Manipulation Can Be Hard in Tractable Voting Systems Even for Constant-Sized Coalitions
aa r X i v : . [ c s . G T ] A ug Manipulation Can Be Hard in Tractable VotingSystems Even for Constant-Sized Coalitions
Curtis Menton and Preetjot Singh ∗ Department of Computer ScienceUniversity of RochesterRochester, NY 14627 USAAugust 9, 2018
Abstract
Voting theory has become increasingly integrated with computational socialchoice and multiagent systems. Computational complexity has been extensivelyused as a shield against manipulation of voting systems, however for several votingschemes this complexity may cause calculating the winner to be computationallydifficult. Of the many voting systems that have been studied with regard to electionmanipulation, a few have been found to have an unweighted coalitional manipula-tion problem that is NP-hard for a constant number of manipulators despite havinga winner problem that is in P. We survey this interesting class of voting systemsand the work that has analyzed their complexity.
Research in voting theory has become increasingly important due to the ubiquity of vot-ing systems. While voting is most typically used for political or organizational elections,recently it has become relevant as a tool in multiagent systems and distributed artifi-cial intelligence. From recommender systems [GMHS99, PHG00] such as those seen inNetflix which makes recommendations based on user activity, to consensus mechanismsfor planning in artificial intelligence [ER91] and search engine and metasearch enginedesign [Lif00, DKNS01], mechanisms that aggregate individual ‘votes’ have far-reachingapplications. Many fields of study in computer science such as mechanism design and ∗ Supported in part by grants NSF-CCF-0915792 and NSF-CCF-1101479.
Voting has long been used as a tool for collaborative decision making, with democraticgovernment known to have existed at least as far back as 6th century BCE in ancientGreece. For nearly as long people have studied voting in an attempt to find the bestelection methods and solve problems related to voting.One milestone in the study of voting is the work of 13th century mystic Ramon Llull.Llull, a prominent Franciscan, was involved with the Catholic church and researchedmethods for electing church officials. Among his extensive body of work, encompassingat least 265 titles on subjects ranging from controversial theological viewpoints to ro-mantic fiction, Llull’s contributions to voting theory stem from three works:
ArtifitiumElectionis Personarum , En qual manera Natana fo eleta a abadessa (Chapter 24 of hisnovel
Blaquerna ) and
De Arte Eleccionis , all featuring variants of a pairwise electionsystem we now know as Condorcet voting [HP01, Szp10].In the eighteenth century, the Marquis de Condorcet developed one of the first criteriafor evaluating voting systems. Condorcet’s criterion is that, for a given election, if thereexists a candidate that beats all other candidates in pairwise contests, then that candi-date must be the winner of the election. It is a somewhat surprising result that such acandidate will not always exist due the possibility of cycles in the pairwise societal pref-erences. Condorcet proposed this criteria and showed that many popular voting systemsdo not meet it. Those that do are called Condorcet methods, or Condorcet-consistent2oting systems. Elections using Condorcet methods can be viewed as a number of pair-wise majority-rule elections or in the case of an election with two candidates, exactlyequivalent to a majority-rule election. Hence the literature often refers to Condorcetmethods as variants of the majority rule [Ris05].Condorcet methods have often been contrasted with Borda voting, which takes com-plete preference orderings as the votes and gives points to each candidate based on thenumber of candidates they are ranked above in each vote. For instance a vote denoting a is preferred to b is preferred to c would give two points to a , one to b , and none to c . Thereare fervent arguments between proponents of the two systems, debating the importanceof the Condorcet criterion [New92, Saa06, Ris05], in a rivalry dating back to the Marquisde Concorcet’s criticism of Borda voting when it was first introduced [Szp10].One persistent concern in elections is that some of the participants may be able to votestrategically thus unfairly gaining an advantage over honest voters. Pliny the Younger’sattempts to manipulate the Roman senate circa 105 CE is possibly the earliest recordedinstance of strategic behavior in elections [Szp10]. The senate, presiding over a murdertrial, were divided into three blocs: those who favored acquittal, banishment, or death forthe accused. The senators were more or less evenly distributed among the three positions,with the acquittal bloc (headed by Pliny) being slightly larger than the other two. Thenormal method of voting was similar to the current justice system in most countries: Thesenate would first vote on the guilt of the accused, followed by a vote on the punishment(banishment or death). Considering how the blocs were aligned, the probable outcomeof the first election would be a decision of guilty, followed by banishment. To give hisfaction an edge, Pliny proposed the senators vote for acquittal, banishment or deathin a single ternary-choice election. However, his strategy backfired. The death penaltyfaction, fearing an acquittal, voted for banishment.Pliny’s story holds more than just strategy and counter-strategy: Pliny convinced thesenate of the fairness of a single ternary-choice election by stating it aligned naturallywith the principle of voting qua sentitits , or according to your true preferences. Hisattempt proved unsuccessful but serves as an excellent example of the problem of gettingpeople to vote honestly. While this problem was recognized by voting theoreticiansthrough history, it was either dismissed or attempts to solve it were limited at best. Forinstance, Llull documents that voters were required to give an oath to vote sincerely,and Jean-Charles de Borda famously dismissed criticism of his system’s vulnerability tomanipulation by saying “My scheme is only intended for honest men” [Bla58].Later work drew from game theory to more formally model voter strategy and toanalyze its possibilities, especially in the work of Allan Gibbard [Gib73] and Mark Sat-terthwaite [Sat75]. We will first explore the work of Kenneth Arrow, who initiated the A majority rule decides on the alternative that receives the majority of the votes.
Arrow’s seminal work in modern social choice theory [Arr50, Arr63] originated in anattempt to formalize an aggregate function for social opinion. Aggregate mechanismsexisted previously in welfare economics: Called welfare functions, they attempted tomeasure societal welfare for a number of alternative possibilities by aggregating the util-ity or welfare of individuals (measuring, for instance, the impact of a change in fiscalpolicy or tax rates). These mechanisms all shared the assumption that as subjectivea concept as individual utility could be compared or quantified. A breakthrough camewith the Bergson-Samuelson social welfare function [Ber38] which inspired Arrow’s ownaggregate mechanism, also called a social welfare function . Like the Bergson-Samuelsonmodel, Arrow broke from previous economic models by considering an individual’s voteto be their ranked preferences rather than a collection of numerical utilities over thealternatives. Thus the output of the social welfare function is a ranked ordering of thealternatives as well. Arrow argues that this is a more appropriate model for aggregatefunctions due to the difficulty of interpersonal comparisons of utility.Arrow’s key result, known as Arrow’s impossibility theorem, shows that no socialwelfare function can satisfy all of a set of five reasonable criteria whenever there are morethan two alternatives. By reasonable criteria we mean these conditions “. . . accord withcommon sense and with our intuition about fairness and the democratic process” [Szp10].In formalizing his aggregate mechanism, Arrow laid down two postulates that directedthe construction of individual preferences, and outlined the aforementioned five charac-teristics.The first postulate states that for any pair of alternatives a, b , every individual willalways have some opinion between them: individuals can be indifferent between them(generally represented as aIb or bIa , denoting indifference between a and b ), or preferone alternative to the other (generally represented as aP b in the case that a is preferredto b ). The second postulate is that an individual’s preferences must be transitive, thusdisallowing cycles in individual preference orderings. Note that this requirement is notuniversally held in voting theory, and intransitive preferences are sometimes consideredreasonable when voters use different criteria to decide between different pairs of alter-natives [Hug80]. Consider the example of an individual Jeff who has to rent a car, andis willing to pay a little more for additional space. Between a compact and a midsizecar, Jeff always chooses the midsize, since he has to pay just a little more for additionalcomfort. Similarly, between a midsize and a fullsize car, Jeff prefers a fullsize car. But Differences between the two functions are elaborated on throughout Arrow’s paper [Arr50].
Unrestricted Domain
By the two aforementioned postulates, an individual’s prefer-ences are represented as an ordering complete over the set of alternatives. Any restrictionon which sets of orderings are permitted as input to the function violates the democraticnature of the mechanism and would fail to satisfy this criterion. Unrestricted domainwould be violated in the example of elections held in a despotic state where only voteswith the current ruler ranked first are allowed.
Nonimposition
The second criterion states that the function should not allow inclu-sion or preclusion of outcomes irrespective of the preferences of the electorate. Thiscriterion implies the social outcome should depend entirely on the set of individual pref-erence orderings. An example of imposition could be an election in a theocracy whereonly candidates from the state religion are permitted to be elected.
