational social choice, thisproblem is often studied through the lens of compu-tational complexity (Bartholdi, Tovey, and Trick 1989;Faliszewski and Procaccia 2010;Faliszewski, Hemaspaandra, and Hemaspaandra 2010).If it is computationally hard for an agent to compute abeneficial misreporting of their preferences for a particularvoting rule, the rule is said to be resistant to manipulation. Ifit is computationally difficult to compute a misreport, agentsmay decide to be truthful, since they cannot always easilymanipulate. Connections have been made between manip-ulation and other important questions in social choice suchas deciding when to terminate preference elicitation anddetermining possible winners (Konczak and Lang 2005).

Surprisingly, there has only been limited considerationof computational aspects of multi-winner elections. Excep-tions include work by Meir et al. (2008) which considerssingle non-transferable voting, approval voting,k-approval,cumulative voting and the proportional schemes of Monroe,and of Chamberlin and Courant. Most relevant to our studyis that for approval voting, Meir et al. prove that manipu-lation with general utilities and control by adding/deletingcandidates are both polynomial to compute, but control byadding/deleting agents is NP-hard. Another work that con-siders computational aspects of multi-winner elections isObraztsova, Zick, and Elkind (2013), but their study is lim-ited to k-approval and scoring rules. Finally, the controland bribery problems forAV and two other approval vot-ing variants are well catalogued by Baumeister et al. (2010),demonstrating thatAV is generally resistant to bribery butsusceptible to most forms of control when voters have di-chotomous utility functions.

The Handbook of Approval Voting discusses var-ious approval-based multi-winner rules including

SAV , PAV and RAV . Another prominent multi-winner rule in the Handbook isminimax approvalvoting (Brams, Kilgour, and Sanver 2007). Each agent’sapproval ballot and the winning set can be seen as abinary vector. Minimax approval voting selects the setof k candidates that minimizes the maximum Ham-ming distance from the submitted ballots. Althoughminimax approval voting is a natural and elegant rule,LeGrand et al. (2007) showed that computing the winnerset is unfortunately NP-hard. Strategic issues and ap-proximation questions for minimax approval voting arecovered in (Caragiannis, Kalaitzis, and Markakis 2010) and(Gramm, Niedermeier, and Rossmanith 2003) where theproblem is known as the “closest string problem.”

The area of multi-winner approval voting is closely re-lated to the study of proportional representation when se-lecting a committee (Skowron et al. 2013b; 2013a). Ideasfrom committee selection have therefore been used in com-putational social choice to ensure diversity when selecting acollection of objects (Lu and Boutilier 2011). Understand-ing approval voting schemes which select multiple winners,as the rules we consider often do, is an important areain social choice with applications in a variety of settingsfrom committee selection to multi-product recommendation(Elkind et al. 2014).

Formal Background

We consider the social choice setting(N,C) whereN ={1, . . . , n} is the set of agents andC = {c1, . . . , cm} isthe set of candidates. Each agenti ∈ N has a completeand transitive preference relation%i overC. Based on thesepreferences, each agent expresses an approval ballotAi ⊂ Cthat represents the subset of candidates that he approves of,yielding a set of approval ballotsA = {A1, . . . , An}. Wewill consider approval-based multi-winner rules that takeasinput (C,A, k) and return the subsetW ⊆ C of sizek thatis the winning set.

Approval Voting (AV )

AV finds a setW ⊆ C of sizek that maximizes the to-tal scoreApp(W ) =

∑i∈N |W ∩Ai|. That is, the set of

AV winners are those candidates receiving the most pointsacross all submitted ballots.AV has been adopted by sev-eral academic and professional societies such as the Amer-ican Mathematical Society (AMS), the Institute of Electri-cal and Electronics Engineers (IEEE), and the InternationalJoint Conference on Artificial Intelligence.

Satisfaction Approval Voting (SAV )

An agent’s satisfaction is the fraction of his or her approvedcandidates that are elected.SAV maximizes the sum of suchscores. Formally,SAV finds a setW ⊆ C of sizek thatmaximizesSat(W ) =

∑i∈N

|W∩Ai||Ai|

. The rule was pro-posed by (Brams and Kilgour 2010) with the aim of repre-senting more diverse interests thanAV .

