Manipulable outcomes within the class of scoring voting rules
MManipulable outcomes within the class of scoringvoting rules ∗ Mostapha Diss † Boris Tsvelikhovskiy ‡ September 28, 2020
Abstract
Coalitional manipulation in voting is considered to be any scenario in which a groupof voters decide to misrepresent their vote in order to secure an outcome they all preferto the first outcome of the election when they vote honestly. The present paper is de-voted to study coalitional manipulability within the class of scoring voting rules. Forany such rule and any number of alternatives, we introduce a new approach allowing tocharacterize all the outcomes that can be manipulable by a coalition of voters. This givesus the possibility to find the probability of manipulable outcomes for some well-studiedscoring voting rules in case of small number of alternatives and large electorates undera well-known assumption on individual preference profiles.
Keywords:
Voting; Scoring Rules; Coalition; Strategic Manipulation; Probability.
JEL classification:
D71, D72
Since the seminal papers of Gibbard (1973) and Satterthwaite (1975) who proved that everynon-dictatorial social choice rule can be manipulated in the presence of at least three alterna- ∗ The authors would like to thank Eric Kamwa for comments and helpful discussions. The first author grate-fully acknowledges the financial supports from Université de Lyon (project INDEPTH Scientific BreakthroughProgram of IDEX Lyon) within the program Investissement d’Avenir (ANR-16-IDEX-0005) and from Univer-sité de Franche-Comté within the program Chrysalide-2020. The second author wishes to express his gratitudeto F. Aleskerov for introducing him to the subject and constant guidance on the early stages of development ofthis project. He also acknowledges the support of the International Laboratory of Decision Choice and Analysis(National Research University Higher School of Economics). † CRESE EA3190, Univ. Bourgogne Franche-Comté, F-25000 Besançon, France. Email:[email protected]. ‡ Department of Mathematics, Northeastern University, Boston, MA, 02115, USA. Email: [email protected]. a r X i v : . [ ec on . T H ] S e p ives, the problem of coalitional manipulation has received a lot of attention in recent decadesin social choice theory. Broadly speaking, a given social choice rule is called coalitionallymanipulable if there exists a given list of voting preferences and a coalition of voters, suchthat the preferences of all the voters outside the coalition remain the same, while the prefer-ences of voters within the coalition can be altered in such a way that the winner changes andeach of the voters from the coalition is ‘happy about the change’.Scoring voting rules, also called positional voting rules , have attracted a considerableamount of attention in the literature dealing with manipulation. This class of voting rulescan be defined as follows: each voter’s preference must be a vector that gives a number ofpoints that the voter assigns to each alternative according to his or her position in the voter’spreference. The points assigned by all voters are summed and the winning alternative has thehighest number of points. A number of studies has been conducted on the evaluation of thedegree of manipulability of various social choice rules, i.e., the extent to which social choicerules are manipulable by a coalition of voters or by an individual voter. The reader mayrefer, for instance, to Aleskerov and Kurbanov (1999), Chamberlin (1985), Diss (2015), ElOuafdi et al. (2020a), El Ouafdi et al. (2020b), Favardin and Lepelley (2006), Favardin et al.(2002), Gehrlein et al. (2013), Kamwa and Moyouwou (2020), Kelly (1993), Kim and Roush(1996), Lepelley and Mbih (1994), Lepelley et al. (2008), Moyouwou and Tchantcho (2017),Nitzan (1985), Peleg (1979), Pritchard and Wilson (2007a), Pritchard and Wilson (2007b),Saari (1990), and Schürmann (2013). The methodology used in this literature consists firstin characterizing the specific conditions that must be required for a given voting rule tobe manipulable by a coalition of voters. The final step consists in the evaluation of the(theoretical) probability of this situation under various assumptions on voters’ preferences.For more details on those probabilistic assumptions and their use in social choice theory,the reader can refer to Diss and Merlin (2020), Gehrlein and Lepelley (2017), Gehrlein andLepelley (2011).However, as pointed out in a recent paper by El Ouafdi et al. (2020a), with some notableexceptions the results appearing in the literature only deal with three-alternative elections,not because it is the most interesting case but due to the difficulties arising when consideringmore than three alternatives. The main goal of this paper is to provide a significant improve-ment in this direction. This is achieved via presenting systems of linear inequalities, whichdetermine manipulable profiles, that is the lists defining the vote of all voters taking part inthe decision process, for the whole class of scoring voting rules independently on the numberof alternatives. More precisely, we focus on a detailed exposition of the new approach forobtaining the list of linear inequalities a profile satisfies if and only if it is manipulable incase of m ≥ alternatives for all scoring rules.The paper is organized as follows. Section 2 describes the basic framework. The coreof the paper is Section 3, where the main results are presented. We start of with providing asimplified illustration of how our methodology works in case of three-alternative elections.Then, the precise list of inequalities for the whole class of scoring rules is presented. Atten- A coalition is defined as any non-empty subset of voters. m = and m = alternatives under the well-known Impartial and Anonymous Cultureassumption (defined later). The results are approximate since they are obtained by Monte-Carlo simulations, but with a high degree of precision, since we used profiles of cardinality · for m = and · for m = alternatives. To