Monotonicity
The third property, monotonicity, states that an individual cannotharm an alternative’s position by ranking it higher. In other words, if an aggregatepreference ordering holds that alternative Ted is preferred to alternative Jeff, then anindividual cannot harm Ted’s aggregate position by ranking him above Jeff in his orher ordering, all other individual orderings being constant. This property implies thatthat the aggregate decision must be responsive to and representative of the individual’spreferences.In the later version of his work, Arrow replaced monotonicity and nonimposition withthe combined property of the Pareto criterion, or unanimity [Arr63]. The Pareto criterionis slightly stronger than monotonicity since it incorporates nonimposition. It states thatfor any two alternatives a and b , if an individual preference ordering prefers a to b withall other individual preferences indifferent between these two alternatives, then the socialoutcome also prefers a to b . Independence of Irrelevant Alternatives
The fourth property, independence of ir-relevant alternatives (IIA), implies that individual preferences for any pair of alternativesshould not be influenced by other alternatives. A famous anecdote attributed to SidneyMorgenbesser [Pou08] illustrates IIA: Morgenbesser, ordering dessert in a restaurant,was informed by the waitress that the dessert choices were apple pie and blueberry pie.Morgenbesser chose apple pie. A few minutes later, the waitress returned and informedhim that cherry pie was also available. “In that case,” said Morgenbesser to the utter5onfusion of the poor waitress, “I’ll have blueberry.”IIA and the possibility of strategic behavior in a voting system are mutually exclusive:the presence of one in a voting model indicates the absence of the other. While anyhonest preference relation between a pair of alternatives would not be influenced byextraneous alternatives, strategic behavior often requires them. Consider a Borda electionbetween two alternatives Jeff and Mike where a certain voter prefers Mike, the strongercandidate, to Jeff. A new candidate, Ted, is introduced that the voter prefers to allothers. The voter, then, instead of his true preference ordering Ted > Mike > Jeff (whereTed > Mike implies that Ted is preferred to Mike), might misrepresent his preferences asTed > Jeff > Mike, in order to weaken Ted’s strongest opponent, Mike. In other words,the introduction of Ted results in the voter switching his preferences for Jeff and Mike.
Nondictatorship
The fifth property states the function should not permit an individ-ual who is a dictator, i.e., for a given profile of individuals, the function cannot reflectany one individual’s preferences, irrespective of the preferences of all others in the profile.Arrow proved that any thusly defined acceptable social welfare function, meetingall these criteria, cannot decisively aggregate the preferences of a profile of individuals ifthere are more than two alternatives. This result essentially meant that any social welfarefunction violated the most basic thresholds for acceptability, thus any such function wouldhave to compromise on meeting at least one of these five criteria. Arrow, in discussing thisproblem opined that compromising on an unrestricted domain was the only reasonablealternative [Arr50].One of the more notable approaches to this problem actually preceded Arrow’s work:In 1948 Scottish economist Duncan Black wrote about an intriguing property of societalpreferences called single-peakedness [Bla48, Bla58]. Consider plotting a preference order-ing with the horizontal axis representing a linear ordering of alternatives and the verticalaxis representing their rank in the ordering. If the resulting curve (drawn from joining allalternative-representing points together) has a single peak (defined to be a point flankedby either lower-ranking points on both sides, or only on one side if the peak starts or endsthe curve) then we can state that the preference ordering is single-peaked with respectto the linear ordering on the horizontal axis. For a given profile of preferences, if thereexists at least one linear ordering such that all votes are single-peaked with reference tothis linear ordering, then we pronounce the profile to be single-peaked, or admitting theproperty of single-peakedness.Black showed that aggregate functions admitting single-peaked preference profiles(with respect to some linear ordering) meet all of Arrow’s criteria except for unrestricteddomain. We lose this criterion since a linear ordering that induces single-peakedness does6 b c d b b b bb b bb bbbb
Figure 1: A preference profile that is single peaked for the ordering abcd . The votes are a >b > c > d (solid), b > c > d > a (dashed), and c > d > b > a (dotted). not exist for every preference profile . Therefore, if an aggregate function is to admit onlysingle-peaked input, it would have to exclude certain preference orderings, restricting thedomain.Another approach by Amartya Sen [Sen69] showed the existence of aggregate mech-anisms that have all of Arrow’s criteria but implement a relaxation of transitivity calledquasi-transitivity. This permits the existence of certain preferences that violate standardtransitivity—for example, for alternatives a, b, c the following preference profile is accept-able: aIb , bIc , aP c . Sen also discusses an aggregate function where the input, instead ofindividual preference orderings, is individual utility functions.Brams and Fishburn showed that approval voting similarly has very desirable prop-erties when we restrict the preference domain [Nie84]. In the case that voters havedichotomous preferences (that is, they can divide the candidates into two groups: onethey they approve of and one they do not), approval voting has many positive properties,including immunity to strategic voting and the Condorcet criterion [BF78]. In the generalcase with unrestricted preferences, the system no longer has these properties [Nie84].The impact of Arrow’s work can be summed up in a quote attributed to Paul Samuel-son [Pou08], “What Kenneth Arrow proved once and for all is that there cannot possiblybe . . . an ideal voting scheme.” The Gibbard-Satterthwaite theorem held even more dis-mal news: In addition to being less-than-ideal, all voting schemes are also vulnerable tomanipulation, unless they admit dictators. The existence of a single-peaked profile would (partially) depend on the set of least-preferredalternatives across all given orderings. For example, in the case of exactly three alternatives a, b, c ,any profile of two or more preference orderings having a total of two alternatives a, b ranked last, linearorderings acb and bca exist relative to which the profile is single-peaked. For methods to determinesingle-peakedness, we refer the reader to the work of Ballester and Haeringer [BH11] and Escoffier etal. [EL ¨O08]. .3 Gibbard-Satterthwaite Theorem In the 1970s Alan Gibbard [Gib73] and Mark Satterthwaite [Sat75] independently ex-tended Arrow’s theorem for voting systems in a model that incorporated strategic mis-representation of preferences. They proved that no strategy-proof voting system existedfor elections with three or more alternatives, unless the voting system allowed dicta-tors. A strategy-proof voting system is one where no manipulating strategy can existin elections using this voting system. While this result revolutionized voting theory, ithad been speculated previously: In 1960, William Vickrey, when discussing individualsstrategically misrepresenting their preferences in Arrow’s model [Vic60], stated that “it isclear that social welfare functions that satisfy the non-perversity [monotonicity] and theindependence [IIA] postulates and are limited to rankings as arguments, are also immuneto strategy.” In addition, the Dummett-Farquharson conjecture of 1961 [DF61] parallelsthe Gibbard-Satterthwaite theorem.The Gibbard-Satterthwaite theorem transformed voting theory for two reasons: onewas the aforementioned result that nondictatorial voting rules were susceptible to strate-gic voting (manipulation) in cases with three or more outcomes, and the second wasthe adoption of the arcane science of game theory from Neumann and Morgenstern’s
Theory of Games and Economic Behavior [vNM44], published just four years prior toArrow’s work. While the influence of game theory was implied (and acknowledged) inArrow’s social welfare function [Wil72], the Gibbard-Satterthwaite theorem proofs weremore explicit in their treatment of voting functions as game-theoretic mechanisms. Thisprovided the field of voting theory with a set of tools to examine a whole range ofscenarios—for example, the motivations for the electorate to vote dishonestly, or theirreaction to changes in the structure of the voting rule, or attempts to form coalitions tofurther their individual utility.In Gibbard’s work, a voting scheme or social choice function is built upon a constructcalled a game form [Gib73]. A game form is similar to a game construct in game the-ory [vNM44] applied to a voting model: players are voters, and any player’s strategiesare the set of all possible orderings of preferences (unless restricted for specific scenar-ios), over a set of alternatives or candidates. Game forms, unlike games, do not havefunctions assigning utilities for each player for a given action (or chosen strategy) or fora scenario of chosen strategies of all the players. Instead the social choice function hasthe concept of an honest or sincere strategy: Among the set of strategies for each playeris a specially marked strategy denoting the honest representation of preferences for thatplayer. This can be used to compare outcomes (which may be a preference ordering or asingle alternative) for a voter’s different strategies, to see if one is less or more preferableto another in the honest ranking of alternatives.This social choice function will be immune to