Trials of Characterizations of Anti-manipulation Method? Michał Ramsza1[0000−0002−0728−5315] and Honorata Sosnowska2[0000−0001−8249−2368] 1 Warsaw School of Economics, Warsaw, Poland [email protected] 2 Warsaw School of Economics, Warsaw, Poland [email protected] Abstract. This paper studies the anti-manipulation voting method in- troduced in [8]. We show that the method does not satisfy the con- sistency condition. The consistency condition characterizes scoring func- tions. Thus, the method is not a scoring function. Also, the method is not any from a family of not scoring functions comprising Copeland method, instant-runoff voting, majority judgment, minimax, ranked pairs, Schulze method. The paper also shows that the choice of a metric, used by the anti-manipulation method, may imply the winner of the voting. Keywords: anti-manipulation method · voting. 1 Introduction Voting is a method of communication. It allows solutions for some social dilem- mas. In this paper, we present some properties of the anti-manipulation voting method, [8], which may be used to aggregate opinions of experts, for example, jurors of classical music competitions. In 2016, the XV International Violin Henryk Wieniawski Competition took place. There were 11 jurors and 7 contestants in the final of the competition. In the final, the Borda Count was used in the inverse version, that is with 1 for the best. In what follows, we shall use the version with 7 for the best and 1 for the worst. Both versions yield the same results. The Borda Count is a particular scoring method. In this scoring method, jurors assign a score, which is a real number, to each contestant. Ties are possible. In the Borda Count, each juror orders contestants from the best to the worst without ties. The best contestant gets a number of points equal to the number of contestants. The contestant next in order gets the same number of points decreased by one. This procedure is repeated. Thus, the last contestant in order gets one point. Then, for every contestant, the sum of points from all jurors is computed. The contestant with the highest total score wins. The Borda Count is a method of voting very sensitive to manipulation. There was some news about the “war of jurors” in the Wieniawski Competition in Polish ? This research was supported by the National Science Centre, Poland, grant number 2016/21/B/HS4/03016 and SGHS19/07/19 2 Michał Ramsza and Honorata Sosnowska daily “Gazeta Wyborcza” (https://www.gazeta.pl) and in the most influential Polish music journal “Ruch Muzyczny”, [6]. Table 1 presents voting results of jurors J1;:::;J11 over contestants A; : : : ; G. Jurors J4;J5;J8 gave the highest scores to contestant A and low scores to con- testant B. Conversely, jurors J2;J3;J7 gave the highest scores to contestant B and low scores to contestant A. Contestant A won the competition. One can think that jurors especially diminished scores for the competitor of their best contestant. Table 1. Votings of jurors in the final of the XV Wieniawski competitions. Source: own calculations based on results of the competition obtained from the Organization Com- mittee. J1 J2 J3 J4 J5 J6 J7 J8 J9 J10J11 A 73277437777 B 47722772565 C 55536655616 D 36451544351 E 14613363434 F 62164216122 G 21345121243 The situation, described above, caused investigations of a method that would be less sensitive to manipulations. Kontek and Sosnowska [8] proposed such a method. The method, called the anti-manipulation method, is constructed as follows. There are k jurors. Each juror assigns scores to the candidates as in the Borda Count method. Then, the following procedure is used. 1. For each contestant, a mean of scores is computed. This results in a vector of the Borda Count means M = (M1;:::;Mm). 2. For every juror, a distance between the vector of means and juror’s vector of scores is computed. 3. 20% jurors with the highest distance of their vector of scores from the mean are removed3. 4. The arithmetic mean M of the remaining jurors is computed as in the first step. 5. The contestant i with the highest mean Mi is a winner. Let us note that the distance and the method of removing jurors may be cho- sen in any way. Kontek and Sosnowska [8] used Manhattan distance. With this distance, R = bk=5c jurors with largest distances from the mean M are removed. When this procedure is applied to the scores in Table 1, a different contestant 3 In fact, as defined later, the jurors are assigned weights with the most distant jurors given zero weight. Trials of Characterizations of Anti-manipulation Method 3 wins voting as opposed to voting without this procedure. These two contestants were subjects of the “jurors’ war” mentioned above. Contestant A won the com- petition, while