the best of our knowledge noneof those results have appeared in the literature, with the exception of the Plurality rule and m = (see El Ouafdi et al. (2020a)). The new results clearly indicate that among the threepositional rules under consideration Antiplurality has a much lesser degree of manipulability(see Tables 5, 6, and 7). The last section presents our conclusions. Let us assume that the number of voters/individuals is denoted by n and the number ofalternatives/candidates by m . The voters’ preferences are assumed to be linear orders whichmeans that voters rank alternatives from most preferred to least preferred and indifference isnot allowed. An anonymous preference profile , hereafter simply called a profile , is an m ! -tuple of non-negative integer numbers ( n , n , . . . , n m ! ) such that n i ≥ for all m ! ≥ i ≥ and m ! (cid:80) i = n i = n . Each n i is equal to the number of voters with preferences of type i . In thelimit as n → ∞ , we consider the normalized profile vectors of the form p = ( p , . . . , p m ! ) with each p i = n i n ≥ and m ! (cid:80) i = p i = . Throughout the paper, the alternatives will be called A , A , . . . , A m and the types of possible preference rankings of the alternatives will alwaysbe listed following an increasing order of j in the notation A j . In addition, the proportionof individuals having each possible preference ranking will be denoted accordingly. Forinstance, in case of m = alternatives, the possible preference rankings are denoted by ( A , A , A ) p , ( A , A , A ) p , ( A , A , A ) p , ( A , A , A ) p , ( A , A , A ) p and ( A , A , A ) p . The notation ( A , A , A ) p , for instance, means that a fraction p ofindividuals have preferences with A being most preferred, A being least preferred andwith A being ranked between them.In the case of m alternatives and individual preferences expressed as linear orders, wecan define the class of scoring voting rules as follows: Definition 2.1.
In the case of linear orders on m alternatives, every scoring voting rule canbe defined by the weight vector w = ( w , w , . . . , w m ) such that w ≥ w ≥ . . . ≥ w m and w > w m . It means that each time an alternative is ranked r -th by one voter it obtains w r points; afterwards we select the alternative(s) obtaining the greatest aggregated score. This is also known as a voting situation in the literature. m alternatives, i.e., w = (
1, 0, . . . , 0 ) . Under Borda rule, we select the alternativewith the highest Borda score such that each first-place vote is worth m − points, eachsecond-place vote is worth m − points, and so on until point to each last-place vote, i.e., w = ( m −
1, m −
2, . . . , 1, 0 ) . Under Antiplurality rule, we return the alternative with thehighest aggregated score such that each voter assigns one point for any one of the m − bestranked alternatives, i.e., w = (
1, 1, . . . , 1, 0 ) .Note that in the presence of three alternatives A , A , and A , we see that any generalscoring rule w = ( w , w , w ) is equivalent to the normalized scoring vector (
1, λ, 0 ) with ≤ λ ≤ . This is obtained by subtracting w and dividing by w − w any component of w . The well-known scoring rules become the Plurality rule with λ = , the Borda rule with λ = , and the Antiplurality rule with λ = .As already mentioned we focus in this paper on the coalitional manipulation which canbe defined as follows: Definition 2.2.
The outcome of an election is said to be coalitionally manipulable if thereexists an alternative such that all members of the electorate for whom she is preferable overthe winning alternative can change their preferences in such a way that this alternative winsthe election. The preferences of the rest of the electorate remain the same.We mainly focus in this paper on the manipulation by coalitions of maximal sizes butwe will also show that the approach that we consider can be adapted when manipulation bycoalitions of smaller sizes of the electorate is considered.We say that the initial arrangement is ( A , A , . . . , A m ) if the collective ranking of thealternatives before manipulation is: A first, A second and so on until alternative A m whichis ranked last. Let us consider an example in order to highlight the notion of manipulation. Example 2.3.
Let m = and the scoring rule be Borda rule, i.e., w = (
2, 1, 0 ) . To show amanipulable outcome, we set p = , p = , and p j = for j =
1, 3 . By the definitionof Borda rule, the initial arrangement is ( A , A , A ) since alternative A wins and gets points, A gets points, and finally A gets points. But, if voters having the preference ( A , A , A ) chose to vote ( A , A , A ) instead, then the final result would be in favor ofalternative A since A gets points, A gets points, and A gets points.With a given initial arrangement, m − different coalitions can be formed. This is dueto the fact that for each alternative, who has not won the election, there may be a group ofvoters who rank it higher than the winner. As any coalition is determined by the alternative itwould prefer to have as the winner of the election, the alternatives’ names will be used for thenotations of the coalitions hereafter. Alternative A i will be called unifying for the coalitiondenoted by Coal A i that would prefer to have A i as the winner and let us denote by C A i theproportion of individuals in this coalition. In the case of m = alternatives, for instance,4wo coalitions can be formed since the unifying alternative can be any alternative except thewinner. In this case, when the winner is alternative A , we have that alternative A is unify-ing for the coalition Coal A formed by voters with preferences ( A , A , A ) , ( A , A , A ) and ( A , A , A ) and alternative A is unifying for the coalition Coal A formed by voterswith preferences ( A , A , A ) , ( A , A , A ) and ( A , A , A ) . It is clear that the preferences ( A , A , A ) and ( A , A , A ) participate in both coalitions Coal A and Coal A .We introduce now the concept of intermediate preferences . Definition 2.4.