manipulation only if each voter has a8 ominant strategy , a strategy that will be at least as good as any other for that voterno matter what any other voter does. Otherwise, if a voter does not have a dominantstrategy, then they might possibly be motivated to change their vote from their truepreferences in order to obtain a better outcome. A social choice function where eachvoter has a dominant strategy (and hence is immune to manipulation) is said to be straightforward .The proof of the Gibbard-Satterthwaite theorem relies on the Vickrey conjecture: avoting scheme (defined with the property of unrestricted domain) is strategy-proof if andonly if it has the properties of IIA and unanimity. Since Arrow showed that no aggregatefunction with more than two outcomes can satisfy all of his model’s criteria, such a votingsystem then is necessarily a dictatorship. In other words, we have that any voting systemwith at least three outcomes will either be a dictatorship or it will be manipulable. Asimilar result, the Duggan-Schwartz theorem [DS00], exists for voting systems that electmultiple candidates.The Gibbard-Satterthwaite theorem presents a problem and accepts a solution sim-ilar to Arrow’s theorem. By Black’s results, voting schemes that permit only single-peaked preferences restrict the domain of the function, but can have all other Arrowcriteria, including unanimity and IIA, which together imply strategy-proofness. Thus,relaxing the condition of unrestricted domain is necessary for voting schemes that re-sist manipulation. Another example of this is Gibbard’s development of probabilisticmechanisms [Gib77, Gib78]. Procaccia took a similar approach by designing strategy-proof probabilistic voting systems that are similar to standard deterministic voting sys-tems [Pro10]. Another solution to the problem of inherent manipulability in voting was proposed byBartholdi along with Tovey, Trick, and Orlin in a series of papers that started the fieldof computational social choice [BTT89a, BTT89b, BO91]. Their approach was to selectvoting schemes where manipulation is computationally difficult to carry out, i.e. wherethe manipulation problem is NP-hard. Our definition of the manipulation problem isthat of constructive coalitional manipulation: i.e., does there exist a set of votes for themanipulating coalition that causes their preferred candidate to win the election? Thissubsumes the case of a single manipulator and contrasts with destructive manipulation,which is concerned with preventing a certain candidate from winning.Bartholdi et al.’s initial work also highlighted the problems of selecting systems withcomplexity. Their impracticality theorem [BTT89b] is another instance of systems meet-ing seemingly reasonable criteria thereby inducing undesirable properties.The theorem states that any fair voting system requires excessive computation to9etermine the winner, making it impractical—a highly disturbing result. This theoremfollowed work by Kemeny [Kem59], Young and Levenglick [YL78] and Gardenfors [Gar76].According to Bartholdi et al., a voting system is fair if it meets the Condorcet cri-terion, the condition of neutrality (symmetry in its treatment of candidates, implied byIIA [GPP09]), and the condition of consistency (if disjoint subsets of the voters votingseparately arrive at the same preference ordering, then voting together always producesthis same preference ordering as well). Their theorem states that computation of thewinner in any such fair voting system is NP-hard.The previously mentioned property of consistency (also called convexity [Woo94] andseparability [Smi73]) has been proven to be present in ranked voting systems (those inwhich a vote is a ranking or ordering of preferences) only if they happen to be scoringprotocols as well (where alternatives receive a certain number of points depending ontheir position in the ordering) [You75]. Scoring protocols are often incompatible with theCondorcet criterion (refer the aforementioned debate on Condorcet versus Borda) thusunsurprisingly so far we know of only one voting system, Kemeny scoring [Kem59], thatmeets all three conditions [YL78]. Kemeny scoring then, is NP-hard, but additionally ithas been shown to be complete for parallel access to NP [HSV05].However, this does not imply other voting systems are immune to having an in-tractable winner problem: systems such as Dodgson meet only two of the three fairnessconditions—that of the Condorcet criterion and neutrality but not consistency—howevercomputation of the winner in Dodgson is known to be not only NP-hard [BTT89b] butcomplete for parallel access to NP [HHR97], similar to the result for Kemeny scoring.Research focusing on tractable voting systems was more promising: While Bartholdiet al. gave us a greedy algorithm that finds a manipulating vote for several tractable vot-ing systems in polynomial time [BTT89a], two voting systems—second-order Copelandand single transferable vote [BTT89a, BO91]—proved to be resistant and were shown tohave a manipulation problem that is NP-hard. Research in this field remained dormantfor the next fifteen years until a revival starting in 2006 brought about results for mostcommon tractable voting systems.In this paper we are concerned with a restricted version of the manipulation problem.We survey tractable voting systems that resist manipulation in the unweighted coalitionalmanipulation (UCM) model with only a constant number of manipulators. This limitedcase subsumes hardness results in the weighted coalitional manipulation (WCM) modelor with variably-sized coalitions, thus making a case for UCM being a stronger class ofmanipulation. We include both the initial work of Bartholdi, Tovey, and Trick [BTT89a]and Bartholdi and Orlin [BO91] that achieved the first results in this area and the recentresurgence of interest in this problem that has resulted in a number of new outcomes. A voting system is tractable if calculating the winner takes at most polynomial time. .5 Terms Defined An election is defined to be an instance of a voting system, comprising a voting rule vr ,a set of candidates C and a set of votes V . A voting rule is a function that takes as inputa set of votes and a set of candidates and outputs a set of winners. Unless explicitlystated otherwise, references to n and m are defined as follows: n = || V || and m = || C || .A vote is defined to be a linear ordering over the set of candidates. The advantage of acandidate c i over c j (hereafter referred to as adv ( c i , c j )) is the number of votes that rank c i ahead of c j , with values ranging from 0 to n . The net advantage of a candidate c i over c j is adv ( c i , c j ) − adv ( c j , c i ), with values ranging from − n to n . A netadv score betweentwo candidates c i and c j is represented as netadv ( c i , c j ). By definition we can see that netadv ( c i , c j ) = − netadv ( c j , c i ), thus one netadv score can represent both directions.UCM (unweighted coalitional manipulation) is defined to be a decision problem asfollows. Given
An election, namely, a voting rule vr , a set of voters V such that V = V NM ∪ V M ,where V M is the subset of voters that form the manipulating coalition and V NM isall other voters, and a set of candidates C containing a distinguished candidate c . Question
Does there exist a set of votes for V M such that vr over the complete set ofvotes yields c as the winner?We use the format UCM Cope to refer to the UCM problem in second-order Copelandand likewise for other election systems.
Single transferable vote (henceforward STV) is a voting system with a long history. Asesteemed a figure as John Stuart Mill said it was “among the greatest improvements yetmade in the theory and practice of government.” It determines the winners with a simplemultiround procedure that redistributes votes placed for less popular candidates. Also,unlike many of the esoteric voting systems studied in voting theory, STV has a history ofbeing used for real-world political elections, in the United States and around the world.The STV vote tallying procedure is as follows. Give a point to each candidate foreach first-place vote it receives. If any candidate is the majority winner (i.e. with morethan half the total points), that candidate will be the only winner of the election. Ifno majority winner exists, then select the candidates with the fewest number of points,remove them from consideration, and for the voters who currently give their supportto these candidates, reallocate their support by giving their points to the next-highestranked candidate on their ballots still under consideration. Repeat this procedure until a11inner is chosen or all candidates are removed. If the latter occurs, then all of candidatesthat were removed in the last round will be the winners.The complex, shifting behavior of STV with multiple candidates is what gives it theresistance to manipulation we discuss here, but it also leads to STV failing to possess somevery desirable voting system characteristics. Notably it does not possess the property ofmonotonicity: It is possible for a voter to increase his or her ranking of a candidate andfor that candidate to subsequently do worse in the election. This was enough for Doronand Kronick [DK77] to refer to it as a “perverse social choice function,” and it certainlyis a flaw of concern.
We show resistance to manipulation through a conventional, if difficult, reduction basedon Bartholdi and Orlin’s work [BO91]. Their work was actually directed towards showingthat the EFFECTIVE PREFERENCE problem is NP-complete. EFFECTIVE PREF-ERENCE is the problem of finding if a single voter can cause the preferred candidateto win an election. This is effectively the same as UCM with a single manipulator, andwe will prove that this problem is NP-hard for STV with a proof based on the afore-mentioned work [BO91]. The proof is structured as a reduction from the exact cover bythree-sets problem.
