Given opinions of members of a society on a set of policies, as ordinal preferences; an approach to polarization is introduced. The concept here is considering polarization in a society as an aggregation of pairwise antagonisms, which depend on identi fication within groups as well as alienation among groups. Among measures which comply to this sort of conceptualization, a class of functions which satisfi es certain plausible properties is introduced for the case of three alternatives. This class coincides with the class characterized for unidimensional spaces in Esteban and Ray (1994)

Ozkes, Ali Ihsan

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Preferential Polarization Measures

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Preferential Polarization MeasuresAli Ihsan OzkesTo cite this version:Ali Ihsan Ozkes. Preferential Polarization Measures. cahier de recherche 2013-22. 2013. <hal-00875949>HAL Id: hal-00875949https://hal.archives-ouvertes.fr/hal-00875949Submitted on 23 Oct 2013HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.     PREFERENTIAL POLARIZATION MEASURES    Ali Ihsan ÖZKES        Cahier n° 2013-22                                   ECOLE POLYTECHNIQUE                       CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE   DEPARTEMENT D'ECONOMIE Route de Saclay 91128 PALAISEAU CEDEX (33) 1 69333033 http://www.economie.polytechnique.edu/ mailto:chantal.poujouly@polytechnique.edu   Preferential Polarization MeasuresAli Ihsan OzkesDepartments of Economics, Ecole Polytechnique and Istanbul Bilgi UniversityE-mail address: ali-ihsan.ozkes@polytechnique.eduAbstractGiven opinions of members of a society on a set of policies, as ordinal preferences; an approach to polarizationis introduced. The concept here is considering polarization in a society as an aggregation of pairwiseantagonisms, which depend on identification within groups as well as alienation among groups. Amongmeasures which comply to this sort of conceptualization, a class of functions which satisfies certain plausibleproperties is introduced for the case of three alternatives. This class coincides with the class characterizedfor unidimensional spaces in Esteban and Ray (1994).Highlights:• We model how to measure polarization with ordinal preferences.• Polarization is seen as the aggregation of pairwise antagonisms.• We propose anonymous and neutral measures that satisfy certain plausible properties.Keywords: polarization measures, ordinal preferences, alienation, identification1. Introduction and the ModelSince polarization is understood to be closely related to social discord and induction of enmity, it isof interest how to understand and conceptualize polarization within a society with ordinal preferences.One approach, not specifically preference based, suggests that it should be considered as the aggregationof pairwise antagonisms within society. In this paper, following Esteban and Ray [6], we suggest thatantagonism felt by an individual towards another depends on the alienation between the two and also onthe feeling of identification he/she enjoys by being in wherever he/she is. Alienation then can be thought ofas a function of the dissimilarity between the two stances. We further postulate that identification can beseen as a function of the support of one’s opinion. In a formal way, this would mean to measure polarizationin a society with a function as follows. ∑j∑ia(I(mi), A(d(oi, oj)))Here the function I(mi) represents how much i is identified with his/her position, as a function of mi,the share of the population who thinks the same with i. The function A(d(oi, oj)) represents the alienationas an increasing function of the distance of opinions of i and j. Finally, the function a(I,A) represents theantagonism felt by i towards j, as a function of identification i enjoys by being wherever i is and alienationbetween i and j. Summing up for all directed pairs is then a measure of polarization as discussed. Weconstrue the set of properties in Esteban and Ray [6] and show that a subclass of the functions abovesatisfies those properties for ordinal preferences over three alternatives. To our knowledge, this