Intermediate preferences are preferences of the form ( ∗ , ∗ , . . . , ∗ ) with each" ∗ " symbol being either an alternative’s name or the " ? " mark, the latter representing the factthat its corresponding rank has yet to be assigned by the voter.Typical examples of intermediate preferences in three-alternative elections are (? , ? , ?) where the concerned voter is undecided on all of the three positions and ( A , ? , ?) wherethe concerned voter ranks alternative A at the first position and he/she is undecided on thesecond and third positions in his/her preference.Let us now introduce the number d ( A k , A i ) , with i = k , which denotes the differ-ence in points between the unifying alternative A k and alternative A i prior to the coalitionparticipants’ arrangement of all the places except the first. In the case of three alterna-tives, when the winner is alternative A for instance, individuals of Coal A (of proportion C A = p + p + p ) vote for alternative A , i.e., the unifying alternative of that coalition. Inaddition, those individuals only assign weight to their first preferred alternative and zeroweight to the other alternatives prior to the coalition participants’ arrangement of the secondand third places. In other words, those individuals have an intermediate preference of type ( A , ? , ?) . Note that the other individuals (of proportion p + p + p ) assign all their pointsand vote sincerely when we focus on the manipulation in favor of A . As a consequence, thescore of alternative A is λp + p + p + p and the score of alternative A is p + p + λp .This leads to d ( A , A ) = p + p + p − p + ( λ − ) p − p . In this section we show that for every coalition there exists a system of inequalities whichcharacterize the set of all the profiles that are manipulable by it. In order to make a simplifiedillustration of the general case that we will provide in Theorem 3.1, let us start first by thecase of three-alternative elections. Let again the alternatives be referred to as A , A , and A and consider the representation w = (
1, λ, 0 ) , with ≤ λ ≤ , that defines all possiblescoring rules with three alternatives. Using all the above definitions, we are ready to providethe list of inequalities which characterizes the set of all the profiles p = ( p , . . . , p ) thatare manipulable by a coalition of voters for three alternatives. The first system indicatesconditions where manipulation is in favour of A whereas the second system is given for a5anipulation in favour of A . p i ≥
0, i ∈ {
1, 2, . . . , 6 } p + p + p + p + p + p = ( − λ ) p + p + ( λ − ) p − p + λp − λp > 0, A beats A λp − λp + p + ( − λ ) p − p + ( λ − ) p > 0, A beats A p + p + p − p + ( λ − ) p − λp > 0, d ( A , A ) > 0 ( − λ )( p + p + p ) + ( − ) p − ( + λ ) p − ( + λ ) p > 0, d ( A , A ) + d ( A , A ) > λ C A p i ≥
0, i ∈ {
1, 2, . . . , 6 } p + p + p + p + p + p = ( − λ ) p + p + ( λ − ) p − p + λp − λp > 0, A beats A λp − λp + p + ( − λ ) p − p + ( λ − ) p > 0, A beats A p + p + p − p + ( λ − ) p − λp > 0, d ( A , A ) > 0p + p + p − p + λp − λp > 0, d ( A , A ) > 0 ( − λ )( p + p + p ) − ( + λ ) p + ( − ) p − ( + λ ) p > 0, d ( A , A ) + d ( A , A ) > λ C A . (3.1)Without loss of generality we assume that the initial arrangement is ( A , A , A ). Thismeans that A has more total points than A who in turn has more total points than A . Thisexplains the two first inequalities of every system. Recall that when the winner is alternative A we have that alternative A is unifying for the coalition Coal A formed by voters withpreferences ( A , A , A ) , ( A , A , A ) , and ( A , A , A ) . If we count the number of pointsfor each alternative prior to the coalition participants’ arrangement of all the places exceptthe first which is given to the unifying alternative A k (i.e., having intermediate preferencesof the form ( A k , ? , ?) ), the unifying alternative A k must have the largest number of points.Thus, the two inequalities d ( A k , A i ) > 0, i = k are necessary with k = and k = for the first and the second systems, respectively. Note that the inequality called ‘ A beats A ’ in the first system leads to d ( A , A ) > 0 . The next step is to understand whenindividuals of the coalition unified by an alternative can successfully manipulate the election.Actually, all the freedom they possess at this point is to give λ points from each participantof the coalition (by assigning the second place) to one of the remaining two alternativeswithout violating the condition that their alternative has more points. This can be done ifand only if the sum of the differences in points between the unifying alternative and eachof the remaining two, prior to the coalition members’ making a choice of their second mostpreferable alternative, is greater than λ times the number of people in the coalition. Thisis equivalent to d ( A , A ) + d ( A , A ) > λ C A and d ( A , A ) + d ( A , A ) > λ C A inthe first and the second systems, respectively. Recalling that we consider a manipulation bycoalitions of maximal sizes (i.e., C A = p + p + p and C A = p + p + p ), this leadsto the last inequality of every system. That analysis gives first insights on the validity of ourapproach since the systems described above are exactly the same as the ones already obtainedby Moyouwou and Tchantcho (2017) who determined necessary and sufficient conditions6or a given profile to be coalitionally manipulable under a general scoring rule with threealternatives.We consider now the general case of m alternatives. Recall that the unifying alternativecan be any alternative except the winner so that m − coalitions can be formed. Here againwe consider the coalitions of maximal sizes, i.e., each coalition consists of all voters withthe unifying alternative ranked higher than the winner in their preferences. The set of suchpreferences has cardinality m ! . Theorem 3.1.