Exact Cover by Three-Sets (X3C)Given
A set D = { d , . . . , d k } and a family S = { S , . . . , S n } of sets of size three ofelements from D . Question
Is it possible to select k sets from S such that their union is exactly D ?In other words, the goal of the problem is to find if there is an appropriately-sizedset of subsets which covers each of the elements in D . Since each S i has exactly threeelements and k such sets from S must be chosen, the set of subsets must be an exactcover with no repeated elements in all subsets chosen. Proof.
We will describe a reduction from an instance of X3C to a instance of the un-weighted manipulation problem for STV. Note that since the reduction will only requirea single manipulator, this shows that UCM
ST V is NP-hard even for only a single manip-ulator.Given an instance of X3C ( D, S ) we construct the election as follows. Let the followingcomprise the candidate set C : • The possible winners c and w ; 12 The set of “first losers” a , . . . , a n and a , . . . a n , one of each corresponding to eachsubset S i ; • The “second line” b , . . . , b n and b , . . . , b n , one of each corresponding to each subset S i ; • The w -bloc d , . . . , d k , each of whose voters will just prefer them to w ; • The “garbage collector” candidates g , . . . , g n .We will now describe the set of voters. Where we use ellipses, the remainder of a voteis arbitrary for our purposes and will not effect the result of the election. • n voters with preferences ( c, . . . ); • n − w, c, . . . ); • n + 2 k voters with preferences ( d , w, c, . . . ); • For each i ∈ { , . . . , k } , 12 n − d i , w, c, . . . ); • For each i ∈ { , . . . , n } , 12 n voters with preferences ( g i , w, c, . . . ); • For each i ∈ { , . . . , n } , 6 n + 4 i − b i , b i , w, c, . . . ) and forthe three j such that d j ∈ S i , 2 voters with preferences ( b i , d j , w, c, . . . ); • For each i ∈ { , . . . , n } , 6 n + 4 i − b i , b i , w, c, . . . ) and 2voters with preferences ( b i , d , w, c, . . . ); • For each i ∈ { , . . . , n } , 6 n + 4 i − a i , g i , w, c, . . . ), 1 voterwith preferences ( a i , b i , w, c, . . . ), and 2 voters with preferences ( a i , a i , w, c, . . . ); • For each i ∈ { , . . . , n } , 6 n + 4 i − a i , g i , w, c, . . . ), 1 voterwith preferences ( a i , b i , w, c, . . . ), and 2 voters with preferences ( a i , a i , w, c, . . . ).This reduction works by requiring the elimination order of a subset of the candidatesto correspond to an exact cover over B in order for c to win the election. Namely, c will winthe election if and only if I = { i | b i is one of the first 3 n candidates to be eliminated } isan exact cover. Furthermore, there is a preference order for a single manipulator that willforce I to be an exact cover if one exists. We will now consider the relevant propertiesof the election and show that this is the case.Since this election is a single-winner STV election with more than two candidates,the scoring process will proceed for a number of rounds and a number of candidates will13e eliminated as the rounds progress. The first 3 n candidates to be eliminated will be a , . . . , a n , a , . . . , a n , and exactly one of b i or b i for every i ∈ { , . . . , n } .Candidate c initially has 12 n votes, while c ’s primary rival w has 12 n − c as their first choice ranks c directly below w , and so c can only gain more votes if w is eliminated. In order to do so, the manipulator mustensure that w does not gain additional votes before it is eliminated, as otherwise w wouldhave gained votes against c and c would have been eliminated first. The manipulatormust consequently make sure that no candidate is eliminated such that any voter mostprefers that candidate and prefers w second-most. This is the case with every voter thatprefers one of the d j candidates, so w will gain a large number of votes if one of themis eliminated. Therefore if any d j candidate is eliminated before w , c cannot possiblywin. For every b i candidate that is eliminated, b i gains a large number of votes, pushingit higher than 12 n in score and preventing it from being eliminated early. Also, every d j ∈ S i gains two votes, pushing them high enough to prevent their early elimination.Thus eliminating b i protects the d j candidates associated with the set S i . Conversely,for every b i that is eliminated, d gains two votes and b i gains a large number of votes,preventing b i from being eliminated early. Thus for every i only one of b i or b i can beeliminated before c or w .The a candidates are the other candidates that can be eliminated early. For every a i that is eliminated, a i gains two votes, b i gains one vote, and g i gains the rest of a i ’s votes.The effect of this is that now a i has been promoted above b i and b i in the overall ranking,and since b i also gained a point over b i , b i will be the next to be eliminated instead of b i . Thus by controlling which of a i or a i is eliminated first, we control which of b i or b i is eliminated early.Hence we can show that the candidate c will win the election if and only if I = { i | b i is one of the first 3 n candidates to be eliminated } is an exact cover. We know that either b i or b i will be among the first 3 n eliminated candidates. If b i is the one eliminated, thenevery d j ∈ S i will gain two votes and will each have at least 12 n votes total, protectingthem from early elimination. If I is an exact cover, this will be true for every d j as eachof them is covered by some selected b i and so each of them will win over w . Also, since d gains two points for every one of the b i eliminated, d will gain at least 2( n − k ) votes andwill receive at least 12 n votes overall, pushing it over the score of w as well. Thus afterthe first 3 n candidates have been eliminated, w will have the least score with 12 n − c will then gain a large number of votes fromthe elimination of w and will go on to win the election.If the set I defined above does not correspond to an exact cover, c cannot win theelection. If I is not an exact cover, some candidate d j will not gain the two points from14 corresponding b i being eliminated. Thus d j will only have 12 n − n candidates are eliminated while the other remaining candidates have at least12 n −
1, leaving d j as the next candidate to be eliminated. The candidate w then gainspoints from d j ’s elimination, preventing c from gaining points against w and winning theelection.If an exact cover exists, a single manipulator can construct it’s preference order asfollows to ensure c is a winner. For an exact cover I , if i ∈ I , let the i th candidate inthe preference order be a i and otherwise a i . The rest of the preference order is arbitrary.This will result in the following order of elimination of the first 3 n candidates: For i ∈ I , a i , b i , a i will be eliminated in that order in positions 3 i −
2, 3 i −
1, and 3 i . For i / ∈ I , a i , b i , a i will instead be the candidates to be eliminated. For i ∈ I , since this preference ranks a i over a i , a i will gain one more vote and thus a i will be eliminated first. This then givestwo votes to a i and one more to b i , making b i the next least-preferred candidate. When b i is eliminated, b i gains a large number of votes, so a i is now the least-preferred candidateand is eliminated next. The rounds proceed similarly in the case that i / ∈ I and a i ispreferred instead. Thus the set { i | b i is one of the first 3 n candidates to be eliminated } will correspond to an exact cover and c will win the election.If no exact cover exists, no matter how a manipulator votes, at least one of the d j candidates will not receive the protective boost from the elimination of the corresponding b candidate as previously described. This candidate will then be eliminated early, leadingto w being boosted past c in votes and preventing c from winning.Thus even just setting the first n rankings for the ballot of a single manipulator forSTV is NP-hard, and the system is resistant even to this very limited case of manipula-tion. Borda voting is a classic voting system dating back at least to the eighteenth century.It was introduced by the French mathematician and engineer Jean-Charles de Bordato remedy the failure of plurality in reflecting the wishes of the electorate when usedwith more than two candidates: In plurality the candidate with the most votes is notnecessarily preferred to all other candidates. Borda voting is very similar to a systemintroduced in the 15th century by Cardinal Nicolaus Cusanus [Szp10].It has a rich and varied history of real-world use: In some form it has been used inpolitical elections in Slovenia and the Micronesian countries of Kiribati and Nauru, inthe Eurovision contest, the election of the board of directors of the X.Org foundation,and even in sports, in the election of the Most Valuable Player award in Major LeagueBaseball. Borda is one of a class of systems known as scoring protocols, where each voteawards points to each candidate depending on their ranking in the vote. The winners are15andidates with the highest sum of points over all the votes. In the case of Borda voting,candidates receive linearly descending points for progressively less favorable positions inthe votes, with the top position awarding m − m − Borda was hard in general, though ma-nipulation with a single manipulator has long been known to be easy [BTT89a]. Agreedy algorithm that can find a set of successful manipulating votes in polynomial timethat is at most one larger than the optimum manipulative coalition size is known aswell [ZPR08]. Recently, Betzler et al. [BNW11] and Davies et al. [DKNW11] provedBorda-manipulation to be NP-hard for instances with two or more manipulators.
