Taking the Final Step to a Full Dichotomy of the Possible Winner Problem in Pure Scoring Rules
aa r X i v : . [ c s . CC ] N ov Taking the Final Step to a Full Dichotomy of thePossible Winner Problem in Pure Scoring Rules ∗ Dorothea Baumeister and J¨org RotheInstitut f¨ur InformatikHeinrich-Heine-Universit¨at D¨usseldorf40225 D¨usseldorf, GermanyMay 23, 2018
Abstract
The P
OSSIBLE W INNER problem asks, given an election where the voters’preferences over the candidates are specified only partially, whether a designatedcandidate can become a winner by suitably extending all the votes. Betzler andDorn [1] proved a result that is only one step away from a full dichotomy of thisproblem for the important class of pure scoring rules in the case of unweightedvotes and an unbounded number of candidates: P
OSSIBLE W INNER is NP-complete for all pure scoring rules except plurality, veto, and the scoring rule withvector ( , ,..., , ) , but is solvable in polynomial time for plurality and veto.We take the final step to a full dichotomy by showing that P OSSIBLE W INNER isNP-complete also for the scoring rule with vector ( , ,... , , ) . The computational complexity of problems related to voting systems is a field of in-tense study (see, e.g., the surveys by Faliszewski et al. [3, 4] and Conitzer [5] and thebookchapters by Faliszewski et al. [6] and Baumeister et al. [7]). For many of the com-putational problems investigated, the voters are commonly assumed to provide theirpreferences over the candidates via complete linear orderings of all candidates. How-ever, this is not the case in many real-life settings: Some voters may have preferencesover some candidates only, or it may happen that new candidates are introduced to anelection after some voters have already cast their votes. As mentioned by Chevaleyreet al. [8] and Xia et al. [9], such a situation may occur, for example, when a com-mittee whose members are to schedule their next meeting date by voting over a set ofproposed dates. After some committee members have cast their votes (and then have ∗ This work was supported in part by DFG grants RO-1202/11-1, RO-1202/12-1, and RO-1202/15-1, theEuropean Science Foundation’s EUROCORES program LogICCC, and the SFF grant “Cooperative Norm-setting” of Heinrich-Heine-Universit¨at D¨usseldorf. A preliminary version appeared as a short paper [2] inthe proceedings of the (ECAI-2010).
OSSIBLE W INNER problem that (for any given election system) asks, given anelection with only partial preferences and a designated candidate c , whether c is awinner in some extension of the partial votes to linear ones. This problem was studiedlater on by Xia and Conitzer [11], Betzler and Dorn [1], and Baumeister et al. [12],and closely related problems have been introduced and investigated by Chevaleyre etal. [8], Xia et al. [9], and Baumeister et al. [12]. In particular, Betzler and Dorn [1]established a result that is only one step away from a full dichotomy result of theP OSSIBLE W INNER problem for the important class of pure scoring rules.Dichotomy results are particularly important, as they completely settle the com-plexity of a whole class of related problems by providing an easy-to-check conditionthat tells the hard cases apart from the easily solvable cases. The first dichotomy resultin computer science is due to Schaefer [13] who provided a simple criterion to dis-tinguish the hard instances of the satisfiability problem from the easily solvable ones.Hemaspaandra and Hemaspaandra [14] established the first dichotomy result related tovoting. Their dichotomy result, which distinguishes the hard instances from the easyinstances by the simple criterion of “ diversity of dislike ,” concerns the manipulationproblem for the class of scoring-rule elections with weighted votes.In contrast, Betzler and Dorn’s above-mentioned result that is just one step awayfrom a full dichotomy is concerned with the P
