In this work, we are interested to prove that for voting systems which can be expressed as scoring rules, the coalitional weighted manipulation problem which is known to be NP-complete is as difficult as solving an integer linear programming problem. For this integer linear programming problem several software solutions exist, and we have found that with a reasonable number of candidates the solution can be found within seconds

Maria Masucci, Antonia

Silva, Alonso

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HAL Id: hal-01086642https://hal.inria.fr/hal-01086642Preprint submitted on 24 Nov 2014HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.An Integer Linear Programming Approach forCoalitional Weighted Manipulation under Scoring RulesAntonia Maria Masucci, Alonso SilvaTo cite this version:Antonia Maria Masucci, Alonso Silva. An Integer Linear Programming Approach for CoalitionalWeighted Manipulation under Scoring Rules. 2014. ￿hal-01086642￿An Integer Linear Programming Approach forCoalitional Weighted Manipulation under ScoringRulesAntonia Maria MasucciOrange Labs38-40 rue du Général Leclerc92130 Issy-Les-MoulineauxFranceEmail: antonia.masucci@orange.comAlonso SilvaBell Labs, Alcatel-LucentCentre de VillarceauxRoute de Villejust91620 NozayFranceEmail: alonso.silva@alcatel-lucent.comAbstract—In this work, we are interested to prove thatfor voting systems which can be expressed as scoring rules,the coalitional weighted manipulation problem which is knownto be NP-complete is as difficult as solving an integer linearprogramming problem. For this integer linear programmingproblem several software solutions exist, and we have foundthatwith a reasonable number of candidates the solution can be foundwithin seconds.I. I NTRODUCTIONVoting systems, also known as electoral systems, havegained increasing interest in recent years. In particular,thecomputational aspects of voting have been object of numerousresearches. Voting systems establish rules to determine thevalue of votes and how to aggregate them.The famous Gibbard-Satterthwaite theorem states that forany voting rule that is not a dictatorship, there are electionsin which at least one of the voters would benefit by lying [1],[2]. A dictatorship is a voting rule where one of the voters (thedictator) single-handedly decides the outcome of the election.The Gibbard-Satterthwaite Theorem implies that in theory,voters are able to manipulate elections. In practice, decid-ing which manipulation to employ may prove to be a hardcomputational problem. Indeed, there are a superpolynomialnumber of possibilities of ranking the candidates. Bartholdi etal. [6] showed a voting rule where manipulation is NP-hard.Conitzer et al. [7] showed that in a variety of voting rules,coalitional manipulation is NP-hard, even if there are onlyaconstant number of candidates. A shortcoming of the result ithat they are worst-case hardness results, and thus provideapoor obstacle against potential manipulators.In this work, we analyze the computational aspects ofmanipulation. We propose an integer linear programming ap-poach and we realize that although the problem is NP-hard, inpractice, with a small number of candidates the resolution ofthe manipulation problem is feasible and rather simple.II. PRELIMINARIESAn election(V , C) consists of a setV = {v1, v2, . . . , vN}of voters and a setC = {c1, c2, . . . , cM} of candidates. Eachvoter submits a ranking of the candidates (which is a totalorder over the setC).The voting setting also includes a voting rule, which is afunction from the set of all possible combinations of votes tothe set of candidates. We shall discuss the voting rules relatedto scoring rules. Whenever the voting rule is based on scores,the candidate with the highest score wins.Examples of voting rules based on scores are [3]:• Plurality: Each voter assigns a point to the preferredcandidate and the candidate who receives the largestscore wins;• Veto: Each voter “vetoes” a candidate and the candi-date that is vetoed by the fewest voters wins;• Borda: Givenm candidates, each voter gives a rankingof all candidates, thej-th ranked candidate score ism − j, and the candidate with largest sum of scorewins.Let the scoresx = (x1, x2, . . . , xM ) be a vector of non-negative integers such thatx1 ≥ x2 ≥ . . . ≥ xM . For eachvoter, a candidate receivesx1 points if it is ranked first bythe voter,x2 if it is ranked second, etc. The scoresm of acandidate is the total number of points the candidate receivs.For example, for Plurality scoring rule we havex = (1, 0, . . . , 0), for Veto scoring rule we havex =(1, 1, . . . , 1, 0), for Borda scoring rule we havex = (m −1,m− 2, . . . , 0), etc.In the constructive coalitional weighted manipulation(CCWM) problem [4], we are given a setC of candidates,with a distinguished candidatep ∈ C, a set of weightedvotersS that already cast their votes (those are the truthfulvoters), and weights for a set of votersT that still have notcast their votes (the manipulators). We are asked whetherthere is a way to cast the votes inT such thatp wins theelection. The constructive coalitional unweighted manipulation(CCUM) problem is a special case of CCWM problem whereall the weights equal1 [5].III. M AIN RESULTSA. Unweighted manipulationConsiderM candidates andN voters. Every vote can beseen as a permutation of the vector of valuesx = (x1, . . . , xM )Twherexm ≥ 0, for all m ∈ {1, . . . ,M} where we considerx1 ≥ x2 ≥ . . . ≥ xMand through translation and scaling without loss of generalitywe can assume thatx1 = 1 andxM = 0 so that1 = x1 ≥ x2 ≥ . . . ≥ xM = 0.We considerσn(x) = (σn(x1), σn(x2), . . . , σn(xM ))the permutation ofx which gives the vote of votern, for n ∈{1, . . . , N}.The final score for a candidatem ∈ {1, . . . ,M}, denotedby sm, can then be written assm =N∑n=1σn(xm)The total final score is denoted ass = (s1, s2, . . . , sM ).Since we are considering the constructive coalitionalweighted manipulation (CCWM) problem, we assume thatwe knowS ∈ N ∪ {0} votes and without loss of generalitywe consider that they are the votes of the first voters, i.e.,σ1(x), σ2(x), . . . , σS(x). We can then choose the remainingN − S votesσS+1(x), . . . , σN (x).We denote byu the combined votes of theS known votes,i.e.,u =(S∑i=1σi(xm))m∈{1,2,...,M}Without loss of generality, we assume that our objective isto elect the first candidate, i.e., the distinguished candidatepcorresponds to the first candidatec1. Thus, the first coordinateof the votes we can choose are given byσi(x1) = 1 ∀S + 1 ≤ i ≤ N.Since we are dealing with a scoring rule, it is always better tochoose the maximum score to the distinguished candidate.Therefore,s1 = u1 + (N − S).To see the differential, denoted bŷx, between the firstcandidate and all the other candidates of the known votes,we considerx̂ = u− s11,where1 = (1, 1, . . . , 1) is them× 1 vector.We notice thatx̂1 = 0. Our objective is that̂xk < 0 forall k ∈ {2, . . . ,M}. Equivalently, our objective is that thereexistsε > 0 such thatx̂k < −ε for all k ∈ {2, . . . ,M}.There areM ! possible permutations ofx. If M is a smallnumber, then we can enumerate the possible permutations in amatrix Â. We are thus looking for a vectorα = (α1, . . . , αM !)such thatÂα ≤ 0 subject to the constraint thatα ·1 = N −S.We assume that we want more than the fact thatε ispositive, we want to maximizeε such thatÂα < −ε1 in orderto maximize the differential with the second closest candidate.or equivalently, by definingA = −Â the constraint becomesAα > ǫThis is an integer linear programming given bymax εAα > ǫα · 1 = N − Sα ∈ N ∪ {0}.B. Weighted manipulationA weight function is a mappingw : V → N. When votersare weighted, the above rules are applied by considering avoter of weight ℓ to be ℓ different voters. The method toconvert the weighter manipulation problem to an integer linearprogramming is similar to the previous method and thus wedo not repeat it here.IV. SIMULATIONSIn this section, we show an example of our integer linearprogramming approach applied to a voting system, based onBorda voting rules, with four candidates. Then, we generalizegeneralize the previous approach to any number of candidatesand we present the proposed algorithm.A. Four candidatesConsider that we have4 candidates, denotedA, B, C andD. The voting rule is Borda, which means that for each vote,the favorite candidate will receive a score of3, the secondfavorite candidate will receive a score of2, the third favoritecandidate, a score of1, and the last favorite candidate willreceive a score of0. Through the transformation of translationand scaling we obtain that it is equivalent to the fact that thescores are given by(1, 2/3, 1/3, 0).Let us suppose that the total number of votes is120000and that the manipulator votes are25% of those, i.e.,30000votes.Consider that we know the remaining90000 truthful votes:• 15000 areB > C > A > D• 15000 areB > D > C > A• 30000 areC > D > A > B• 30000 areD > A > B > CAnd consider that we want to make the first candidate towin the election, i.e., candidateA. Following the algorithmproposed in the previous section, the first step is to computethe current scores of each candidate.It is easy to see that the scores