Both Betzler et al. [BNW11] and Davies et al. [DKNW11] prove their results by re-duction from the problem of 2-numerical matching with target sums, a known NP-hardproblem [YHL04] that closely corresponds to the problem of allocating points to thenonfavored candidates in the election.
A sequence a , . . . , a k of positive integers with P ki =1 a i = k ( k +1) and 1 ≤ a i ≤ k . Question
Are there permutations ψ and ψ of 1 , . . . , k such that ψ ( i ) + ψ ( i ) = a i for1 ≤ i ≤ k ? Preliminaries
For manipulation to work, nonfavored candidates must be ranked lowenough in manipulative votes such that the number of points they gain by said votes donot prevent the preferred candidate from winning. To that end we define the gap to bethe maximum number of points nonfavored candidates can gain by all manipulative voteswhile still allowing the preferred candidate to win. In any Borda instance with a favoredcandidate c ∗ , m other candidates, and t manipulators, the gap g i for a candidate c i is score ( c ∗ ) + t · m − score ( c i ). Here score ( c ) refers to the Borda score for the candidate c over the nonmanipulative votes. We assume these gap values g , . . . , g m to be ordered ina nondecreasing fashion. Result 1
Recall that the awarded points for the last j candidates in a vote will rangefrom j − j ( j − /
2. Thus for any successfulmanipulation instance, we must have that P ji =1 g i ≥ t · j ( j − / j ∈ { , . . . , m } .We define an instance to be tight if P ji =1 g i = t · j ( j − /
2. Thus, in a tight instance, formanipulation to be successful, the number of points a nonfavored candidate gains frommanipulative votes must be exactly equal to its gap value.
Proof.
Given any instance of 2NMTS we construct a UCM
Borda instance (
C, V, p ) as16ollows. The candidate set C consists of candidates c , . . . , c k and the preferred candidate p , and thus the range of Borda points is from 0 to k . The set of votes V consists of amanipulating coalition of size two and a set of nonmanipulating votes of size three. Theinstance is constructed such that the gap g i for any nonfavored candidate c i is 2 k − a i .In this context, the Borda problem can be considered as follows: for every nonfavoredcandidate c i , can we assign a position in each of the manipulating votes such that thepoints c i gains from said votes is ≤ g i ? This constructed instance of Borda manipulationwill have a solution if and only if the 2NMTS instance has a solution. Direction 1
Given a solution to 2NMTS, we can obtain a solution for the Borda instanceas follows: Preferred candidate p is placed in the first position in both manipulatingvotes. A solution to 2NMTS exists so we have two orderings ψ ( i ) , ψ ( i ) such that ψ ( i ) + ψ ( i ) = a i . For every candidate c i , (1 ≤ i ≤ k ) set its position to ψ ( i ) + 1in the first manipulative vote, and ψ ( i ) + 1 in the second manipulative vote. Thecorresponding Borda points are obtained from subtracting this position number from || C || = k + 1. Therefore the points c i has gained from both manipulative votes is( k + 1 − ( ψ ( i ) + 1)) + ( k + 1 − ( ψ ( i ) + 1)) which equals 2 k − a i = g i , permitting p towin. Direction 2
Given a solution to the Borda instance, we have a solution for 2NMTS asfollows: By construction, k X i =1 g i = k X i =1 (2 k − a i ).Since k X i =1 a i = k ( k + 1), k X i =1 (2 k − a i ) = k ( k − j = k and t = 2 (from Result 1) and conse-quently each nonfavored candidate c i gains exactly g i points from the manipulative votes.If pos i (1) and pos i (2) are the positions for c i in the two manipulative votes, points gainedfrom these positions total ( k + 1 − pos i (1)) + ( k + 1 − pos i (2)) = g i = 2 k − a i , which yields pos i (1) + pos i (2) = a i + 2. Therefore setting ψ ( i ) = pos i (1) − ψ ( i ) = pos i (2) − g i = 2 k − a i . This requires construct-ing the set of nonmanipulative votes such that the deficits for each candidate relativeto the preferred candidate map precisely to the target sums in the original problem.Executing this involves complicated construction and the addition of a large number of“dummy” candidates to pad out the remaining positions and precisely set the requireddeficits for the primary candidate set [BNW11] (in Davies et al., such padding is donewith voters [DKNW11]). Thus the constructed instance has a much larger candidate setthan a voter set (though it remains polynomially bounded), but it suffices to prove thedesired hardness result. 17 UCM in Copeland Elections
Copeland voting is a voting system with a long history. One version of the system wasdiscovered by the 13th century mystic Ramon Llull, and then another variation wasdiscovered by A.H. Copeland in the 1950s. It is a Condorcet voting system.Copeland voting is in fact a family of voting systems, parametrized on how tiesare handled. In Copeland α , the score of a candidate c in an election E is wins E ( c ) + α · ties E ( c ), where wins E ( c ) denotes the number of pairwise victories and ties E ( c ), thenumber of ties of the candidate c in the election E . Llull’s system is Copeland , while“Copeland voting” has been used to describe Copeland . or to describe Copeland .Different parameter values subtly alter the behavior of the system and complicate thetask of analyzing its computational properties, as we will explore. α UCM
Cope α is in P when there is only one manipulator [BTT89a], but it is known to beresistant to manipulation even with two manipulators for α ∈ [0 , . ∪ (0 . ,
1] [FHS10,FHS08]. The complexity of UCM
Cope α for α = 0 . α for α ∈ { , } [FHS10]. Both these cases were proved by Faliszewski et al. [FHS10]through similar reductions from X3C. Exact Cover by Three-Sets (X3C)Given
A set D = { d , . . . , d k } and a family S = { S , . . . , S n } of sets of size three ofelements from D . Question
Is it possible to select k sets from S such that their union is exactly D ? Proof.
The proof constructs a UCM
Cope α instance from an X3C instance ( D, S ) usinggraph representations to equate both problem instances. A election for the reduction canbe constructed without an explicit collection of votes as such a collection can be elicitedfrom the set of netadv or adv scores for all pairs of candidates . The graph representationof the election can be constructed from the set of netadv or adv scores as well . Boththese constructions are polynomially bounded in the size of the set of candidates and thevalue of the netadv function. Thus, using these techniques we can construct the electiongiven a partial set of significant candidates, the netadv scores for all pairs of candidates, Refer Appendix A. Refer Appendix B. bbbb b bb bbb b
222 2 22 c − ( n − k ) c i, ( − c i, ( − c i, ( − d i, ( − d i, ( − d i, ( − d ′ i, ( − d ′ i, ( − d ′ i, ( − S i (3) Figure 2: Gadget used in the Copeland manipulation NP-hardness proof [FHS10]: The gadgetis constructed for each S i with numbers for each candidate showing the lead in Copeland scorethe preferred candidate has over them. Given any instance of an X3C problem, the proof defines the set of candidates as theelements of sets D and S , one preferred candidate p , and one main adversary candidate c (the candidate holding the highest Copeland score prior to manipulation) as well asauxiliary candidates used for padding and constructing the mathematical gadgets. Suchgadgets (also called widgets [CLRS01]) are a common feature in reduction construc-tions: They are typically intricate subgraphs that are constructed for each element of themapped-from problem. They align calculations and enforce constraints so as to ensure atight mapping between instances of the two problems.The numbers of the election are constructed in such a way that prior to manipulation,the elements of D each beat p by 1 unit of Copeland score and c beats p by n − k unitsof Copeland score. However, the elements of S each lose to p by 3 units. Hence anysuccessful set of manipulating votes must cause the elements of D and c to lose justenough pairwise contests against the elements of S to erode their lead over p withoutmaking any S i a possible winner. This involves selecting a subset of S that beats everycandidate from D but still leaves enough elements of S to lower c ’s score by n − k points.The construction of such a vote corresponds to selecting a k -sized subset of S that coversexactly the elements of D , that is, a solution to the X3C problem. Hence, a solution toUCM Cope α gives us a solution to X3C. Solutions for breaking ties in Copeland voting attempt to choose the more “powerful”candidate as the winner, which can be defined in a number of ways. One such method issecond-order Copeland. It selects the candidate whose set of defeated opponents (here-after referred to as DO c for a candidate c ) has the higher sum of Copeland scores. We19ill let sum score ( S ) for a set of candidates S be the sum of the Copeland scores forcandidates in S , and so sum score ( DO c ) gives the second-order Copeland score for acandidate c .Second-order Copeland has been used by the National Football League and the UnitedStates Chess Federation to break ties, and has a special place in voting theory: it wasthe first tractable voting system for which the manipulation problem was shown to beNP-complete, even for just one manipulator [BTT89a]. The problem of unweighted coalitional manipulation in second-order Copeland, hereafterreferred to as UCM Cope , is NP-complete even for one manipulator [BTT89a]. Verifying agiven solution is clearly in P as we simply have to calculate Copeland scores and second-order Copeland scores for each candidate. To prove UCM Cope is NP-hard we show apolynomial-time reduction from 3,4-SAT, a known NP-hard problem [Tov84].