OSSIBLE W INNER problem for pure scor-ing rules with unweighted votes and any number of candidates [1]. In particular, theyshowed NP-completeness for all but three pure scoring rules, namely plurality, veto,and the scoring rule with scoring vector ( , , . . . , , ) . For plurality and veto, theyshowed that this problem is polynomial-time solvable, but the complexity of P OSSI - BLE W INNER for the scoring rule with vector ( , , . . . , , ) was left open. Taking thefinal step to a full dichotomy result, we show that P OSSIBLE W INNER is NP-completealso for the scoring rule with vector ( , , . . . , , ) . An election ( C , V ) is specified by a set C = { c , c , . . . , c m } of candidates and a list V =( v , v , . . . , v n ) of votes over C . In the most common model of representing preferences,each such vote is a linear order of the form c i > c i > · · · > c i m where { i , i , . . . , i m } = Formally, a linear order L on C is a binary relation on C that is (i) total (i.e., for any two distinct c , d ∈ C ,either cLd or d Lc ); (ii) transitive (i.e., for all c , d , e ∈ C , if cLd and d Le then cLe ); and (iii) asymmetric , , . . ., m } , and c i k > c i ℓ means that candidate c i k is (strictly) preferred to candidate c i ℓ . A voting system is a rule to determine the winners of an election Scoring rules(a.k.a. scoring protocols) are an important class of voting systems. Every scoring rulefor m candidates is specified by a scoring vector ~ a = ( a , a , . . . , a m ) with a ≥ a ≥· · · ≥ a m , where each a j is a nonnegative integer. For an election ( C , V ) , each voter v ∈ V gives a j points to the candidate ranked at the j th position in his or her vote. Summingup all points a candidate c ∈ C receives from all votes in V , we obtain score ( C , V ) ( c ) , c ’sscore in ( C , V ) . Whoever has the highest score wins the election. If there is only onesuch candidate, he or she is the unique winner. Betzler and Dorn [1] focus on so-calledpure scoring rules. A scoring rule is pure if for each m ≥
2, the scoring vector for m candidates can be obtained from the scoring vector for m − a ≥ a ≥ · · · ≥ a m . Wewill study only the pure scoring rule that for m ≥ ( , , . . . , , ) : In each vote the first candidate gets two points, the last candidategets zero points, and the m − OSSIBLE W INNER problem is defined for partial rather than linear votes. Fora set C of candidates, a partial vote over C is a transitive, asymmetric (though notnecessarily total) binary relation on C . For any two candidates c and d in a partial vote,we write c ≻ d if c is (strictly) preferred to d . For any two sets A , B ⊆ C of candidates,we write A ≻ B to mean that each candidate a ∈ A is preferred to each candidate b ∈ B ,i.e., a ≻ b for all a ∈ A and b ∈ B . As a shorthand, we write a ≻ B for { a } ≻ B and wewrite A ≻ b for A ≻ { b } .A linear vote v ′ over C extends a partial vote v over C if v ⊆ v ′ , i.e., for all c , d ∈ C ,if c ≻ d in v then c > d in v ′ . A list V ′ = ( v ′ , v ′ , . . . , v ′ n ) of linear votes over C is an extension of a list V = ( v , v , . . . , v n ) of partial votes over C if for each i , 1 ≤ i ≤ n , v ′ i ∈ V ′ extends v i ∈ V .Given a voting system E , Konczak and Lang [10] define the following problem: E -P OSSIBLE W INNER
Given:
A set C of candidates, a list V of partial votes over C , and a designatedcandidate c ∈ C . Question:
Is there an extension V ′ of V to linear votes over C such that c is a winnerof election ( C , V ′ ) under voting system E ? This defines the problem in the nonunique-winner case; for its unique-winner vari-ant, simply replace “a winner” by “the unique winner.” We focus on the nonunique-winner case here, but mention that the unique-winner case can be handled analogouslyas described by Betzler and Dorn [1]. We may drop the prefix “ E -” and simply writeP OSSIBLE W INNER when the specific voting system used is either clear from the con-text or not relevant in the corresponding context. (i.e., for all c , d ∈ C , if cLd then d Lc does not hold). Note that asymmetry of L implies irreflexivity of L (i.e., for no c ∈ C does cLc hold). The Final Step to a Full Dichotomy Result
Theorem 3.2 below shows that P
OSSIBLE W INNER for the scoring rule with vector ( , , . . . , , ) is NP-hard. Our proof of this theorem uses the notion of maximumpartial score defined by Betzler and Dorn [1]. Fix any scoring rule. Let C be a set ofcandidates, c ∈ C a candidate we want to make win the election, and let V = V ℓ ∪ V p be a list of votes over C , where V ℓ contains only linear votes and V p contains partial(i.e., incomplete) votes such that c ’s score is fixed, i.e., the exact number of points c receives from any v ∈ V p is known, no matter to which linear vote v is extended. Foreach d ∈ C − { c } , define the maximum partial score of d with respect to c (denotedby s max p ( d , c ) ) to be the maximum number of points that d may get from (extending tolinear votes) the partial votes in V p without defeating c in ( C , V ′ ) for any extension V ′ of V to linear votes. Since the score of c is the same in any extension V ′ of V to linearvotes, it holds that s max p ( d , c ) = score ( C , V ′ ) ( c ) − score ( C , V ℓ ) ( d ) . The following lemma will be useful for our proof of Theorem 3.2.