Proportional Approval Voting (PAV )

In PAV , an agent’s satisfaction score is1+1/2+1/3 · · ·1/jwherej is the number of his or her approved candidates thatare selected inW . Formally,PAV finds a setW ⊆ C of sizek that maximizes the total scorePAV (W ) =

∑i∈N r(|W ∩

Ai|) wherer(p) =∑p

j=11j. PAV was proposed by the

mathematician Forest Simmons in 2001 and captures theidea of diminishing returns — an individual agent’s pref-erences should count less the more he is satisfied.

Reweighted Approval Voting (RAV )

RAV convertsAV into a multi-round rule, selecting a can-didate in each round and then reweighing the approvals forthe subsequent rounds. In each of thek rounds, we se-lect an unelected candidate to add to the winning setWwith the highest “weight” of approvals. In each round wereweight each agents approvals, assigning for alli ∈ Nthe weight 1

1+|W∩Ai|to agenti. RAV was invented by

the Danish polymath Thorvald Thiele in the early 1900’s.RAV has also been referred to as “sequential proportionalAV” (Brams and Kilgour 2010), and was used briefly inSweden during the early 1900’s.

Tie-breaking is an important issue to consider when in-vestigating the complexity of manipulation and winner de-termination problems as it can have a significant impact onthe complexity of reasoning tasks (Obraztsova et al. 2011,Obraztsova and Elkind 2011, Aziz et al. 2013). We make theworst-case assume that a tie-breaking rule takes the form ofa linear order over the candidates that is given as part of theproblem input and favors the preferred candidate; as is com-mon in the literature on manipulation (Bartholdi et al. 1989,Faliszewski and Procaccia 2010, Faliszewski et al. 2010).Note that many of our proofs are independent of the tie-breaking rule, in which case the hardness results transfer toany arbitrary tie-breaking rule.

Winner DeterminationWe first examine one of the most basic computational ques-tions, computing the winners of a voting rule.

Name: WINNER DETERMINATION (WD).Input: An approval-based voting ruleR, a set of approvalballotsA over the setC of candidates, and a committee sizek ∈ N.Question: What is the winning set,W ⊆ C, with |W | = k?

Firstly, we observe that WD is polynomial-time com-putable for SAV , RAV , and AV . Although RAV ispolynomial-time to compute, it has been termed “computa-tionally difficult” to analyze in (Kilgour 2010). We providesupport for this claim by showing that computing a best re-sponse forRAV is NP-hard (Theorem 11). We close thecomputational complexity of WD forPAV in this section.

Theorem 1 WD for PAV is NP-complete, even if eachagent approves of two candidates.

Proof: The problem is in NP since we merely need as wit-ness a set of candidates withPAV scores.

To show hardness we give a reduction from the NP-hardINDEPENDENT SET problem (Garey and Johnson 1979):Given (G, t), whereG = (V,E) is an arbitrary graph andt an integer, is there an independent set of sizet in G. Anindependent set is a subset of verticesS ⊆ V such that noedge ofG has both endpoints inS. For a graphG, we build aPAV instance for which a winning committee of sizet cor-responds to an independent set inG of sizet, and vice-versa.

Consider a graphG = (V,E), and define the followingPAV instance,(N,C,A, k): We have a set of agentsN anda set of candidatesC. For each vertexv ∈ V , we createdeg(G)− deg(v) ‘dummy’ candidates inC, wheredeg(G)is the maximum degree ofG, deg(G) > 1, anddeg(v) thedegree of vertexv. For eachv ∈ V , we also create anothercandidate inC, labeledCv. We create an agent inN foreach edgee ∈ E. For each vertex we also createdeg(G)−deg(v) agents. Each of the edge agents approves of the twocandidates corresponding to the vertices connected by theedge. Each vertex agent associated with vertexv approvesof Cv and one of the dummy candidates associated withv,thus each dummy candidate has exactly one agent approvingof him. We also setk = t.