Let m ≥ and suppose that the initial arrangement is ( A , A , . . . , A m ) under the positional voting rule having weight vector w = ( w , w , . . . , w m ) . A profile p = ( p , . . . , p m ! ) is manipulable by the coalition with unifying alternative A k if and only if p satisfies the following system of inequalities: Preliminary Inequalities (PI) m ! (cid:80) i = p i = i ≥ The initial arrangement is ( A , A , . . . , A m ) Strategic Inequalities (SI) (cid:80) i = k d ( A k , A i ) > ( w + . . . + w m − + w m )C A k (cid:80) i = k d ( A k , A i ) − M > ( w + . . . + w m − + w m )C A k . . . (cid:80) i = k d ( A k , A i ) − m − (cid:80) j = M j > ( w m − + w m − + w m )C A k (cid:80) i = k d ( A k , A i ) − m − (cid:80) j = M j > ( w m − + w m )C A k M m − > w m C A k , (3.2) where M = max { d ( A k , A i ) | i = k } , M = max { d ( A k , A i ) | i = k } , . . . , M m − = max m − { d ( A k , A i ) | i = k } . The notation max s { d ( A k , A i ) | i = k } stands for the s th largestvalue in { d ( A k , A i ) | i = k } .Proof. First let us show that the SI inequalities in (3.2) are necessary. This will be checkedby induction on m , the base being m = . Notice that it is clearly reasonable for all themembers of the coalition to start filling their profiles by updating their intermediate prefer-ences to ( A k , ? , . . . , ?) . The first inequality in the SI system simply guarantees that after thatupdate the coalition participants can distribute the remaining votes they have to give withoutviolating the fact that A k has more points than any other alternatives with no restrictions onthe distribution of the votes. The second to last inequalities of the SI system can be seen tobe those for the positional rule with w = ( w , w , . . . , w m − , w m ) and m − alternatives7alternative A s with M = d ( A k , A s ) is eliminated). The statement follows.Now we establish the sufficiency of inequalities (3.2). We argue by induction on the numberof alternatives m , the base being m = . The sufficiency of inequalities (3.2) is estab-lished via the step by step procedure of updating the intermediate preferences of coalitionparticipants presented below. Step 1 . All the members of the coalition start filling their preferences by putting A k as the mostpreferrable alternative. At this point the preferences of all participants of Coal A k areidentical and equal to ( A k , ? , . . . , ?) , where the question marks stand for the places yetto be assigned. Step 2 . Let A s be the alternative with d ( A k , A s ) = M . We will distinguish between twocases: case (a) M > w C A k . Then all participants of Coal A k update their intermediate pref-erences to ( A k , A s , ? , . . . , ?) and the list of inequalities in the SI system of (3.2)without the second one from the top is easily seen to be the one for the positionalrule with m − alternatives and weight vector ( w , w , . . . , w m ) ; case (b) w σ C A k < M < w σ − C A k for some ≤ m . We would like to point outthat the case M < w m C A k can never occur as it would imply (cid:80) i = k d ( A k , A i ) < ( m − ) w m C A k < ( w + . . . + w m − + w m )C A k and violate the first (SI) inequality.Let ε > 0 be an arbitrary small positive constant, satisfying( (cid:63) ) ε does not exceed the smallest difference between the left-hand side andright-hand side of the SI inequalities of (3.2).It will serve as the remaining difference in points between A k and A s after theupdate of preferences described below. Set t := M − w σ C Ak w σ − − w σ − ε and let t ‘par-ticipants’ update their intermediate preferences to ( A k , ? , . . . , ? , A sσ − , ? , . . . , ?) andthe remaining C A k − t to ( A k , ? , . . . , ? , A sσ , ? , . . . , ?) . Introduce the weight w σ − := (C A k − t ) w σ − + tw σ C A k (3.3)and set w = ( w , . . . , w σ − , w σ − , w σ + , . . . , w m ) . (3.4)After updates of coalition participants’ intermediate preferences and eliminationof alternative A s , the system of SI inequalities in (3.2) becomes the one for the8ositional rule with m − alternatives and weight vector w : (cid:80) i = k,s d ( A k , A i ) > ( w + . . . + w σ − + . . . + w m )C A k (cid:80) i = k,s d ( A k , A i ) − M > ( w + . . . + w σ − + . . . + w m )C A k . . . (cid:80) i = k,s d ( A k , A i ) − σ − (cid:80) j = M j > ( w σ − + . . . + w m )C A k (cid:80) i = k,s d ( A k , A i ) − σ − (cid:80) j = M j > ( w σ + + . . . + w m )C A k . . . (cid:80) i = k,s d ( A k , A i ) − m − (cid:80) j = M j > ( w m − + w m − + w m )C A k (cid:80) i = k,s d ( A k , A i ) − m − (cid:80) j = M j > ( w m − + w m )C A k M m − > w m C A k , (3.5)which, using that M = d ( A k , A s ) = ( w σ − + w σ − w σ − )C A k , can be rewrittenas (cid:80) i = k d ( A k , A i ) > ( w + . . . + w σ − + w σ + . . . + w m )C A k + ε (cid:80) i = k d ( A k , A i ) − M > ( w + . . . + w σ − + w σ + . . . + w m )C A k + ε. . . (cid:80) i = k d ( A k , A i ) − σ − (cid:80) j = M j > ( w σ − + w σ + . . . + w m )C A k + ε (cid:80) i = k d ( A k , A i ) − σ − (cid:80) j = M j > ( w σ + + . . . + w m )C A k . . . (cid:80) i = k d ( A k , A i ) − m − (cid:80) j = M j > ( w m − + w m − + w m )C A k (cid:80) i = k d ( A k , A i ) − m − (cid:80) j = M j > ( w m − + w m )C A k M m − > w m C A k (3.6)Recalling that M is greater than any other M i by definition and using the ( (cid:63) ) assumption on ε , we see that each of the inequalities in (3.6) follows from thecorresponding SI inequality in (3.2).9et us now illustrate the results of this theorem using an example. Example 3.2.