Lemma 3.1 (Betzler and Dorn [1])
Let ~ a = ( a , a , . . . , a m ) be any scoring rule, letC be a set of m ≥ candidates with designated candidate c ∈ C, let V p be a list ofpartial votes in which the score of c is fixed, and let s max p ( c ′ , c ) be the maximum partialscore with respect to c for all c ′ ∈ C − { c } . Suppose that the following two propertieshold:1. There is a candidate d ∈ C − { c } such that s max p ( d , c ) ≥ a | V p | .2. For each c ′ ∈ C − { c } , the maximum partial score of c ′ with respect to c can bewritten as a linear combination of the score values, s max p ( c ′ , c ) = (cid:229) mj = n j a j , withm = | C | , n j ∈ N , and (cid:229) mj = n j ≤ | V p | .Then a list V ℓ of linear votes can be constructed in polynomial time such that forall c ′ ∈ C − { c } , score ( C , V ℓ ) ( c ′ ) = score ( C , V ′ ) ( c ) − s max p ( c ′ , c ) , where V ′ is an extensionof V p to linear votes. Theorem 3.2 P OSSIBLE W INNER (both in the nonunique-winner case and in theunique-winner case) is NP -complete for the pure scoring rule with scoring vector ( , , . . . , , ) . Proof.
Membership in NP is obvious. Our NP-hardness proof uses a reduction fromthe NP-complete H
ITTING S ET problem (see, e.g., [15]), which is defined as follows: H ITTING S ET Given:
A finite set X , a collection S = { S ,..., S n } of nonempty subsets of X (i.e., /0 = S i ⊆ X for each i , 1 ≤ i ≤ n ), and a positive integer k . Question:
Is there a subset X ′ ⊆ X with | X ′ | ≤ k such that X ′ contains at least oneelement from each subset in S ? ( X , S , k ) be a given H ITTING S ET instance with X = { e , e , . . . , e m } and S = { S , S , . . . , S n } . From ( X , S , k ) we construct a P OSSIBLE W INNER instancewith candidate set C = { c , h } ∪ { x i , x i , x i , . . . , x ni , y i , y i , . . . , y ni , z i , z i , . . . , z ni | ≤ i ≤ m } and designated candidate c . The list of votes V = V ℓ ∪ V p consists of a list V ℓ of linearvotes and a list V p of partial votes. V p = V p ∪ V p ∪ V p consists of three sublists:1. V p contains k votes of the form h ≻ C − { h , x , x , . . . , x m } ≻ { x , x , . . . , x m } .2. V p contains the following 2 n + i , 1 ≤ i ≤ m : v i : h ≻ C − { h , x i , y i } ≻ { x i , y i } , v ji : y ji ≻ C − { y ji , z ji , h } ≻ h for 1 ≤ j ≤ n , w ji : x ji ≻ C − { x ji , y j + i , z ji } ≻ y j + i for 1 ≤ j ≤ n − , w ni : x ni ≻ C − { x ni , z ni , h } ≻ h . V p contains the vote T j ≻ C − { T j , h } ≻ h for each j , 1 ≤ j ≤ n , where T j = { x ji | e i ∈ S j } . For each i , 1 ≤ i ≤ m , and j , 1 ≤ j ≤ n , the maximum partial scores with respect to c are set as follows: s max p ( x i , c ) = | V p | − s max p ( x ji , c ) = | V p | + s max p ( y ji , c ) = s max p ( z ji ) = | V p | s max p ( h , c ) ≥ | V p | . This means that each x i must take at least one last position, which is possible in thevotes from V p and the votes v i , 1 ≤ i ≤ m , from V p . Since the candidates x ji cannever take a last position, they may take at most one first position. For y ji and z ji , themaximum partial scores with respect to c are set such that for each first position theytake, they must also take at least one last position. Finally, h can never beat c . ByLemma 3.1, we can construct a list of votes V ℓ such that all candidates other than c canget only their maximum partial scores with respect to c in the partial votes.We claim that ( X , S , k ) is a yes-instance of H ITTING S ET if and only if c is apossible winner in ( C , V ) , using the scoring rule with vector ( , , . . . , , ) .From left to right, suppose there exists a hitting set X ′ ⊆ X with | X ′ | ≤ k for S . Thepartial votes in V p can then be extended to linear votes such that c wins the election asfollows: 5 i ∈ X ′ e i X ′ V p : h > · · · > x i V p : v i : h > · · · > x i > y i h > · · · > y i > x i v ji , ≤ j ≤ n : y ji > · · · > z ji z ji > y ji > · · · > hw ji , ≤ j < n : z ji > x ji > · · · > y j + i x ji > · · · > y j + i > z ji w ni : z ni > x ni > · · · > h x ni > · · · > h > z ni V p : x ji > · · · > h for some j ∈ { ℓ | e i ∈ S ℓ } Every x i takes one last position and get his or her maximum partial score withrespect to c . For e i ∈ X ′ , all y ji take exactly one first, one last, and a middle positionin all remaining votes. For e i X ′ , all y ji take middle positions only. So they alwaysget their maximum partial scores with respect to c . The candidates z ji also get theirmaximum partial scores with respect to c , since they always get one first position, onelast position, and a middle poisition in all remaining votes. Every candidate x ji gets atmost one first position and therefore does not exceed his or her maximum partial scorewith respect to c . Since no candidate exceeds his or her maximum partial score withrespect to c , candidate c is a winner in this extension of the list V p of partial votes.Conversely, assume that c is a possible winner for ( C , V ) . Then no candidate mayget more points in V p than his or her maximum partial score with respect to c . Sinceat most k different x i may take a last position in V p , at least n − k different x i must takea last position in v i . Fix any i such that x i is ranked last in v i . We now show that it isnot possible that a candidate x ji then takes a first position in any vote of V p . Since x i takes the last position in v i , y i takes a middle position in this vote and gets one point.The only vote in which the score of y i is not fixed is v i . Without the points from thisvote, y i already gets | V p | − y i cannot get two points in v i , and z i takesthe first position in v i . Without the points from w i , z i gets | V p | points and must takethe last position in w i . The first position in w i is then taken by x i , so x i cannot take afirst position in any vote from V p . Candidate y i gets one point in w i , and by a similarargument as above, x i is placed at the first position in w i . Repeating this argument, wehave that for each j , 1 ≤ j ≤ n , x ji is placed at the first position in w ji and thus cannottake a first position in a vote from V p . This means that all first positions in the votesof V p must be taken by those x ji for which x i takes the last position in a vote from V p .This is possible only if the x ji are not at the first position in w ji . Thus z ji must take thisposition. Due to z ji ’s maximum partial score with respect to c , this is possible only if z ji takes the last position in v ji . Then y ji takes the first position in this vote. This ispossible, since y ji can take a middle position in v i for j =
1, and in v ji for 2 ≤ j ≤ n .Hence all x ji , where x i takes the last position in the votes of V p , may take the firstposition in the votes of V p . Thus, by the definition of V p (which, recall, contains thevote T j ≻ C − { T j , h } ≻ h for each j , 1 ≤ j ≤ n , where T j = { x ji | e i ∈ S j } ), the elements e i corresponding to those x i must form a hitting set of size at most k for S . ❑ Conclusions and Future Research
In this paper, we have taken the final step to a full dichotomy theorem for the P