are given as follows:A = 30000 ∗ (2/3) + 45000 ∗ (1/3) = 35000B = 30000 ∗ (1) + 30000 ∗ (1/3) = 40000C = 30000 ∗ (1) + 15000 ∗ (2/3) + 15000 ∗ (1/3) = 45000D = 30000 ∗ (1) + 45000 ∗ (2/3) = 60000.According to the previous algorithm, the manipulator voteswill always start withA, so the final score forA will besA = 35000 + 30000 = 65000The difference between the final score and the currentscores ofB, C, andD are 25000, 20000, and 5000 respec-tively.Since we have imposed that the favorite candidate of themanipulator votes isA, these votes can only beA > B > C > DA > B > D > CA > C > B > DA > C > D > BA > D > B > CA > D > C > B.We are looking for the number of votes of each type, thatwe denoteα1, α2, α3, α4, α5, α6 respectively.The system of equations then becomesα112/31/30+ α212/301/3+ α311/32/30+α411/302/3+ α5102/31/3+ α6101/32/3≤3000025000200005000.with equality for the first equation, under the constraintsα1, α2, α3, α4, α5, α6 ≥ 0 andα1, α2, α3, α4, α5, α6 ∈ Z.This system of equations can be analyzed for a standardmixed integer linear program using as objective function thefunction 0. We used the software R, in a i7 machine usingUbuntu 12.04 and the time it took to make the computationwas2 ms.B. Any number of candidatesWe can generalize the previous analysis to consider anynumber of candidates.The algorithm is as follows:Under the algorithm described above, we have done aMontecarlo simulation. The simulation consisted on a Bordaelection with increasing number of candidates (from4 to 9 can-didates) with100 votes,20 of which were manipulated and80Algorithm 1 ILP CCWM1: Compute the combined votes of theS known votes,denotedu2: Compute the score of the distinguished candidatep3: Compute the differential between the score of the distin-guished candidate and the combined votes of theS knownvotes, denoted̂x4: Compute all the permutations of votes in a matrixA5: Create a linear program withm! variables6: Set the objective function to be zero7: repeat8: Add the constraint from the column ofA and the vectorx̂9: until we reach the last column ofA10: Set lower bound to be zero11: Set the solution to be integer12: Solve the integer linear program13: Return the list of votes needed (if they exist)truthful votes. We run the simulation100 times and we presentthe mean, the minimum and the maximum computational time(in seconds) of the simulations (see Figure 1).Fig. 1. Computational time (in seconds) vs No. of candidates. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 4  5  6  7  8  9Computation time (in seconds)No. of candidatesComputation time vs No. of candidatesV. CONCLUSIONSIn the present work, we have considered voting systemswhich can be expressed as scoring rules. In particular, we havanalyzed the coalitional weighted manipulation problem andwe have provided an integer linear programming approach tosolve it. The approach allows us to solve in some millisecondsa problem which is known to be NP-hard.ACKNOWLEDGMENTThe work presented in this paper has been partially carriedout at LINCS (http://www.lincs.fr).REFERENCES[1] A. Gibbard, “Manipulation of voting schemes: a general result”, Econo-metrica 41 (4): 587-602, 1973.[2] M. Satterthwaite, “Strategy-proofness and Arrow’s conditions: Existenceand correspondance theorems for voting procedures and social welfarefunctions.” Journal of Economic Theory, 10:187–217, 1975.[3] O. Lev and J. S. Rosenschein, “Convergence of Iterative Voting”, theEleventh International Joint Conference on Autonomous Agents andMultiagent Systems (AAMAS 2012), June 2012, Valencia, Spain, 611-618.[4] M. Zuckerman, A. D. Procaccia, and J. S. Rosenschein, “Algorithms forthe coalitional manipulation problem”, Journal of Artificial Intelligence,173(2):392-412, February 2009.[5] N. Betzler, R. Niedermeier, G. Woeginger, “Unweighted coalitionalmanipulation under the Borda rule is NP-hard”, Proceedingsof the 22ndInternational Joint Conference on Artificial Intelligence(IJCAI 2011),International Joint Conference on Artificial Intelligence.[6] J. Bartholdi, C. A. Tovey, and M. A. Trick, “The computational difficultyof manipulating an election”, Social Choice and Welfare, 6:157–165,1989.[7] V. Conitzer, T. Sandholm, and J. Lang, “When are elections with fewcandidates hard to manipulate?”, Journal of the ACM, 54(3):1–33, 2007.

An Integer Linear Programming Approach for Coalitional Weighted Manipulation under Scoring Rules

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An Integer Linear Programming Approach for Coalitional Weighted Manipulation under Scoring Rules

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