A set U of Boolean variables, a collection of clauses Cl , each clause composed ofdisjunctions of exactly three literals , which may be a variable or its complement,and each variable occurs in exactly four clauses. Question
Does there exist a Boolean assignment over U such that each clause in Cl contains at least one literal set to true?To facilitate the reduction we construct a graph representation of a second-orderCopeland election that encodes the given 3,4-SAT instance. We will use a election graphrepresentation with vertices representing candidates and directed edges representing theresult of pairwise contests. Proof.
Given a 3,4-SAT instance, we create a second-order Copeland election graph asfollows: Every clause ( C to C || Cl || ) and every literal is a candidate, represented as a ver-tex in our graph. The manipulating coalition’s chosen candidate is a separate candidate C . All pairs of vertices have directed edges between them, representing decided pairwisecontests, except for any variable and its complement. The decided contests cannot beoverturned by our manipulators, while undecided contests can be shifted in either direc-tion according to the manipulating vote. Clauses beat (that is, have a directed edge to)literals they contain, and lose to all other literals.In addition to these candidates derived from the 3,4-SAT instance, we pad the electionwith a number of auxiliary candidates in such a way to achieve the desired Copelandscores and second-order Copeland scores for each of the candidates. We will have that20 bbb b bb bb b C C C C x ¬ x x ¬ x x ¬ x Figure 3: A partial representation of the resultant election graph: Clauses represented are C ( x ∨ x ∨ x ) , C ( x ∨ ¬ x ∨ x ) , C ( ¬ x ∨ ¬ x ∨ x ) and C ( x ∨ x ∨ ¬ x ) . Each clausevertex beats the variables (or their complements) that are its component literals. The dotted linesindicate undecided (second-order Copeland) contests between variables and their complements. all the clause candidates and C are tied with the highest Copeland score. We will alsohave that each clause candidate C i has sum score ( DO C i ) = sum score ( DO C ) − C will be independent of the variable-complementcontests. For all possible outcomes of the variable-complement contests, C still beatsevery candidate except for the clause candidates. Recall that the elements for everyclause candidate’s defeated-opponent set are their component literals. Their second-order Copeland scores and the final result of the election will then depend on how eachof the variable-complement contests are decided.Consider if any clause candidate C i ’s literals win all their contests. The sum score ( DO C i )increases by 3 points and C i is tied with C for first place. Therefore, in order for C tobe the unique winner, at least one element of each clause candidate’s defeated-opponentset must lose one of their contests. For any variable x , we can interpret a directed edgefrom x to ¬ x as setting x to true . A vote that allows C to win, then, would correspondexactly to each clause having at least one literal evaluating to true. In other words, wehave a solution for UCM Cope if and only if we have solution for 3,4-SAT. ThereforeUCM Cope is NP-hard with just a single manipulator.
Maximin voting, also known as the Simpson-Kramer method, is a typical Condorcetvoting system. As such it deals with contests between pairs of candidates, specificallytheir netadv scores. To find the winner under maximin, given a netadv function overthe set of candidates, we first select the lowest netadv score for each candidate k in C ,21.e., we select the minimum score for netadv ( k, k ′ ) for all k ′ in C such that k ′ = k . Thewinner is the candidate with the highest such score. We can trivially see that a candidatewith a minimum netadv score greater than 0 will be the Condorcet winner and there canonly be one such candidate, thus maximin is a Condorcet voting system.Calculating the maximin winner is easily seen to be polynomial in the size of theelection, but Xia et al. [XCPR09] prove that for two or more manipulators the problemof UCM maximin is NP-complete. UCM maximin is NP-complete for two or more manipulators [XCPR09]: Verifying an in-stance is easily seen to be polynomial in the size of the election as we can calculate the win-ner in polynomial time. To prove UCM maximin is NP-hard we construct a polynomial-timereduction from the vertex-disjoint-two-path problem, known to be NP-complete [LR78].
Vertex-Disjoint-Two-Path problem (VDP )Given A directed graph G and two sets of vertices u, u ′ and v, v ′ such that all fourvertices are unique. Question
Do there exist two paths u → u → . . . → u j → u ′ and v → v → . . . → v k → v ′ in G such that each path is a set of vertices disjoint from the other? Proof.
To facilitate our reduction construction from a graph problem such as the vertex-disjoint-two-path problem to maximin, we first construct a graph representation of max-imin elections. A complete set of votes is not required to represent an election . We canconstruct the same given just a netadv (or adv ) function. Also, there exists a bijectionbetween the netadv function and directed edges of a complete antisymmetric graph suchthat given one, we can represent it in terms of the other . P coal is the set of votes of the manipulating coalition and P noncoal is the set of all othervotes. The term netadv noncoal indicates the netadv score obtained by considering onlythe noncoalitional votes with netadv coal similarly defined. M is the size of the coalitionand c is the candidate supported by the manipulating coalition.Given a VDP instance, that is a graph G ( V G , E ) and vertices u, u ′ , v, v ′ ∈ V , weobtain a graph G ′ using the following constructions and assumptions: • Every vertex in our graph is reachable from u or v . • There are no directed edges u → v ′ or v → u ′ . Refer Appendix A. Refer Appendix B. We add special edges u ′ → v and v ′ → u such that E G ′ = E ∪ { ( u ′ , v ) , ( v ′ , u ) } .Our UCM maximin instance then is as follows: • Set of candidates C = V G . • The set of P noncoal preferences. – ∀ c ′ ∈ C : c ′ = c, netadv ( c, c ′ ) = − M ; – netadv ( u, v ′ ) = netadv ( v, u ′ ) = − M ; – For all other edges ( x, y ) in E , netadv ( y, x ) = − M − – For all other vertex pairs a, b , netadv ( a, b ) = 0.Regarding P coal votes, we can freely assume that every coalition vote will rank c first, thus giving netadv coal ( c, c ′ ) = M and netadv noncoal ∪ coal ( c, c ′ ) = − M . Thus, in ourconstruction, scores for netadv ( c, c ′ ) are fixed for all candidates c ′ ∈ C . In order for c to be the winner, at least one netadv score must be less than − M for every othercandidate. We will see that in our construction this can occur if and only if there aretwo vertex-disjoint paths in G ′ . Direction 1
The existence of vertex-disjoint paths u → u → . . . → u j → u ′ and v → v → . . . → v k → v ′ yields a P coal that makes c the winner. In order to construct themanipulative preferences, we will make use of a connected subgraph over G ′ containingall the vertices, but with u → . . . → u ′ → v → . . . → v ′ → u as the only cycle in thegraph.We can construct P coal votes in 3 parts as follows: Each manipulator vote will rank c the highest, followed by the vertex-disjoint-path vertices, followed by the other vertices. Other-vertex ordering:
These vertices will be ordered in the votes based on a linearorder extracted from the single-cycle subgraph.