We will show that there is a committee of sizek = t scor-ing a total approval of at leasts = deg(G) · t if and only ifGhas an independent set of sizet. First, note that adding a can-didate to a committee increases the total score of the com-mittee by at mostdeg(G), since at mostdeg(G) agents seetheir satisfaction score rise by at most one. Also, if addinga candidatec to a committee increases the total score of thecommittee by exactlydeg(G), thenc corresponds to a ver-tex inG, since each dummy vertex is approved by only oneagent, and the vertex corresponding toc is not adjacent toa vertex corresponding to any other candidate in the com-mittee. Thus, the candidates in a committee of sizek = tscoring a total approval ofs correspond to an independentset of sizet in G and vice-versa. ✷

The reduction in this proof actually implies a stronger re-sult, namely that, unless FPT=W[1], WD forPAV cannot besolved in timef(k) ·mO(1), for any functionf , even if eachagent approves of two candidates. This is because it is a pa-rameterized reduction where the parameterk is a function ofthe parametert for INDEPENDENTSET, which is W[1]-hardfor parametert (Downey and Fellows 2013). Thus, even forrelatively small committee sizes, a factormk in the runningtime seems unavoidable.

Corollary 1 WD for PAV is W[1]-hard.

Strategic VotingAs in the single winner case, agents may benefit from mis-reporting their true preferences when electing multiple win-ners. We consider the special case of dichotomous prefer-ences where each agent has utility 0 or 1 for electing eachparticular candidate. In this case, we say that a multi-winnerapproval-based voting rule isstrategyproof if and only ifthere does not exist an agent who has an incentive to approve

a candidate with zero utility and does not have an incen-tive to disapprove a candidate for whom the agent has utility1. We note that fordichotomous preferences, AV is strate-gyproof (if lexicographic tie-breaking is used). However,itis polynomial-time manipulable for settings with more gen-eral utilities (Meir, Procaccia, and Rosenschein 2008). Onthe other hand,SAV , PAV andRAV are not.

Theorem 2 SAV , PAV and RAV are not strategyproofwith dichotomous preferences.

Proof: We treat each case separately. We assume that tiesare always broken lexicographically witha ≻ b ≻ c, e.g.,{a, b} is preferred to{a, c}.

(i). For SAV , assumek = 2, C = {a, b, c}, and agent1 hasnon-zero utility only fora andb. Let,A2 = {a}, A3 = {a}, A4 = {a}, A5 = {c}, A6 = {b, c}.

The outcome is{a, c} if A1 = {a, b}, but if agent1 onlyapprovesb, the outcome is{a, b} which has the maximumutility and is preferred by tie-breaking.

(ii). For PAV , consider the same setting but now with the fol-lowing votes:A2 = {b}, A3 = {a, c}, A4 = {a, c}, A5 = {c}.

The outcome{a, b} is only possible if agent 1 approvesonly b. Otherwise it is{a, c}.

(iii). For RAV , consider the same setting but now with the fol-lowing votes:A2 = {a}, A3 = {a}, A4 = {a}, A5 = {c}, A6 = {b, c}.

The outcome is{a, c} for all reported preferences of agent1 A1 = {b}, in which case the outcome isW = {a, b}.

This completes the proof. ✷

With SAV , PAV andRAV , it can therefore be beneficialfor agents to vote strategically. Next, we consider the com-putational complexity of computing such strategic votes.

Name: WINNER MANIPULATION (WM)Input: An approval-based voting ruleR, a set of approvalballotsA over the setC of candidates, a winning set sizek, anumber of agentsj still to vote, and a preferred candidatep.Question: Are therej additional approval ballots so thatpis in the winning setW underR?

Name: WINNING SET MANIPULATION (WSM).Input: An approval-based voting ruleR, a set of approvalballotsA over the setC of candidates, a winning set sizek, a number of agentsj still to vote, and a set of preferredcandidatesP ⊆ C.Question: Are therej additional approval ballots such thatP is the winning set of candidates underR?

We note if WM or WSM is NP-hard for a single agent(j = 1), then the more general problem of maximizing theutility of an agent is also NP-hard. ForAV , the utility max-imizing best response of a single agent can be computed inpolynomial time (Meir et al. 2008). We note our definitionshave additive utilities, and the question is to castj votes soas to maximize thetotal utility. This is more general thanWM/WSM, since a simple reduction from gives utility 1 tothe candidates in P (or{p}), and 0 to all the other candidates.