contestant B would have won had the anti-manipulation method been used. First, let us note that A is the Condorcet winner. So, the anti-manipulation method is not a Condorcet method. There are many not Condorcet methods, some very popular, for example the Borda Count, the Litvak method ([10], [2], [15]). Second, the anti-manipulation method uses the procedure of removing jurors. An experiment, described in [8], shows that removing jurors results in a lower manipulation level than the elimination of opinions deviating significantly from the mean. Also, 20% threshold is obtained as a compromise between the Borda Count and Majority Criterion, [8]. There are a lot of voting methods, [14], [5]. There is, to the best of our knowl- edge, none that removes voters in a non-random way. This raises the question of possible properties that may characterize the anti-manipulation method. We use the concept of equivalence of voting methods as a tool. Two voting methods are equivalent when for the same profile of voters’ pref- erences and the same set of alternatives, both methods lead to the same re- sults, that is, give the same winners. We compared the properties of the anti- manipulation method with the properties of the Borda Count and scoring meth- ods. We got that the anti-manipulation method is not equivalent to the Borda Count or any scoring method. We used the consistency condition as the primary tool of the analysis. Consistency condition guarantees that the result of voting in subgroups, should it be the same, is the result of voting in a whole group composed of these subgroups. The anti-manipulation method does not satisfy the consistency con- dition, which is one of the axioms for the Borda count and scoring method. Thus, this method cannot be equivalent to the Borda Count or any scoring method. Also, using the consistency condition, we showed that the anti-manipulation method is not equivalent to any method from a group on non-scoring methods. The analyzed group comprised the Copeland method, instant-runoff voting, the Kemeny-Young method, majority judgment [1], minimax, ranked pairs, and the Schulze method. Additionally, we showed a dependence of the anti-manipulation method on the particular choice of a metric. The paper contains five sections. Section 2 presents the axiomatizations of Borda Count and scoring methods. In Section 3, we show an example of voting where the anti-manipulation method does not fulfill the consistency condition and give different winners according to chosen metrics, namely, the Manhattan and the Euclidian metrics. In Section 4, we present examples of voting where the consistency condition is not fulfilled for the Copeland method, instant-runoff voting, majority judgment, minimax, ranked pairs, and the Schulze method. Section 5 offers conclusions. 4 Michał Ramsza and Honorata Sosnowska 2 Axiomatization of the Borda Count and scoring functions Young, [22,23], provided axiomatizations of the Borda Count and scoring func- tions. In this section, we recall these results. Let A = fa1; : : : ; amg be a set of alternatives (in our examples contestants) where m is a number of alternatives, N set of voters (in our examples jurors). Voter’s preferences are linear orders on A. A function from a finite subset V of N into linear orders on A is called a profile. A social choice function f assigns to each profile w a nonempty subset of A, called a choice set. A scoring method may be defined in the following way [23]. There is a se- quence s = (s1; : : : ; sm) of scores, which are real numbers. A vector s is called a scoring vector. Every voter ranks alternatives from best to worst and assigns scores to them. Score si is assigned to alternative on i-th position. For each al- ternative, the sum of scores is computed. The alternative with the highest sum of scores wins. The Borda Count is a special scoring function. The best alterna- tive gets the score of m, the second-best alternative receives the score of m − 1, and each consecutive alternative gets the score diminished by one. The worst alternative receives the score of 1. Formally, let Λ be a set of all linear orders over A, p 2 Λ. Ep is m × m permutation matrix with 1 at the (i; j)-th position if and only if ai is the j-th most preferred in the preference order p. For every x 2 Rm! we define the matrix P D(x) = p2Λ xp · Ep. Let Di(x) be i-th row of D(x). A simple scoring function fs is defined as follows ai 2 fs(x) if and only if hDi(x)jsi ≥ hDj(x)jsi for all j, where h·|·i is scalar product. For anonymous social choice function g and scoring vector s, the composition fs ◦ g of fs with g is defined as follows ai 2 fs ◦ g(x) if and only if ai 2 g(x) and hDi(x)jsi ≥ hDj(x)jsi for all j such that aj 2 g(x).