Vertex-disjoint-path orderings:
We have two vertex-disjoint-path orderings: u → . . . → u ′ → v → . . . → v ′ and v → . . . → v ′ → u → . . . → u ′ and thus two possible voteconstructions for P coal . We construct M − . Thus netadv ( c, c ′ ) increases by M points but every other netadv scoreincreases by less than M points, making c the winner. The calculations are as follows forthe complete (coalitional and noncoalitional) set of votes: • netadv ( u, v ′ ) = − M + ( M − − − M − • netadv ( v, u ′ ) = − M + 1 − ( M −
1) = − M + 2 Switching these orderings results in the same outcome. c ′
6∈ { c, u, v } , we can see that there exists some candidate d inevery vote of P coal that beats c ′ , i.e., the lowest netadv score for c ′ is netadv coal ( c ′ , d ) = − M ,and thus for the complete set of votes the (lowest) netadv for any such candidate c ′ is nomore than − M − − M = − M −
2. All the above netadv scores are less than − M for all values of M ≥
2, thus c is the winner. Direction 2
The existence of a P coal that makes c a winner yields a positive VDP instance in the graph G ′ :Since c is the winner, we know that for any other candidate c ′ in C , there exists acandidate d that beats c ′ such that: • netadv ( c ′ , d ) < − M ; • There exists an edge ( d, c ′ ) in G ′ ; • d is ranked higher than c ′ in a majority of the total votes and in at least one votein P coal —the proof of this is as follows.Consider such an edge ( d, c ′ ) : either ( d, c ′ ) is one of the special edges ( v ′ , u ), ( u ′ , v ′ )or ( d, c ′ ) ∈ E . If ( d, c ′ ) is a special edge, then at least one vote in P coal must prefer d to c ′ (since netadv ( c ′ , d ) < − M ). If ( d, c ′ ) ∈ E , then all M votes in P coal must prefer d to c ′ . For this d , we can choose a candidate that beats it with sufficient margin, and continueto find such a candidate for the previous choice of d . That is, we find a d for c ′ startingwith u and then continue to find such a candidate after setting d to c ′ . There is only onepossible d for c ′ as either u or v . Thus, we obtain a set of chained pairs recursively. This,coupled with the facts that any vertex is reachable from u, v and the existence of specialedges (both by construction), we obtain a cycle of vertices which breaks into disjointsets along the special edges. Obtaining such a set of pairs with netadv less than − M is not possible without the existence of a ( c -winner-making) P coal (as per the third itemabove). Tideman ranked pairs (TRP) was conceived by Nicolaus Tideman in 1987 when attempt-ing to define a voting system that “almost always” has the property of independence ofclones [Tid87]. It is defined as follows: given a netadv function over the set of candidates,create a list by ranking the pairs in descending order of their scores. In the case of a tiebetween two netadv pairs, e.g., netadv ( a, b ) = netadv ( x, y ), we break ties by ordering If there exists more than one such d we choose one arbitrarily. Choosing such a d is formalized in the proof of Xia et al. [XCPR09] as a composite function f . G if the resultant graph does not contain acycle. Otherwise we skip this pair and move on to the next one in the list. We continueuntil we have considered all pairs. Since we now have a directed acyclic graph, theremust exist a source vertex, which we state to be the TRP winner. Xia et al. [XCPR09] found that UCM
T RP is NP-complete even for one manipulator. Wecan easily see that verifying a given solution to UCM
T RP is in P. UCM
T RP was provento be NP-hard by a polynomial-time reduction from 3SAT.
Three-Conjunctive-Normal-Form Satisfiability (3SAT)Given
A set U of Boolean variables, a collection of clauses Cl , each clause composed ofdisjunctions of exactly three literals , which may be a variable or its complement. Question
Does there exist a Boolean assignment over U such that each clause in Cl contains at least one literal set to true? Proof.
Given a 3SAT instance, we construct a UCM
T RP election graph as follows. Clauses C . . . C || Cl || are vertices as is the coalition’s preferred candidate c . For each clause C i , weconstruct six other special clause candidates—three for the literals it contains and threefor their complements. C i beats (has a directed edge to) the special clause candidatescorresponding to the literals it contains, which in turn beat the literals they correspondto. We also have a C ′ i such that it is beaten by the complement of the literals in C i .Candidate c starts out beating the C i candidates but gets defeated (though by a smallermargin) by the C ′ i candidates. The intuition of this correspondence is that the edgesbeating c are so weak and so far down the ordering that they are not added, leaving c tobe the source vertex (and TRP winner) if and only if there exists a solution to 3SAT. Thusthe UCM problem is NP-complete even in the case of a single manipulator. The proof ofXia et al. [XCPR09] relies on mathematical gadgets to achieve this correspondence. Our survey of UCM results can be seen as qualifying election systems by a single metric.Determining which election system is superior is an ongoing debate often reflecting differ-ing philosophies. Pierre-Simon Laplace in his lectures at the Ecole Normale Superieurein 1795 attacked the ´el´ection par ordre de m´erite (election by ranking of merit) systemof his contemporary Jean-Charles de Borda [Szp10], later proposing a variation of themajority rule in its place. Much of the modern literature on voting theory is still devoted As usual, V G = C and directed edges represent netadv scores. oting rule Coalition size = 1 Coalition size ≥ α (0 < α < . . < α <
1) P [BTT89a] NP-complete [FHS08]Copeland α ( α = { , } ) P [BTT89a] NP-complete [FHS10]Copeland α ( α = { . } ) P [BTT89a] ?Second-order Copeland NP-complete [BTT89a] NP-complete [BTT89a]Single Transferable Vote NP-complete [BO91] NP-complete [BO91]Maximin P [BTT89a] NP-complete [XCPR09]Tideman Ranked Pairs NP-complete [XCPR09] NP-complete [XCPR09]Borda P [BTT89a] NP-complete [BNW11, DKNW11]Bucklin P [XCPR09] P [XCPR09]Plurality with Runoff P [ZPR08] P [ZPR08]Veto P [BTT89a] P [ZPR08]Cup P [CSL07] P [CSL07] Table 1: Table of UCM results for common tractable voting systems. to advocacy for particular voting systems, arguing their superiority by one metric oranother [New92, Saa06, Ris05].In our survey we showcase manipulation results for a particular class of voting sys-tems, namely those with a tractable winner problem but where unweighted coalitionalmanipulation is hard for a constant coalition size. The complexity of this case of UCMhas been determined for most common voting rules, though a few remain: Copeland . remains unsolved even as results for all other parameter values have been found [FHS10].Several related areas of research, however, remain more or less uncharted. The mostsignificant simplification in the literature is that most of the hardness results achievedare just worst-case. Several papers have studied whether voting systems are difficult tomanipulate in a large fraction of instances, finding that manipulation can be easy inthe average case while being hard in the worst case [CS06, PR07b, PR07a]. Addition-ally, approximation algorithms exist for several worst-case hardness results. Brelsfordet al. [BFH +
08] formalized manipulation as an optimization problem and then studiedwhether this version of the problem is approximable. Zuckerman et al. [ZPR08] dis-covered an approximation algorithm for Borda manipulation before it was known to beNP-hard, and Davies et al. [DKNW11] gave several other approximation algorithms forthis problem. In other results, approximation algorithms for manipulation of maximin aswell as families of scoring protocols exist [ZLR10, XCP10]. Other techniques include theuse of relatively efficient algorithms for the NP-complete integer partitioning problem tosolve manipulation instances [Lin11].Conitzer et al. [CSL07] qualified the manipulation problem with an additional metric:the minimum number of candidates that must be present for manipulation to be NP-hard. Additionally their work breaks from the standard model and studies whethermanipulation is hard for cases where manipulators do not have complete information of26ll of the votes. Slinko explored how often elections will be manipulable based on thesize of the manipulative coalition [Sli04].Research into how often elections can be manipulated [FKN08], and more general ar-eas such as parametrization of NP-hard problems [Nie10] and phase transitions [CKT91,KS94, Zha01] lead to a more nuanced approach to problem classification. Phase transi-tions have been examined in the manipulation problem for the veto rule [Wal09].Another issue is that votes are most commonly represented in the literature as tran-sitive linear preference orderings over the set of candidates and the concept of irrationalvotes has only been sparsely dealt with. Irrational (by which we mean intransitive)votes, may be more apt for any number of real-world scenarios where voters tend torank candidates according to multiple criteria. Irrational votes are not represented asa linear ordering but as a preference table which holds the voter’s choice for any pairof candidates. For Copeland α for α ∈ { , . , } , manipulation is in P in the irrationalvoter model, while it is known to be NP-hard for α ∈ { , } in the standard votermodel [FHS10]. Thus voting systems may have different behavior with regard to manip-ulation in the irrational voter model and it deserves more study. Another convention isthat the default definition of UCM is constructive—i.e., efforts are directed to making apreferred candidate a winner, rather than preventing a certain candidate from winning.Variations of UCM with a destructive approach is another area rich with possibilities.UCM instances presented in this paper typically have a large number of candidatesand a smaller constant-sized coalition of manipulators. In contrast, cases with a smallnumber of candidates and