Satisfaction Approval Voting (SAV )WM underSAV is polynomial-time solvable. The agentscast an approval ballot for just the preferred candidate. Thisis the best that they can do. If the preferred candidate doesnot win in this situation, then the preferred candidate cannever win. It follows that we can also construct the set ofcandidates that can possibly win in polynomial time. It ismore difficult to decide if a givenk-set of candidates canpossibly win. With certain voting rules, this problem sim-plifies if the optimal strategy ofj manipulating agents needto cast only one form of vote. This is not the case withSAV .

Theorem 3 To ensure a given set of candidates is selectedunder SAV , the manipulating coalition may need to cast aset of votes that are not all identical.

Proof: Supposek = 3 andC = {a, b, c, d, e, f, g}, oneagent approves botha andb, and three agents approved, e,f andg. If there are two more agents who wanta, b andc tobe elected, then one agent needs to approvec and the otherbotha andb, or one agent needs to approvea andb, and theothera andc. ✷

This makes it difficult to decide how a coalition of agentsmust vote. In fact, it is intractable in general to decide ifa given set of candidates can be made winners. We omitthe proof for space but observe that it is a reduction to thepermutation sum problem as in the NP-hardness proof forBorda manipulation with two agents (Davies et al. 2011).

Theorem 4 WSM is NP-hard for SAV .

The proof requires both the number of agents and the sizeof the winning set to grow. An open question is the computa-tional complexity when we bound either or both the numberof agents and the size of the winning set. We can also showthat it is intractable to manipulateSAV destructively.

We can adapt the proof for Theorem 4 to show the follow-ing statement as well.

Theorem 5 For SAV , it is not possible for a single manip-ulator to compute in polynomial time a vote that maximizeshis utility, unless P=NP.

Hence, in the case of multi-winner voting rules, de-structive manipulation can be computationally harderthan constructive manipulation. This contrasts tothe single winner case where destructive manipula-tion is often easier than constructive manipulation(Conitzer, Sandholm, and Lang 2007). It also followsfrom Theorem 5 that it is intractable to manipulateSAV toensure a given utility or greater.

We next turn to the special cases of a single agent and apair of agents. Winning set manipulation is polynomial witheither one or two agents left to vote. This result holds evenif the size of the winning set is not bounded (e.g.k = m/2).The proofs are one agent is omitted for space, however weobserve that the proof of the following Theorem can be ex-tended for the case where a setP has to be a subset of thewinning set.

Theorem 6 If two agents remain to vote, WSM is polyno-mial for SAV .

Proportional Approval Voting (PAV )The proof of the NP-hardness of WINNER DETERMINA-TION for PAV can be adapted to also show that basic ma-nipulation problems are coNP-hard forPAV .

Theorem 7 For PAV , WM and WSM are coNP-hard,even if there is no manipulator.

In Theorem 7 the hardness of WM and WSM really comesfrom the hardness of WD, demonstrated by requiring nomanipulators. This result motivates us to investigate thesituation where a “real” manipulation is necessary, that is,whether a single manipulator can include a particular can-didate in the winning set, even if WD is polynomial-timecomputable for the underlyingPAV instance. While weconjecture this is hard, we can formally prove the follow-ing, slightly weaker, statement.

Theorem 8 For PAV , it is not possible for a single manip-ulator to compute in polynomial time a vote that maximizeshis utility, unless P = NP .

Reweighted Approval Voting (RAV )In RAV the decision for a single agent of whom to votefor in order to maximize his utility is not straightforward.Suppose we are selecting a committee of sizek = 2 withC = {a, b, c, d}:

A2 = {b, d}, A3 = {c, d}, A4 = {a, b, c, d}

A5 = A6 = {b, c, d}, A7 = {a, b}, A8 = {c}, A9 = {a}.

If the agent wants to electa to the committee then he mayneed to express preference formore than just his choice set.In the above example, if agent1, casts the ballotA1 = {a}then in Round 1b is elected, in Round 2c is elected. How-ever, if the agent casts the ballot{a, d} then in Round 1d iselected, and in Round 2a is elected.