is the firstattempt at this question.1Let A be the set of alternatives with |A| = m. There are m! possible linear orders and the set of allthose is denoted L(A). Let P¯ = (P 1, P 2, ..., Pm!) be an arbitrary listing of the set L(A). A preferenceprofile for a society of N individuals is a member of L(A)N . Given P¯ and PN ∈ L(A)N while lettingN P¯i = |{j ∈ N : Pj = P i}| we have a representing preference distribution (pi, P¯ ) where pi = (pi1, pi2, ..., pim!)is such that pii = NP¯i /N . Hence we have that∑m!i=1 pii = 1.A polarization measure is a mapping P : L(A)N → R+. Denoting the set of all preference distributions(pi, P¯ ) by D, we have an equivalent depiction of polarization measures; P : D → R+.In the following analysis, we will be employing a particular metric on linearly ordered preferences,namely Kemeny metric.2 Given any P, P ′ ∈ L(A), the Kemeny distance between P and P ′ is defined asdκ(P, P′) = |P\P ′| where P\P ′ is the symmetric difference. The graph for three alternatives is as follows.Figure 1: Kemeny graph, three alternativesNow we introduce a set of properties on preferential polarization measures following the axiomatizationin Esteban and Ray [6].3Definition 1 (Property 1). Let pii > pij = pik be the only masses of a profile such that pij and pik are atleast as close to each other as they are to pii. A polarization measure is said to satisfy Property 1 if joiningthe two smaller masses at a point at least as further as the average distance to pii increases polarization.1Baldiga and Green [2] analyze social choice functions in terms of assent maximizing attributes. Alcalde-Unzu and Vorsatz[1] provide ways to measure consensus where their analysis which pertains to a consequentialist approach is not compatiblewith framework of this paper.2This widely known and used metric is due to Kemeny [7] for which Bogart [3] and Kemeny and Snell [8], inter alia, provideaxiomatic characterizations. Can and Storcken [5] provides a refinement of previous characterizations. For a general treat ofweighted distances between preferences, see Can [4].3In fact, Property i here may be matched with Axiom i in Esteban and Ray [6], although we keep in mind that there is nounique translation of each axiom into the current setting.2Figure 2: Property 1Figure 2 depicts the two situations Property 1 apply to for three alternatives. The two small masses onthe right can be joined in a middle way further away than the larger mass (as in the move 1 in the figure),or they can be imbricated at one of the positions with small densities (as in the move 2 in the figure). Thesetypes of moves do not decrease (and in case of the move 2, do not increase) the average distance, hencealienation, but increase within group support.Definition 2 (Property 2). A polarization measure satisfies Property 2 if moving a mass that is opposedmildly towards only smaller masses increases polarization.Figure 3: Property 2Property 2 basically requires that polarization increases whenever an in between stance is moved towardsthe side which has a smaller support. This idea is explained in Figure 3 for three alternatives. What iscarefully required is that the move should be towards only smaller poles and is not carried by a stronglyopposed (this strength depends on the whole profile) mass.Definition 3 (Property 3). A polarization measure satisfies Property 3 if for any profile of preferenceswith three consecutive masses, dissolution of the middle mass equally into two sides increases polarization.3Figure 4: Property 3Property 3 is an exact counterpart of Axiom 3 in Esteban and Ray [6] and the intuition is clear. Thefollowing homotheticity property requires that the size of the society does not matter. Let λPN denote aprofile where NλP¯i = λNP¯i for all i ≤ m!.Definition 4 (Homotheticity). Let PN and P′N˜be two profiles with possibly distinct sizes; N and N˜ . Apolarization measure P satisfies homotheticity if P(PN ) ≥ P(P ′N˜ ) implies P(λPN ) ≥ P(λP ′N˜ ) for all λ > 0.Finally, bipolarity, as a regularity property, requires that full concentration at one order is the leastpolarized case and that bipolar society where one half is at the exact opposite of the other is the mostpolarized.Definition 5 (Bipolarity). Suppose PN is such that piQ =12 = pi−Q for some Q ∈ L(A) and that P ′Nis such that pi′Q′ = 1 for some Q′ ∈ L(A). A polarization measure P is said to satisfy bipolarity ifP(PN ) ≥ P(P˙N ) ≥ P(P ′N ) for all N ∈ N and P˙N ∈ L(A)N .2. ResultsProposition 2.1. The polarization measure defined asP∗α(pi, P¯ ) =m!