OSSIBLE W INNER problem with unweighted votes and an unbounded number of candidates inpure scoring rules. Our result complements the results of Betzler and Dorn [1] byshowing that P
OSSIBLE W INNER is NP-complete for the pure scoring rule with vector ( , , . . . , , ) , the one missing case in [1].Besides establishing this dichotomy theorem, our result has also other conse-quences. Since P OSSIBLE W INNER is a special case of the S
WAP B RIBERY problemintroduced by Elkind et al. [16], Theorem 3.2 implies that this problem is NP-hard forthe pure scoring rule with vector ( , , . . . , , ) as well. Informally put, in a S WAP B RIBERY instance an external agent seeks to make a distinguished candidate c win theelection by bribing some voters so as to swap adjacent candidates in their preferenceorders (see [16] for formal details).On the other hand, the P OSSIBLE W INNER problem generalizes the C
OALITIONAL U NWEIGHTED M ANIPULATION problem where a group of strategic voters, knowingthe preferences of the nonstrategic voters, seeks to make their favorite candidate winby reporting insincere preferences. An instance of this manipulation problem can beseen as a P
OSSIBLE W INNER instance in which all nonstrategic voters report (sincere)complete linear orderings of all candidates, whereas all strategic voters initially haveempty preference lists, and the question is whether they can extend them to completelinear orderings of all candidates such that their favorite candidate wins.The NP-hardness result of Theorem 3.2 has no direct consequence for the complex-ity of this more special problem, and neither so for other more special variants of P OS - SIBLE W INNER , such as P
OSSIBLE W INNER WITH RESPECT TO THE A DDITION OF N EW C ANDIDATES (see Chevaleyre et al. [8], Xia et al. [9], and Baumeister et al. [12]).Note that the complexity of the C
OALITIONAL W EIGHTED M ANIPULATION problem,where all votes are weighted and the weights of all manipulators are known initiallyin addition to the weights and preferences of the nonmanipulators, is well understood(see the work of Conitzer et al. [17]), and even a dichotomy theorem for scoring rulesdue to Hemaspaandra and Hemaspaandra [14] is known for weighted votes. However,the complexity of C
OALITIONAL U NWEIGHTED M ANIPULATION is still unknown formany voting systems, including many scoring rules. Only recently Betzler et al. [18]and Davies et al. [19] independently showed that C
OALITIONAL U NWEIGHTED M A - NIPULATION , even for only two manipulators, is NP-complete for Borda elections,where Borda with m candidates is the scoring rule with vector ( m − , m − , . . . , ) .Further complexity results regarding the C OALITIONAL U NWEIGHTED M ANIPULA - TION problem for various voting systems are due to Faliszewski et al. [20, 21], Narodyt-ska et al. [22], Xia et al. [23, 24], and Zuckerman et al. [25, 26]. None of these papersestablishes a dichotomy theorem for manipulation in the unweighted case, althoughdichotomy results for scoring rules are now known for two of its generalizations, theC
OALITIONAL W EIGHTED M ANIPULATION problem (see [14]) and the (unweighted)P
OSSIBLE W INNER problem (see [1] and this paper). For future research, we pro-pose to tackle the open problem of finding a dichotomy result for C
OALITIONAL U N - WEIGHTED M ANIPULATION in scoring rules.7 cknowledgments
We thank the anonymous ECAI-2010 and IPL reviewers for their expert comments onthis paper that helped improving its presentation.