a relatively large manipulating coalition might be consideredmore natural. Betzler et al. [BNW11] mention a specific open problem in this area:whether there exists a combinatorial algorithm to solve Borda efficiently with few can-didates and an unbounded coalition size. Another open problem is solving a UCM Borda instance having a coalition of size 2 in less than O ( || C || !).Another approach to the manipulation problem taken by Conitzer and Sandholm [CS03]and Elkind and Lipmaa [EL05] is modifying voting systems to give them greater resis-tance to manipulation. Both add an extra initial round of subelections between subsetsof the candidates. Conitzer and Sandholm [CS03] describe techniques that can makemanipulation NP-hard or even PSPACE-hard for these modified voting systems. Elkindand Lipmaa [EL05] present a version of this technique that uses one-way functions toconstruct the initial-round schedule from the set of votes. Reversing the one-way functionis computationally hard, preventing election organizers from gaming the initial round andforcing their desired result in polynomial time. These techniques essentially constructnew voting systems by structurally augmenting standard systems to imbue them withcomplexity.Other related work includes the study of electoral control, which encompasses attempts27y an election organizer to change the result by modifying the election structure in vari-ous ways [BTT92, FHHR09, EF10b, HHR09], which also encompasses cloning, or addingcandidates very similar to existing candidates in an attempt to split their support [Tid87,EFS10]. Other ways to influence elections include bribery and campaign management,where in both cases a briber attempts to sway the result of an election by paying off a setof voters to change their votes [FHH09, FHHR09, Fal08, EFS09, EF10a, SFE11]. These,too, are problems endemic to many voting systems to which complexity can serve as adefense.Another possible response to the problems presented by Arrow’s theorem and theGibbard-Satterthwaite theorem is to reconsider the standard model of the aggregatefunction. Balinski and Laraki [BL07] introduce a model where voters give candidatesindependent grades, such as the letter grades F to A or { good, average, bad } , similar toapproval voting or range voting, rather than ranking them in a linear order. In a sense thisrepresents a reversion to the pre-Bergson-Samuelson model of welfare functions. Balinskiand Laraki’s method defines the aggregate grade of each candidate to be the mediangrade over all votes, unlike range voting where the aggregate grade is the average. Wecan obtain a complete aggregate preference ordering of the candidates with this methodprovided that ties can be broken. Balinski and Laraki give a tie-breaking mechanism thatsuccessively removes one of the median-score-awarding voters from the votes for each tiedcandidate and recomputes the median grades until they are no longer tied [BL07]. Theirapproach does not rely on complexity but instead redesigns the election model to becomestrategy-proof in a limited case defined by the authors.Faliszewski et al. [FHHR11] showed that with a restriction to single-peaked pref-erences, a wide range of manipulation and control instances that are NP-hard in thegeneral case turn out to be easy (though not any of the results we describe here). Forsome voting systems these problems remain easy even with a partial relaxation of thesingle-peaked model that allows for a small number of “mavericks”, whose votes are notaligned with the single-peaked ordering [FHH11]. The single-peaked model is considered“ the canonical setting for models of political institutions” [GPP09], so this work calls thesignificance of a number of hardness results into question.After the birth of research in the manipulation problem with the work of Bartholdi etal. [BTT89a], most research moved towards the weighted voter model and many resultsfor the weighted coalitional manipulation problem (WCM) were achieved [CSL07, HH07],until the resurgence of interest in the UCM problem [BNW11, DKNW11, FHS10, FHS08,XCPR09]. It can be argued that as compared to WCM, UCM is a better test of a votingsystem’s vulnerability to manipulation. UCM serves as a special case of WCM andhence subsumes its hardness results. In other words, if an election system is resistantto manipulation in the UCM case, it will resist manipulation in the WCM case, but theother direction does not necessarily follow. With this problem solved for most common28oting systems, we look forward to the resolution of the remaining open problems as wellas new avenues of research into the manipulability of voting systems. A Constructing an Election Given a netadv
Function: the Mc-Garvey Method
While the traditional representation of an election requires a set of votes, we can constructthese votes given a pairwise relation denoting preference over the set of candidates. Thismethod was given by McGarvey [McG53], and can be applied with very little modificationto a netadv function.
A.1 From a Preference Pattern to a Set of VotesTheorem A.1. (McGarvey’s Theorem)
Given a preference pattern we can elicit a set ofvotes (defined to be strict and complete preference orderings over the set of candidates)such that (the ordering derived from) the preference pattern is the result of the election.
Definition 1. A preference pattern is a set of relations over the set of candidates. Therelations are a preference relation (expressed as aP b viz. a is preferred to b ) and anindifference relation ( aIb viz. a is neither preferred to b nor is b preferred to a ). Both relations are distinct for any pair of candidates - i.e., aP b implies ¬ bP a , and aIb implies bIa . Thus we will have m ( m − / m is thenumber of candidates. McGarvey’s method constructs a set of votes as follows:For each pair aP b with remaining candidates c , . . . , c m − , we construct two preferenceorderings abc . . . c m − and c m − . . . c ab . For each pair aIb , we construct abc . . . c m − and c m − . . . c ba The idea is that on evaluation for these preferences, rankings of allcandidates besides a, b from these orderings will be equal, and the rankings of a, b reflectthe preference relation under consideration. Consider the example C = a, b, c, d . The sixpairs we consider are aP b, aP c, aP d, bP c, bP d, cP d .To represent aP b we construct two votes abcd and dcab . Evaluating these two votesin the context of pairwise rankings leads to two votes for aP b and no votes for any otherpair over the set of candidates. Similarly, for aP c , we construct acbd and dbac and so on.The idea is that for all candidates besides the ones under consideration, preferences forand against them cancel each other out. Hence the need for two votes for each pair. Thetotal number of thus-constructed votes is twice the cardinality of the preference pattern.Thus in our example, we obtain a set of votes which yield exactly the relations in thegiven preference pattern. Where candidates are listed in order of decreasing preference. .2 From a netadv Function to a Set of Votes
We consider the following useful property of the netadv function when constructing thecorresponding election:
Theorem A.2.
For all pairs of candidates c i , c j where c i = c j , the values of netadv ( c i , c j ) are either all even or all odd.Proof. Consider two blocs of votes where each bloc takes one side in a pairwise election.Let the sizes of the blocs be x and y such that x + y = n .If n is even: • Then x, y are either both odd or either both even since the sum components of aneven number are either both even, or both odd. • The difference between two even numbers or two odd numbers is always even.If n is odd: • Then x, y are either odd and even, or even and odd, respectively, since the sumcomponents of an odd number are always a combination of even and odd. • The difference between an even and odd number is always odd.Thus, all the netadv values are either all even or all odd.We can see that the netadv function corresponds to elements of a preference pattern: netadv ( c i , c j ) > c i P c j , netadv ( c i , c j ) = 0 corresponds to c i Ic j , and netadv ( c i , c j ) < c j P c i . However, the key difference between the netadv functions and preference-pattern elements is that netadv has scores, which we must factorinto our construction.Since we construct two votes for each pair of candidates, the problem of applyingMcGarvey’s method to a netadv relation with an even score is trivial—for each of thetwo preference orderings constructed we simply have n/ netadv function, if one netadv score is even, then (1) the number of votes will be even,and (2) every netadv score in that set will be even. The converse also applies: if one netadv score in a given set is odd, then the number of votes will be odd, and every netadv score in the given set will be odd.Applying McGarvey’s method to an odd set of netadv scores requires a slight tweak.We first must select some arbitrary ordering of the candidates. Then for any netadv score s = netadv ( a, b ) where a precedes b in the ordering, we construct s − s − / s = netadv ( b, a ) where a precedes b , we will instead create ( s + 1) / s + 1 for b over a . To obtain the last points, we construct one preference ordering correspondingto the previously chosen ordering of the candidates. This will give one net vote to everypair a, b where a precedes b , and minus one vote where b precedes a . Thus we achievethe desired odd numbers for each netadv input. Number of Constructed Votes
In the above two cases ( netadv scores being even orodd) we can see the upper bound on the number of votes constructed is one for everyunit of score across all netadv values. The size of the set of votes will be boundedby P c i ,c j ∈ C | netadv ( c i , c j ) | . Thus given a netadv function with bounded value, we canconstruct a reasonably small set of votes. B Graph Representation of the netadv
Function
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