∑j=1m!∑i=1pi1+αi pij(dK(Pi, P j))satisfies properties 1-3 for any α ∈ (0, α∗) with α ' 1.6 and m = 3.Note that this function reduces to Gini Index with Kemeny distance when α = 0 and the higher α isthe higher the sensitivity to identification. In what follows, let’s fix P¯ = ((abc) = P 1, (bac) = P 2, (acb) =P 3, (bca) = P 4, (cab) = P 5, (cba) = P 6) and denote the initial profile by PN and the profile after a describedmove by P ′N .Proof. Property 1: Without loss of generality (wlog), three cases may apply.(i) pi1 > pi4 = pi6, and move pi4 → P 6: We have P(P ′N )− P(PN ) = pi4(pi1+α1 − pi1+α4 ) + pi1+α4 (6pi1(2α − 1)).(ii) pi1 > pi2 = pi4, and move pi2 → P 6: We have P(P ′N )−P(PN ) = pi4(pi1+α1 − pi1+α4 ) + pi1+α4 (4pi1(2α − 1)).(iii) pi1 > pi4 = pi5, and move (iii.a) pi4 → P 5 or (iii.b) {pi4, pi5} → P 6: (iii.a) induces that P(P ′N )−P(PN ) =4pi1pi1+α4 (2α − 1) whereas (iii.b) induces P(P ′N )−P(PN ) = 2pi4pi1+α1 + 2pi1pi1+α4 (3 · 2α − 2). In all cases, wehave P(P ′N ) > P(PN ) if α > 0.4Property 2: Let pi4 move to P6 to induce P ′N . We have P(P ′N ) − P(PN ) > pi4(pi1+α1 + pi1+α2 − pi1+α3 −pi1+α5 − pi1+α6 ) + pi1+α4 (pi1 + pi2 − pi3 − pi5 − pi6) which is positive if the move is only towards smaller poles,or formally both pi1 > pi6 and pi2 > pi5, for any support pi3 of opposition to P4 smaller than ∗ where∗ = min{pi1 + pi2 − pi5 − pi6, (pi1+α1 + pi1+α2 − pi1+α5 − pi1+α6 )11+α } whenever α > 0.Property 3: The question reduces to that of the Axiom 3 in Esteban and Ray [6], and the proof that α isbounded above approximately by 1.6, which pertains to that maxz≥0[(1 + α)(z − zα2 − z1+α)− 12 ] < 0, canbe found in the last paragraph of the proof of the Theorem 1 in that paper, p. 837.Furthermore, it is quick to observe that the measure P∗ is anonymous and neutral in the sense thatit treats individuals and preferences equally. A polarization measure P is said to be anonymous if forany profile PN , for any permutation σ : N → N of individuals, we have that P(PN ) = P(PσN ), wherePσN = (Pσ(i))i∈N . Similarly, a polarization measure P is said to be neutral if for any profile PN , forany permutation δ : A → A of alternatives, we have that P(PN ) = P(δPN ) where δPN = (δPi)i∈N withaPib ⇐⇒ δ(a)δPiδ(b).Proposition 2.2. The polarization measure P∗α is anonymous, neutral and homothetic for any α ≥ 0 andm ∈ N.Proof. Take any N ∈ N and let σ : N → N be an arbitrary permutation. Since NPi = NσPi , we havepii = pi′i for all i ∈ Nm! where pi′i = NσPi /N , which demonstrates anonymity. For neutrality it sufficesto observe that dκ(P, P′) = dκ(δP, δP ′) for any P, P ′ ∈ L(A), under any permutation δ : A → A.4 Thehomotheticity follows the fact that the summation is taken over the supports as percentages.Finally, if we take α as 1, which reflects a simple antagonism function where alienation is represented asthe distance and identification as just the relative societal support of one’s preference5, the measure satisfiesthe bipolarity condition.Proposition 2.3. The measure P∗1 satisfies bipolarity, for m = 3.Proof. The measure is zero if we have full concentration at one order, and strictly positive in all othercases. For any two mass profile, increasing distance in between increases polarization clearly. Once they areat exact opposites, making them equal in density will increase the measure.Lemma 2.1. The bipolar case is more polarized than any three mass profile under P∗1 .Proof. Suppose first that the three masses are not equidistant to each other. Two cases are possible. (Casei) One is further away from the two. Let, w.l.o.g., the support be {pi1, pi2, pi5}. If pi5 > pi2, moving pi1 toP 2 increases the value if pi25pi1 + pi21(pi5 − pi2) + 6pi1pi2pi5 > pi22pi1 which is true. If pi2 > pi5, moving pi1 to P 5increases the value if pi22pi1 + pi21(pi2 − pi5) + 3pi1pi2pi5 > pi25pi1 which is true. (Case ii) One is at unit distanceto both. Let, w.l.o.g., the support be {pi1, pi2, pi3} and that pi2 > pi3. Moving pi1 to P 3 increases the value ifpi22pi1 + pi21(pi2 − pi3) + 4pi1pi2pi3 > pi23pi1 which is true.Now suppose the three masses are equidistant to each other. Let, wlog, the support be {pi1, pi4, pi5} and wemove pi1 to P5. For this to increase polarization it is enough to have pi4 to be the greatest of the three. If4A characterization of Kemeny distance incorporating neutrality can be found in Bogart [3].5Formally, T (I, a) = a · I where I(pii) = pii and a(d(P i, P j)) = dK(P i, P j).5they are all equal size 13 , we have P∗1 (PN ) = 3(13 )2(2 13 + 2 13 ) = 49 < 2( 12 )2(3 12 ) = 34 where the latter is of thebipolar case.The following lemma shows that from any five or six mass profile, we can reduce the domain to fourmasses and increase polarization.Lemma 2.2. For any six (five) support profile, there exists a move that reduces the support of the profileto five (four) and increases the value of P ∗1 .Proof. Let masses at each (or all but one) location be positive. Wlog, either (i) pi1 > pi4 + pi5 + pi6 or(ii) pi5 > pi4 + pi2 + pi1 is true. Suppose the first is true and move the mass pi2 to P4.6 The differenceP(P ′N ) − P(PN ) = pi2[pi21 + pi23 − (pi24 + pi25 + pi26)] + pi22[pi1 + pi3 − (pi4 + pi5 + pi6)] is clearly positive. Thisis regardless of pi3 being non-zero or not (in other words, the support being five or six masses) hence thelemma demonstrated.Corollary 2.1. For any six support profile there exists a consecutive pair of moves that induces a foursupport profile with higher value under P∗1 .7The final lemma below concludes the proof of the proposition by showing that from any (out of threepossible) four mass profile we can reduce to a two mass profile with higher measure.Lemma 2.3. The bipolar profile is more polarized than any PN with four masses under P∗1 .Proof. Let Σ ⊆ L(A) be the orders that have nonzero support in PN . Wlog, we have three cases.(a) Σ = {P 1, P 3, P 4, P 6}. We have three distinct cases; either (i) pi4 ≥ pi3 and pi1 ≥ pi6 , or (ii) pi4 < pi3and pi1 < pi6 , or (iii) pi4 > pi3 and pi1 < pi6 (not distinctively pi4 < pi3 and pi1 > pi6 is also possible.). Forthe first two cases, moving pi4 to P6 and pi3 to P1 increases polarization if (pi4 − pi3)(pi1 − pi6) ≥ 0, whichis true. For the third case, moving pi1 to P4 and pi6 to P3 increases polarization if (pi4 − pi3)(pi6 − pi1) ≥ 0,which is true.(b) Σ = {P 1, P 4, P 5, P 6}. Moving pi6 to P 1 induces P(P ′N )−P(PN ) > 3pi1pi6(pi1−pi6) > 0 if pi1 ≥ pi6. Ifotherwise, moving pi4 to P2 and pi5 to P3 induces P(P ′N )−P(PN ) = (pi4+pi5)(pi26−pi25)+(pi24+pi25)(pi6−pi1) > 0which leaves us with the exact situation as before and hence moving pi1 to P6 increases polarization.(c) Σ = {P 1, P 2, P 4, P 6}. Moving pi2 to P 3 induces P(P ′N )−P(PN ) = 2pi2pi4(pi2 +pi4) which is positiveand leaves us with the case (a).3. References[1] Alcalde-Unzu, J. and Vorsatz, M. [2011]. Measuring consensus: Concepts, comparisons, and properties, ConsensualProcesses pp. 195–211.[2] Baldiga, K. and Green, J. [2011]. Assent-maximizing social choice, Social Choice and Welfare pp. 1–22.[3] Bogart, K. [1973]. Preference structures i: Distances between transitive preference relations, Journal of MathematicalSociology 3(1): 49–67.[4] Can, B. [2012]. Weighted distances between preferences, Technical report, Maastricht: METEOR, Maastricht ResearchSchool of Economics of Technology and Organization.[5] Can, B. and Storcken, T. [2013]. A re-characterization of the kemeny distance, mimeo .[6] Esteban, J. and Ray, D. [1994]. On the measurement of polarization, Econometrica: Journal of the Econometric Societypp. 819–851.[7] Kemeny, J. [1959]. Mathematics without numbers, Daedalus 88(4): 577–591.[8] Kemeny, J. and Snell, L. [1962]. Preference ranking: an axiomatic approach, Mathematical models in the social sciencespp. 9–23.6If the second is true, moving pi6 to P 4 will do the same.7Proof left to the reader, since it only incorporates application of the move in the Lemma 2.4 consecutively.6

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