Consider the Borda rule with weight vector w = (
3, 2, 1, 0 ) for m = alternatives A , A , A , and A and the voting profile given in Table 1 below. In this example,the initial arrangement is ( A , A , A , A ) as it is shown in the last row. We would liketo determine if the chosen profile is vulnerable to manipulability by the largest possiblecoalition with unifying alternative A , i.e., Coal A of cardinality C A = p + p + p = .Table 1: Voting profile and corresponding point distributionPreference Fraction A A A A ( A , A , A , A ) p = ( A , A , A , A ) p = ( A , A , A , A ) p = ( A , A , A , A ) p = ( A , A , A , A ) p = ( A , A , A , A ) p = Other p i = (cid:80) In order to answer this question we will adhere to the step by step procedure given in theproof of Theorem 3.1 above. The intermediate preferences of the participants in the coalitionafter the first update are presented in Table 2.Table 2: Step intermediate preferences and corresponding point distributionPreference Fraction A A A A ( A , ? , ? , ?) C A = ( A , A , A , A ) p = ( A , A , A , A ) p = ( A , A , A , A ) p = Other p i = (cid:80) This allows to evaluate d ( A , A i ) for every i = : d ( A , A ) = − = ( A , A ) = − = ( A , A ) = − = M = max { d ( A , A i ) | i = k } = d ( A , A ) = and since w C A = = w C A , we are in case ( b ) of Step ( ) above. The next stepis to compute t = − · − − ε = − ε and update the intermediate profiles accordingly (see Table 3). At this point we have reducedthe problem to the case of three alternatives ( A , A , and A ) and positional rule determinedby the weight vector (
3, w , 0 ) , where w = ( + ε ) · + ( − ε ) · = − = − Table 3: Step intermediate preferences and corresponding point distributionPreference Fraction A A A A ( A , A , ? , ?) t = − ε 6/9 −
2ε 1 −
3ε 0 0 ( A , ? , A , ?) + ε 2/9 + ε 6/9 +
3ε 0 0 ( A , A , A , A ) p = ( A , A , A , A ) p = ( A , A , A , A ) p = Other p i = (cid:80) − ε 15/9 8/9 1 At this stage the question of manipulability is resolved according to whether or not theinequality d ( A , A ) + d ( A , A ) > w C A is satisfied. Plugging in the corresponding values we obtain + = − ε = ( − ) , securing the affirmative answer.One possible arrangement of the complete profiles is presented in Table 4 and this con-cludes our example. 11able 4: Complete strategic profile and resulting point distributionPreference Fraction A A A A ( A , A , A , A ) ( − ε ) · − − − ( A , A , A , A ) ( + ε ) · + + + ( A , A , A , A ) ( − ε ) · − − − ( A , A , A , A ) ( + ε ) · + + + ( A , A , A , A ) p = ( A , A , A , A ) p = ( A , A , A , A ) p = Else p i = (cid:80) − ε 75/45 61/45 + + Since w = w = . . . = w m = under the Plurality rule, the SI inequalities reduce to d ( A k , A i ) > 0 for i ∈ {
1, . . . , m | i = k } . Then the following corollary can be deduced fromTheorem 3.2. Corollary 3.3.
Consider the Plurality voting rule with m ≥ alternatives. A profile p =( p , . . . , p m ! ) is manipulable by a coalition with unifying alternative A k if and only if p satisfies the following system of inequalities: m ! (cid:80) i = p i = i ≥ The initial arrangement is ( A , A , . . . , A m ) d ( A k , A i ) > 0 for i ∈ {
1, . . . , m | i = k } ( SI ) . (3.7)We would like to bring the reader’s attention to the fact that the SI inequalities in (3.7)are the same as inequalities ( ) in Theorem of Lepelley and Mbih (1987), where the char-acterization of coalitional manipulable profiles under Plurality rule was first derived. We alsobring the reader’s attention to the fact that Lepelley and Valognes (2003) and Favardin et al.(2002) deal with the coalitional manipulation of the Antiplurality rule ( λ = ) and the Bordarule ( λ = ), respectively, in the case of three-alternative elections. The approach that weuse in this paper to derive our list of inequalities is different from those used by Lepelley andValognes (2003) and Favardin et al. (2002). This explains why the systems given in thesepapers (Lemma 2 in Lepelley and Valognes (2003) and Lemma 4 in Favardin et al. (2002))are not a direct consequence of the systems given in Theorem 3.1. However, our results inSection 4 for the case of three alternatives allow us to show that the two characterizations areequivalent.As already noticed, the approach that we consider in the main result of the paper canbe adapted when only manipulation by small coalitions is considered. Indeed, under some12rofiles a small coalition may reverse the relative ranking of two alternatives as it is shownin the following example. Example 3.4.