References [1] N. Betzler, B. Dorn, Towards a dichotomy for the possible winner problem in electionsbased on scoring rules, Journal of Computer and System Sciences 76 (8) (2010) 812–836.[2] D. Baumeister, J. Rothe, Taking the final step to a full dichotomy of the possible winnerproblem in pure scoring rules, in: Proceedings of the 19th European Conference on Artifi-cial Intelligence, IOS Press, 2010, pp. 1019–1020, short paper.[3] P. Faliszewski, E. Hemaspaandra, L. Hemaspaandra, Using complexity to protect elections,Communications of the ACM 53 (11) (2010) 74–82.[4] P. Faliszewski, A. Procaccia, AI’s war on manipulation: Are we winning?, AI Magazine31 (4) (2010) 53–64.[5] V. Conitzer, Making decisions based on the preferences of multiple agents, Communica-tions of the ACM 53 (3) (2010) 84–94.[6] P. Faliszewski, E. Hemaspaandra, L. Hemaspaandra, J. Rothe, A richer understanding ofthe complexity of election systems, in: S. Ravi, S. Shukla (Eds.), Fundamental Problems inComputing: Essays in Honor of Professor Daniel J. Rosenkrantz, Springer, 2009, Ch. 14,pp. 375–406.[7] D. Baumeister, G. Erd´elyi, E. Hemaspaandra, L. Hemaspaandra, J. Rothe, Computationalaspects of approval voting, in: J. Laslier, R. Sanver (Eds.), Handbook on Approval Voting,Springer, 2010, Ch. 10, pp. 199–251.[8] Y. Chevaleyre, J. Lang, N. Maudet, J. Monnot, Possible winners when new candidatesare added: The case of scoring rules, in: Proceedings of the 24th AAAI Conference onArtificial Intelligence, AAAI Press, 2010, pp. 762–767.[9] L. Xia, J. Lang, J. Monnot, Possible winners when new alternatives join: New resultscoming up!, in: Proceedings of the 10th International Joint Conference on AutonomousAgents and Multiagent Systems, IFAAMAS, 2011, pp. 829–836.[10] K. Konczak, J. Lang, Voting procedures with incomplete preferences, in: Proceedings ofthe Multidisciplinary IJCAI-05 Workshop on Advances in Preference Handling, 2005, pp.124–129.[11] L. Xia, V. Conitzer, Determining possible and necessary winners under common votingrules given partial orders, in: Proceedings of the 23rd AAAI Conference on Artificial In-telligence, AAAI Press, 2008, pp. 196–201.[12] D. Baumeister, M. Roos, J. Rothe, Computational complexity of two variants of the pos-sible winner problem, in: Proceedings of the 10th International Joint Conference on Au-tonomous Agents and Multiagent Systems, IFAAMAS, 2011, pp. 853–860.[13] T. Schaefer, The complexity of satisfiability problems, in: Proceedings of the 10th ACMSymposium on Theory of Computing, ACM Press, 1978, pp. 216–226.[14] E. Hemaspaandra, L. Hemaspaandra, Dichotomy for voting systems, Journal of Computerand System Sciences 73 (1) (2007) 73–83.
