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Tie-Breaking the Highest Median : Alternatives to the Majority Judgment

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Tie-Breaking the Highest Median : Alternatives to the Majority Judgment Noname manuscript No. (will be inserted by the editor) Tie-breaking the Highest Median : Alternatives to the Majority Judgment Adrien Fabre the date of receipt and acceptance should be inserted later Keywords : Voting system; Election; Ranking; Majority Judgment; Acknowledgements I am grateful to Jean-François Laslier, Rida Median; Grades; Evaluative voting; Evaluation; Typical judgment; Laraki as well as three anonymous reviewers for their comments Central Judgment. and suggestions. I am thankful to François Durand for the proofrea- ding. I thank Maria Sarmiento for the grammar check. JEL classification : D71; D72. Replication files All the code, figures and data are Résumé The paper deals with voting rules that require available on github.com/bixiou/highest_median. voters to rate the candidates on a finite evaluation scale and then elect a candidate whose median grade is maxi- mum. These rules differ by the way they choose among candidates with the same median grade. Call proponents (resp. opponents) of a candidate the voters who rate this candidate strictly above (resp. strictly below) her median grade. A simple rule, called the typical judgment, orders tied candidates by the difference between their share of proponents and opponents. An appealing rule, called the usual judgment, divides this difference by the share of median votes. An alternative rule, called the central judg- ment, compares the relative shares of proponents and opponents. The usual judgment is continuous with res- pect to these shares. The majority judgment of Balinski & Laraki(2007) considers the largest of these shares and loses continuity. A result in Balinski & Laraki(2014) aims to characterize the majority judgment and states that only a certain class of functions share some valuable cha- racteristics, like monotonicity. We relativize this result, by emphasizing that it only holds true for continuous scales of grades. Properties remaining specific to the ma- jority judgment in the discrete case are idiosyncratic fea- tures rather than universally sought criteria, and other median-based rules exist that are both monotonic and continuous. Paris School of Economics, Université Paris 1. [email protected]. 48 bd Jourdan 75014 Paris. ORCID : 0000-0001-5245-7772 Address(es) of author(s) should be given 2 Adrien Fabre 1 Introduction In a series of papers and a book, Balinski & Laraki (2007; 2011; 2014; 2016) have developed a “theory of mea- suring, electing and ranking”. This theory demonstrates 5 valuable properties of a voting system based on evalua- tions rather than on rankings or single marks, where the choice (or candidate) obtaining the highest median eva- luation is elected. Balinski & Laraki (BL) have propo- sed an elegant set of rules electing a choice with the hi- 10 ghest median, and coined this voting system the “ma- jority judgment”. Ties between choices with the same median grades are resolved by comparing the shares of grades above and below the median, respectively : the choice with the largest of these shares wins if the grades of FIGURE 1 Example of vote outcome where each choice A, B, C or D wins according to one of the four tie-breaking rule studied : res- 15 that largest share are above the median, and loses if they pectively difference (d ), relative share (s ), normalized difference (n), are below. The majority judgment (MJ) holds remarkable and majority judgment (m j ). properties, making it the ideal voting system according to its advocates. 1 It is independent of irrelevant alterna- c α p q d s n m j tives, it reduces manipulability, and it does not fall into c c c c c c c 30 10 20 20 20 30 20 the scope of Arrow’s impossibility theorem (Balinski & A 0 100 100 100 80 120 100 11 1 10 10 10 11 Laraki, 2007). These properties emerge from the expres- B 0 100 100 100 24 176 100 siveness of the information contained in each vote, as 40 27 13 13 13 40 C 0 100 100 100 134 66 100 well as from the election of the highest median. They do D 0 45 43 2 2 2 45 not rely on the particular tie-breaking rules of the majo- 100 100 100 176 24 100 TABLE 1 Comparison of different tie-breaking rules on an example : 25 rity judgment. Contrarily to what Balinski & Laraki(2014) the score of the winner is in bold and blue for each rule. convey, neither does the property of monotonicity, which ensures that a choice can only benefit from an increase of its grades. Indeed, the