Let m = and the scoring rule be Borda rule having w = (
2, 1, 0 ) . As inExample 2.3, we set p = , p = , and p j = for j =
1, 3 . The initial arrangementis ( A , A , A ) since alternative A wins and gets points, A gets points, andfinally A gets points. But, if a fraction p = of voters chose ( A , A , A ) instead of ( A , A , A ) , and the others kept their preferences unchanged, then the final result would bein favor of alternative A since A gets points, A gets points, and A gets points. In other words, a manipulation in favor of A is also possible with a coalition ofvoters smaller than the maximal size (see Example 2.3 for a manipulation by a coalition of amaximal size). Corollary 3.5.
Let m ≥ and suppose that the initial arrangement is ( A , A , . . . , A m ) under the positional voting rule having weight vector w = ( w , w , . . . , w m ) . A profile p = ( p , . . . , p m ! ) is manipulable by some coalition not exceeding a proportion p ≤ of the electorate with unifying alternative A k if and only if p satisfies the following system ofinequalities: Preliminary Inequalities (PI) m ! (cid:80) i = p i = i ≥ ≤ e p j ≤ p j for j, s.t. A ≺ j A k (cid:80) A ≺ j A k e p j = p The initial arrangement is ( A , A , . . . , A m ) Strategic Inequalities (SI) (cid:80) i = k d ( A k , A i ) > ( w + . . . + w m − + w m )p (cid:80) i = k d ( A k , A i ) − M > ( w + . . . + w m − + w m )p . . . (cid:80) i = k d ( A k , A i ) − m − (cid:80) j = M j > ( w m − + w m − + w m )p (cid:80) i = k d ( A k , A i ) − m − (cid:80) j = M j > ( w m − + w m )p M m − > w m p , (3.8) where by A ≺ j A k we understand that A k has a higher ranking than A in preference oftype j and e p j denotes the share of ‘participants’ with preference of this type. The added PI inequalities correspond to the fact that each e p j can not exceed p j . As it is sufficient toconsider coalitions of the largest allowed proportion, we add the equality (cid:80) A ≺ j A k e p j = p . Remark 3.6.
Theorem 3.1 does not hold true in case of finite number of voters. Considerthe case of m = alternatives A , A and A and the scoring rule with weight vector w =(
1, 0.9, 0 ) . Let there be n = voters and the profile p = (
6, 7, 8, 0, 0, 0 ) . Then initially A gets points, A gets points and A gets points satisfying the arrangementassumptions. Next we show that the remaining two inequalities in the system, responsiblefor the possibility of manipulation by Coal A , hold true for the profile p as well. Indeed, d ( A , A ) = and d ( A , A ) + d ( A , A ) = = λ C A . However, ifa single member of the coalition updated his/her profile to ( A , A , A ) then λ = = d ( A , A ) would lead to alternative A getting ahead of A . On the other hand itis not possible for all people in Coal A to update their profiles to ( A , A , A ) either, as d ( A , A ) = = λ C A . Hence, the profile is not manipulable by Coal A .We finish this part with a remark on the manipulability of the Antiplurality rule. Remark 3.7.
Suppose that the initial arrangement is ( A , A , . . . , A m ) . The group of vot-ers having the last alternative prior in their preferences ( Coal A m ) can never manipulate anelection under Antiplurality rule whichever the number of alternatives m is. Otherwise, asafter such a manipulation the last alternative A m would have the same number of points (hisnumber of points is not influenced by the manipulation) all the other alternatives would haveto undergo a simultaneous loss of points. That is clearly impossible, because the number ofrearranged points equals to the number of points initially arranged between the alternatives A , . . . , A m − . The aim of this section is to compute the share of manipulable profiles for three well-studiedscoring rules under the assumption of Impartial and Anonymous Culture (IAC) in the limitcase as n → ∞ . We consider this hypothesis because it has been widely used in manypapers that we cite in the introduction. When n → ∞ , this assumption indicates that allprofile vectors of the form p = ( p , . . . , p m ! ) , with each p i ≥ and m ! (cid:80) i = p i = , are equallylikely to occur. This probability distribution was first introduced by Fishburn and Gehrlein(1976). For more details on the IAC condition and others, we refer the reader to Diss andMerlin (2020), Gehrlein and Lepelley (2011), Gehrlein and Lepelley (2017).It is well-known that the limiting probability of every voting event (here the manipulationof voting outcomes) under IAC can be simply reduced to volume computations. Since thenormalized profile vectors are of the form p = ( p , . . . , p m ! ) with each p i ≥ and m ! (cid:80) i = p i = ,the polytope that describes all possible profile vectors defines the standard simplex R m ! .The volume of R d is equal to ( d − )! , which, since in our case d = m ! , becomes14 m !− . What remains to compute is the volumes of the polytopes P A k (with k =
1, . . . , m )cutting out the regions of profiles manipulable by