15] M. Garey, D. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman and Company, 1979.[16] E. Elkind, P. Faliszewski, A. Slinko, Swap bribery, in: Proceedings of the 2nd InternationalSymposium on Algorithmic Game Theory, Springer-Verlag
Lecture Notes in ComputerScience , 2009, pp. 299–310.[17] V. Conitzer, T. Sandholm, J. Lang, When are elections with few candidates hard to manip-ulate?, Journal of the ACM 54 (3) (2007) Article 14.[18] N. Betzler, R. Niedermeier, G. Woeginger, Unweighted coalitional manipulation under theBorda rule is NP-hard, in: Proceedings of the 22nd International Joint Conference on Arti-ficial Intelligence, IJCAI, 2011, pp. 55–60.[19] J. Davies, G. Katsirelos, N. Narodytska, T. Walsh, Complexity of and algorithms for Bordamanipulation, in: Proceedings of the 25th AAAI Conference on Artificial Intelligence,AAAI Press, 2011, pp. 657–662.[20] P. Faliszewski, E. Hemaspaandra, H. Schnoor, Copeland voting: Ties matter, in: Proceed-ings of the 7th International Joint Conference on Autonomous Agents and Multiagent Sys-tems, IFAAMAS, 2008, pp. 983–990.[21] P. Faliszewski, E. Hemaspaandra, H. Schnoor, Manipulation of Copeland elections, in: Pro-ceedings of the 9th International Joint Conference on Autonomous Agents and MultiagentSystems, IFAAMAS, 2010, pp. 367–374.[22] N. Narodytska, T. Walsh, L. Xia, Manipulation of nanson’s and baldwin’s rules, in: Pro-ceedings of the 25th AAAI Conference on Artificial Intelligence, AAAI Press, 2011, pp.713–718.[23] L. Xia, V. Conitzer, A. Procaccia, A scheduling approach to coalitional manipulation, in:Proceedings of the 11th ACM Conference on Electronic Commerce, ACM Press, 2010, pp.275–284.[24] L. Xia, M. Zuckerman, A. Procaccia, V. Conitzer, J. Rosenschein, Complexity of un-weighted coalitional manipulation under some common voting rules, in: Proceedings ofthe 21st International Joint Conference on Artificial Intelligence, IJCAI, 2009, pp. 348–353.[25] M. Zuckerman, A. Procaccia, J. Rosenschein, Algorithms for the coalitional manipulationproblem, Artificial Intelligence 173 (2) (2009) 392–412.[26] M. Zuckerman, O. Lev, J. Rosenschein, An algorithm for the coalitional manipulationproblem under maximin, in: Proceedings of the 10th International Joint Conference onAutonomous Agents and Multiagent Systems, IFAAMAS, 2011, pp. 845–852., 2009, pp. 299–310.[17] V. Conitzer, T. Sandholm, J. Lang, When are elections with few candidates hard to manip-ulate?, Journal of the ACM 54 (3) (2007) Article 14.[18] N. Betzler, R. Niedermeier, G. Woeginger, Unweighted coalitional manipulation under theBorda rule is NP-hard, in: Proceedings of the 22nd International Joint Conference on Arti-ficial Intelligence, IJCAI, 2011, pp. 55–60.[19] J. Davies, G. Katsirelos, N. Narodytska, T. Walsh, Complexity of and algorithms for Bordamanipulation, in: Proceedings of the 25th AAAI Conference on Artificial Intelligence,AAAI Press, 2011, pp. 657–662.[20] P. Faliszewski, E. Hemaspaandra, H. Schnoor, Copeland voting: Ties matter, in: Proceed-ings of the 7th International Joint Conference on Autonomous Agents and Multiagent Sys-tems, IFAAMAS, 2008, pp. 983–990.[21] P. Faliszewski, E. Hemaspaandra, H. Schnoor, Manipulation of Copeland elections, in: Pro-ceedings of the 9th International Joint Conference on Autonomous Agents and MultiagentSystems, IFAAMAS, 2010, pp. 367–374.[22] N. Narodytska, T. Walsh, L. Xia, Manipulation of nanson’s and baldwin’s rules, in: Pro-ceedings of the 25th AAAI Conference on Artificial Intelligence, AAAI Press, 2011, pp.713–718.[23] L. Xia, V. Conitzer, A. Procaccia, A scheduling approach to coalitional manipulation, in:Proceedings of the 11th ACM Conference on Electronic Commerce, ACM Press, 2010, pp.275–284.[24] L. Xia, M. Zuckerman, A. Procaccia, V. Conitzer, J. Rosenschein, Complexity of un-weighted coalitional manipulation under some common voting rules, in: Proceedings ofthe 21st International Joint Conference on Artificial Intelligence, IJCAI, 2009, pp. 348–353.[25] M. Zuckerman, A. Procaccia, J. Rosenschein, Algorithms for the coalitional manipulationproblem, Artificial Intelligence 173 (2) (2009) 392–412.[26] M. Zuckerman, O. Lev, J. Rosenschein, An algorithm for the coalitional manipulationproblem under maximin, in: Proceedings of the 10th International Joint Conference onAutonomous Agents and Multiagent Systems, IFAAMAS, 2011, pp. 845–852.