uniqueness property of one of and Table1 show the voting profiles of four choices — A, the various theorems that BL have established in their B, C and D— evaluated by 100 voters. All choices share 30 theory turns out to be inapplicable. Actually, Balinski & the same median grade α 0. Ties are resolved by looking Laraki(2014) do not make clear that their theorem only = at what we abusively call the proponents and the oppo- applies to a continuous set of grades, whereas in all prac- nents to each choice c : the voters attributing to c, respec- 55 tical application, the set of grades is finite. As a conse- tively, a higher and a lower grade than c’s median grade. quence, other voting systems electing the highest median The shares of proponents and opponents are respectively 35 are as credible as the majority judgment. Rather than dis- noted p and q —in this simple example, they correspond crediting the theory elaborated by BL, this observation to the votes 1 and 1. For example, A has 30 proponents enriches it by a variety of simpler voting rules. We will and 10 opponents,− so that p 0.3 and q 0.1. Choice 60 present three such rules in Section3, after describing the A = A = D wins the majority judgment, because its share of pro- setting and notations of the paper in Section2. Then, the ponents exceeds all other shares of proponents or oppo- 40 main result is exposed in Section4 : we clear up ambiguity nents. A wins with the difference tie-breaking rule relying of a proposition of BL on the uniqueness of monotoni- on the score d , because its difference between propo- city by showing that our three voting rules are counter- nents’ and opponents’ shares is the highest. B wins the 65 examples. In Section5, we compare the properties of the tie-breaking rule relying on s , because its ratio of propo- different tie-breaking rules, and show that the majority nents (or equivalently, of pB qB ) over non-median gra- 45 judgment is overly sensitive to small fluctuations in the ders (p q ) is the highest. C−wins with n, because its dif- grades. Finally, Section6 provides a practical understan- B + B ference between proponents’ and opponents’ shares nor- ding of the difference between these voting systems, by malized by its share of median grades (1 p q ) is the 70 exhibiting how they behave on four real-world examples. C C highest. This example shows that the tie-breaking− − rules Before the formal analysis, let us grasp an intuition of described can lead to different outcomes. It is worth no- 50 the tie-breaking rules with a graphical example. Figure1 ticing that, as MJ only takes into account the largest of the 1. Nuanced reviews of MJ have also been published (Felsenthal two groups of non-median grades, it is more sensitive to a & Machover, 2008; Brams, 2011; Laslier, 2018). small variation in their sizes when these are close : in our 75 Tie-breaking the Highest Median : Alternatives to the Majority Judgment 3 n n example, if 3% of grades shifts from 0 to 1 for D, the hi- For n 1;G 1 , we denote by pc (resp. qc ) the − 2 J − K ghest group of D becomes its opponents and D is ranked proportion of c’s grades at or above αc + n (resp. at or n 1 last instead of first (in which case C wins MJ). However, a below αc n): pc := V v g c ,v αc + n (resp. n 1 similar variation would not impact the other tie-breaking − 2 V j ≥ 125 qc := V v g c ,v αc n ). For simplicity, we abu- 80 rules as heavily. 2 V j ≤1 − sively qualify pc := pc as the share of proponents and 1 qc := qc as the share of opponents to c. Finally, a rule is a total preorder on , function of the C 2 Setting profile Φ; while a tie-breaking rule is an order on res- tricted to profiles with the same median grade. RulesC of- 130 Let be a finite set of choices, a finite set of V vo- ten rely on scores, which are real-valued functions of the ters, andC a finite 2 ordered set of VG grades. We assume grades of a choice. We call tie-breaking score a function that G 3,G as all relevant rules boil down to approval vo- of p and q; primary score a function of α, p and q (more ≥ precisely the sum of α and of a tie-breaking score); and 85 ting when G = 2. a complementary score any other function of the grades 135 The grade (or evaluation) of choice c by voter v is no- which is used to rank choices sharing the same primary ted g c ,v . The family Φ = g c ,v is called the (c , v ) score. Without further precision, score refers to primary voting profile2 G . In AppendixB, we allow for2C ×V partial absten- score. tion, i.e. g c ,v , where g c ,v = indicates that voter v 90 does not attribute2 G [f;g any grade to c.
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