Coal A k inside the standard simplex . Forinstance, in the case of three alternatives, the volumes of the polytopes P A and P A cuttingout the regions of profiles manipulable by Coal A and Coal A (see systems in (3.1)) insidethe standard simplex are required. Then the total volume of the region of manipulableprofiles is given by Vol ( P A )+ Vol ( P A )− Vol ( P A ∩P A ) . Afterwards the result is dividedby the volume of the standard simplex and multiplied by (all possible final arrangements ofalternatives) to produce the final answer. The approach described in Cervone et al. (2005) tells us an effective way to tackle this problem for three-alternative elections. When we havemore than three alternatives, however, those methods become too complicated in practice.Indeed, due to dimension reasons, the precise probabilities seem to be difficult to obtain for m = and it is impossible to compute for m ≥ at this stage using the existing algorithms.In case the precise volume cannot be found, it is natural to obtain a sufficiently good ap-proximation of it. One of the most commonly used procedures for this purpose is the MonteCarlo volume estimation. The share of manipulable outcomes for Plurality, Antiplurality andBorda rule in case of m =
3, 4, 5 alternatives are presented in Table 5 (exact results), andTables 6 and 7 (simulated results).Table 5: Share of manipulable outcomes with m = alternativesRule Manipulable by Coal A by Coal A Plurality % % %Antiplurality % % %Borda % % %Table 6: Share of manipulable outcomes with m = alternativesRule Manipulable by Coal A by Coal A by Coal A Plurality % % % %Antiplurality % % % %Borda % % % %Table 7: Share of manipulable outcomes with m = alternativesRule Manipulable by Coal A by Coal A by Coal A by Coal A Plurality % % % % %Antiplurality % % % % %Borda % % % % % See also Moyouwou and Tchantcho (2017). m = for the Plurality rule was studied in ElOuafdi et al. (2020a). The precise value of % for general manipulability was obtained(this corresponds to % appearing in the top left corner of Table 6). In other words,our approximation differs by %. Note also that the zero value in the last column of theAntiplurality rule is consistent with Remark 3.7.Finally, the Python code for carrying out the computation of the share of manipulableoutcomes in case of Borda rule with m = alternatives is provided in Appendix A . Adetailed outline on the execution complimentary to the comments inside the code is given.The recipe for finding the corresponding results for other scoring voting rules with than m ≥ alternatives is provided. At least three main extensions can emerge from the present study. First, it would be in-teresting to investigate other voting rules not yet considered, especially in the class of theso-called iterative scoring rules , also known as multi-stage or sequential elimination scoringrules . This class of multi-round procedures are based on the same scoring principle previ-ously defined but proceed by eliminating one or more alternatives at each round, until thereis only one alternative left who is considered as the winner. Second, taking into account thepossibility of reactions of voters outside the coalition that manipulates the vote may also be agood extension of this study. This appears to have a considerable impact on the manipulabil-ity shares and on the induced hierarchy of voting rules when analyzing strategic voting (seefor instance Favardin and Lepelley (2006)). Third, our probabilistic investigations may alsobe extended to the framework with small sizes of manipulating coalitions using Corollary3.8. 16 ppendix A: Python code used for computations in case of Borda countwith m = alternatives In this section we present the code (written in Python) used for carrying out the algorithmfor finding the share of manipulable outcomes in case of Borda rule with m = alternatives.With a few modifications to the code provided one can perform analogous computations forscoring voting rules with different weights. Step 1 . Create the array of possible preferences with the alternatives corresponding to numbers to arranged in increasing order. This array has elements. Each element of thepreferences’ array is an array of its own with elements (see lines − in the code).preferences [ ] = { [ ] , [ ] , [ ] , [ ] } , preferences [ ] = { [ ] , [ ] , [ ] , [ ] } ,. . . preferences [ ] = { [ ] , [ ] , [ ] , [ ] } . After this the point distribution array is created, i.e., the element ( i, j ) (here ≤ i ≤ and ≤ j ≤ ) is the number of points alternative j gets from preference i (lines − .) Step 2 . Now we are in position to store the inequalities responsible for the initial arrangementof alternatives. As usual we assume that this arrangement is (