Theorem 9 Under RAV , an agent who wants to includea single candidate in the committee may have incentive toapprove more candidates than P .

Furthermore, if the agent is attempting to fill a committeewith a preferred set of candidates, he may have incentivenot to approve some candidates so that they may be elected.Suppose we are selecting a committee of sizek = 3 withC = {a, b, c, d}, using lexicographic tie-breaking:

A2 = {b, d}, A3 = {c, d}, A4 = A5 = A6 = {b, c, d}

A7 = {b}, A8 = {c}, A9 = A10 = {a}.

If the agent has favored set{a, b, d} and he approves all ofthem, then in Round 1b is elected, in Round 2c is elected,and in Round 3a is elected. However, if the agent casts theballot {a, d} then in Round 1d is elected, in Round 2a iselected, and in Round 3b is elected, exactly the favored set.

If a manipulator wants to elect exactly a favored setPthen he must approve eitherP , or a subset of it.

WD WM WSM

AV in P in P in PSAV in P in P NP-hPAV NP-h coNP-h coNP-hRAV in P NP-h -

Table 1: Summary of computational results for approval-based multi-winner rules forWinnerDetermination,WinnerManipulation, andWinningSetManipulation.

Theorem 10 Under RAV , an agent who wants to elect anexact set of candidates will never have an incentive to ap-prove a superset of his preferred candidates, though he mayhave an incentive to approve a subset of them.

Theorem 11 For RAV , WM is NP-hard.

Proof: To show thatRAV is NP-hard to manipulate we re-duce from 3SAT. Given a instance of 3SAT withw variablesΦ = {φ1, . . . , φw}, t clausesΨ = {ψ1, . . . , ψt}, inducing2w literals{l1, . . . l2w}. We construct an instance ofRAV ,(C,A, k) where a manipulator’s preferred candidatep is inthe winning set if and only if there is an assignment to thevariables inΦ such that all clauses are satisfied.

For each variableφi introduce 2 candidates inC, corre-sponding to the positive and negative literal of that variable,and2n − i agents approving of the 2 candidates; note thatn ≫ w + t. For each clauseψj introduce two additionalnew candidates, corresponding to the clause being satisfiedor unsatisfied, along with2n − w − j new agents approv-ing of both the two new candidates. Additionally for eachclauseψj , we add an agent inA approving each of the can-didates that correspond to the positive and negative literalsin ψj ; this ensures that both the positive and negative lit-eral have the same weight of approval in the set of agents.We also need to add 2 agents approving of the candidatecorresponding to the negation of the clause to maintain theweighting. Finally, add an extra 2 candidates toC, a andb.We add 2 agents approving of the candidate correspondingto a clause being unsatisfied, and 2 agents approving of thethe candidate corresponding to each clause being satisfiedand approving ofb

Add t agents approving ofa. The size of the winningsetk is equal to|Φ| + |Ψ| + 1. Intuitively, the manipulatormust approve of a setting of all the variables in the original3SAT instance that satisfies all the clauses, plus the preferredcandidate. We can now see that the manipulating agent isonly capable of ensuring candidate a is elected by computinga solution to the initial 3SAT instance. ✷

The above proof also shows it is NP-hard to determine ifP can be made a subset of the winning set,P ⊆W .

ConclusionsWe have studied some basic computational questions regard-ing three prominent voting rules that use approval ballots toelect multiple winners. We closed the computational com-plexity of computing the winner forPAV and studied the

computational complexity of computing a best response fora variety of approval voting rules. In many settings, weproved that it is NP-hard for an agent or agents to computehow best to vote given the other approval ballots. To com-plement this complexity study, it would be interesting to un-dertake further axiomatic and empirical analyses ofPAV ,RAV , andSAV . Such an analysis would provide furtherinsight into the relative merits of these rules.

Acknowledgements NICTA is funded by the AustralianGovernment through the Department of Communicationsand the Australian Research Council through the ICT Cen-tre of Excellence Program. Serge Gaspers is the recipient ofan Australian Research Council Discovery Early Career Re-searcher Award (project number DE120101761). JoachimGudmundsson is funded by the Australian Research Coun-cil (project number FT100100755).

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