0, 1, 2, 3 ) and the PIinequalities responsible for that are obtained (lines − and − ). Step 3 . Next generate a sample of n (‘trials’ in the code) random points in the simplex .For this choose random numbers on the closed interval [ ; ] , arrange them in anondecreasing order and take the subsequent differences. The points created this wayhave a uniform distribution in the simplex (see Chapter V.2 in Devroye (1986)). Thecorresponding lines of the code are − . Step 4 . The coalition arrays are established. These are arrays of elements, each elementequal to or , depending on whether preference i belongs to the coalition or not (lines − ). Step 5 . The difference arrays are generated (lines − ). Step 6 . The remaining part of the code stores the SI inequalities (3.2) responsible for the profilebeing manipulable by each of the coalitions separately (recall that inequalities (3.2) arewritten in terms of the elements of difference arrays established on the previous step)and then checks for all sample points to determine how many of them satisfy the initialarrangement conditions and are manipulable by at least one coalition.17 import random InitArrangement = 3 * [0] for i in range(0, 3): InitArrangement[i] = [] for j in range(0, 24): InitArrangement[i].append(0) preferences = [] for i in range(1, 25): preferences.append(i) counter = 0 for i in range(0, 4): for j in range(0, 4): for k in range(0, 4): for l in range(0, 4): if (i != j and i != k and i != l and j != k and j != l and k != l): preferences[counter] = [] preferences[counter].append(i) preferences[counter].append(j) preferences[counter].append(k) preferences[counter].append(l) counter += 1 PointDistArr = [] for i in range(0, 4): PointDistArr.append(i) for i in range(0, 4): PointDistArr[i] = [] for j in range(0, 24): PointDistArr[i].append(0) for i in range(0, 24): for j in range(0, 4): s = preferences[i][j] PointDistArr[s][i] = 3 - j for j in range(0, 24): InitArrangement[0][j] = PointDistArr[0][j] - PointDistArr[1][j] InitArrangement[1][j] = PointDistArr[1][j] - PointDistArr[2][j] InitArrangement[2][j] = PointDistArr[2][j] - PointDistArr[3][j] trials = 8000000 TrialPts = trials * [0] S = 24 * [0] for i in range(0, trials): TrialPts[i] = [] for j in range(0, 23): S[j] = random.uniform(0, 1) S[23]=1 TrialPts[i] = sorted(S) for k in range(1, 24): TrialPts[i][24 - k] -= TrialPts[i][23 - k] CoalB = 24 * [0] CoalC = 24 * [0] CoalD = 24 * [0] for j in range(0, 24): if PointDistArr[0][j] < PointDistArr[1][j]: CoalB[j] = 1 if PointDistArr[0][j] < PointDistArr[2][j]: CoalC[j] = 1 if PointDistArr[0][j] < PointDistArr[3][j]: CoalD[j] = 1 dBA = 24 * [0] dBC = 24 * [0] dBD = 24 * [0] dCA = 24 * [0] dCB = 24 * [0] dCD = 24 * [0] dDA = 24 * [0] dDB = 24 * [0] dDC = 24 * [0] for j in range(0, 24): if CoalB[j] == 1: dBA[j] = 3 dBC[j] = 3 dBD[j] = 3 else: dBA[j] = PointDistArr[1][j] - PointDistArr[0][j] dBC[j] = PointDistArr[1][j] - PointDistArr[2][j] dBD[j] = PointDistArr[1][j] - PointDistArr[3][j] for j in range(0, 24): if CoalC[j] == 1: dCA[j] = 3 dCB[j] = 3 dCD[j] = 3 else: dCA[j] = PointDistArr[2][j] - PointDistArr[0][j] dCB[j] = PointDistArr[2][j] - PointDistArr[1][j] dCD[j] = PointDistArr[2][j] - PointDistArr[3][j] for j in range(0, 24): if CoalD[j] == 1: dDA[j] = 3 dDB[j] = 3 dDC[j] = 3 else: dDA[j] = PointDistArr[3][j] - PointDistArr[0][j] dDB[j] = PointDistArr[3][j] - PointDistArr[1][j] dDC[j] = PointDistArr[3][j] - PointDistArr[2][j] counter = 0 B = 3*[0]
C = 3*[0]
D = 3*[0]
Sb = trials*[0]
Sc = trials*[0]
Sd = trials*[0]
BmanipCheck = False
CmanipCheck = False
DmanipCheck = False
CoalBsize = 0
CoalCsize = 0
CoalDsize = 0 for i in range(0, trials): check = False
CoalBsize = 0
CoalCsize = 0
CoalDsize = 0
Sb[i] = []
Sc[i] = []
Sd[i] = [] for j in range(0, 3): s = 0 check = True for k in range(0, 24): s += TrialPts[i][k] * InitArrangement[j][k] if s < 0: check = False break if check is True:
BmanipCheck = False
CmanipCheck = False
DmanipCheck = False for j in range(0, 3):
B[j] = 0
C[j] = 0
D[j] = 0 for k in range(0, 24):
CoalBsize += TrialPts[i][k] * CoalB[k]
CoalCsize += TrialPts[i][k] * CoalC[k]
CoalDsize += TrialPts[i][k] * CoalD[k]
B[0] += TrialPts[i][k] * dBA[k]
B[1] += TrialPts[i][k] * dBC[k]
B[2] += TrialPts[i][k] * dBD[k]
C[0] += TrialPts[i][k] * dCA[k]
C[1] += TrialPts[i][k] * dCB[k]
C[2] += TrialPts[i][k] * dCD[k]
D[0] += TrialPts[i][k] * dDA[k]
D[1] += TrialPts[i][k] * dDB[k]
D[2] += TrialPts[i][k] * dDC[k]
Sb[i] = sorted(B)
Sc[i] = sorted(C)
Sd[i] = sorted(D) if (Sb[i][0] > 0) and (Sb[i][0] + Sb[i][1] > CoalBsize) and (Sb[i][0]+ Sb[i][1] + Sb[i][2] > 3 * CoalBsize): BmanipCheck = True if (Sc[i][0] > 0) and (Sc[i][0] + Sc[i][1] > CoalCsize) and (Sc[i][0]+ Sc[i][1] + Sc[i][2] > 3 * CoalCsize):
CmanipCheck = True if (Sd[i][0] > 0) and (Sd[i][0] + Sd[i][1] > CoalDsize) and (Sd[i][0]+ Sd[i][1] + Sd[i][2] > 3 * CoalDsize):
DmanipCheck = True if check is True and (BmanipCheck is True or CmanipCheck is True orDmanipCheck is True): counter += 1 ans = 0.0 ans = float(24 * counter / 10000) print(str(ans) + ’%’)
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