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Agreement, Diversity, and Polarization Indices for Approval Elections
Piotr Faliszewski, Jitka Mertlová, Krzysztof Sornat, Stanisław Szufa, Tomasz Wąs
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Summary
The paper introduces a framework for measuring agreement, diversity, and polarization in approval elections using normalized indices. A core requirement is saturation independence, ensuring indices remain consistent regardless of the average number of approved candidates. The authors propose six agreement indices (Approval Agreement, Central Agreement, and four Pairwise variants), from which diversity and polarization indices are derived via voter clustering. These metrics are evaluated on synthetic models and real-world datasets like Pabulib and Preflib to construct a saturation-independent map of approval elections, enabling more nuanced comparisons of electoral data.
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Krzysztof Sornat → isauthorof → Agreement, Diversity, and Polarization Indices for Approval Elections
confidence 99% · Krzysztof Sornat, sornat@agh.edu.pl, AGH University of Kraków, Poland
Piotr Faliszewski → isauthorof → Agreement, Diversity, and Polarization Indices for Approval Elections
confidence 99% · Piotr Faliszewski, faliszew@agh.edu.pl, AGH University of Kraków, Poland
Jitka Mertlová → isauthorof → Agreement, Diversity, and Polarization Indices for Approval Elections
confidence 99% · Jitka Mertlová, mert- lova.jita@gmail.com, Czech Technical University in Prague, Czech Republic
Tomasz Waś → isauthorof → Agreement, Diversity, and Polarization Indices for Approval Elections
confidence 99% · Tomasz W ̨as, tomasz.was@cs.ox.ac.uk, University of Oxford, United Kingdom
Stanisław Szufa → isauthorof → Agreement, Diversity, and Polarization Indices for Approval Elections
confidence 99% · Stanisław Szufa, s.szufa@gmail.com, University of Geneva, Switzerland
Preflib → issourcefor → Approval Election
confidence 98% · show similarities and differences between various real-life elections from Pabulib, Preflib and other sources.
Pabulib → issourcefor → Approval Election
confidence 98% · show similarities and differences between various real-life elections from Pabulib, Preflib and other sources.
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Abstract
Abstract:An index is a function that given an election outputs a value between 0 and 1, indicating the extent to which this election has a particular feature. We seek indices that capture agreement, diversity, and polarization among voters in approval elections, and that are normalized with respect to saturation. By the latter we mean that if two elections differ by the fraction of candidates approved by an average voter, but otherwise are of similar nature, then they should have similar index values. We propose several indices, analyze their properties, and use them to (a) derive a new map of approval elections, and (b) show similarities and differences between various real-life elections from Pabulib, Preflib and other sources.
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- Source: https://arxiv.org/abs/2605.14983v1
- Canonical: https://arxiv.org/abs/2605.14983v1
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Agreement, Diversity, and Polarization Indices for Approval Elections Piotr FaliszewskiJitka MertlováKrzysztof SornatStanisław SzufaTomasz W ̨as Abstract: An index is a function that given an election outputs a value between 0 and 1, indicating the extent to which this election has a particular feature. We seek indices that capture agreement, diversity, and polarization among voters in approval elections, and that are normalized with respect to saturation. By the latter we mean that if two elections differ by the fraction of candidates approved by an average voter, but otherwise are of similar nature, then they should have similar index values. We propose several indices, analyze their properties, and use them to (a) derive a new map of approval elections, and (b) show similarities and differences between various real-life elections from Pabulib, Preflib and other sources. Code: https://github.com/Project-PRAGMA/Approval-DAP-IJCAI-2026 1Introduction Our main goal is to design three functions that given an approval election—i.e., an election where voters give yes/no answers regarding their support for particular candidates—indicate the extent to which the votes are in agreement, are diverse, or are polarized. Following Hashemi and Endriss [2014], we refer to functions that quantify election features (by giving values between 0 and 1) as election indices. Such indices would be valuable because they would allow us to compare approval elections coming from different sources. For example, it is not uncommon to use participatory budgeting (PB) elections from Pabulib [Faliszewski et al., 2023a] outside of the PB context and it is important to validate if this data remains relevant. 1 Further, as recently shown by Faliszewski et al. [2025b], for ordinal elections (where the voters rank the candidates) having good agreement, diversity, and polarization indices leads to appealing maps of elections (i.e., visualizations of election datasets where each election is a dot and distances between these dots give an idea of their elections’ similarity). While a map of approval elections exists [Szufa et al., 2022], it seems to be less useful than those for ordinal ones (see the argument below). One characteristic issue when dealing with approval elections is that they may largely differ in terms of their saturation, i.e., the average number of candidates approved by a voter, while being very similar in nature. As an extreme example, consider one election where all voters approve the same 10% of candidates, and a second one, where all voters approve the same 50% of candidates. Both of these elections represent perfect agreement (and complete lack of diversity and polarization) but according to the map of Szufa et al. [2022], they are quite far from each other. Indeed, that map has two main dimensions, election saturation and the position on the agreement-disagreement spectrum. We seek saturation-independent agreement, diversity, and polarization indices, so that a map based on them would be capable of discovering more nuanced relations between data. Our general strategy is to focus on designing a saturation-independent agreement index. Given such an index, we derive a diversity one by clustering the voters into as like-minded groups as possible and claiming Authors’ Information: Piotr Faliszewski, faliszew@agh.edu.pl, AGH University of Kraków, Poland; Jitka Mertlová, mert- lova.jita@gmail.com, Czech Technical University in Prague, Czech Republic; Krzysztof Sornat, sornat@agh.edu.pl, AGH University of Kraków, Poland; Stanisław Szufa, s.szufa@gmail.com, University of Geneva, Switzerland; Tomasz W ̨as, tomasz.was@cs.ox.ac.uk, University of Oxford, United Kingdom. 1 E.g., for year 2025 Google Scholar points to 18 papers that cite the Pabulib database and perform numerical experiments. Among these, only 12 are indeed focused on participatory budgeting. 1 arXiv:2605.14983v1 [cs.GT] 14 May 2026 that the election is diverse if the agreement within these groups is low. For polarization–which we define as the presence of two groups with conflicting preferences—we take the difference between the overall agreement and the agreements in two voter groups after clustering (the overall agreement in a polarized election would be low, but it would be high in the like-minded groups). This approach was proposed by Faliszewski et al. [2023b] in the ordinal case, for the special case of measuring agreement using the Kemeny rule Kemeny [1959]. We extend their idea. Altogether, our contributions are as follows. First, we propose a number of agreement indices and analyze them theoretically and experimentally, paying careful attention to saturation independence. In particular, we verify if their results are intuitive on a number of selected election instances. Then, using these indices, we derive and test diversity and polarization ones (using both the general scheme and some other ideas). Finally, we identify three indices—capturing agreement, diversity, and polarization—that work best together in a certain measurable way, and use them to form a map of approval elections. Our map contains both synthetic and real-life elections, e.g., coming from Preflib [Mattei and Walsh, 2013] and Pabulib [Faliszewski et al., 2023a], but also from other sources, including some new ones. We discuss related work throughout the paper, in relevant places. Omitted proofs and discussions are in the appendix. 2Preliminaries For an integert, we write [t] to denote1, . . . , t . For a vectorx∈R t , we writex[i] to denote itsi-th coordinate, andx= 1 t (x[1]+ . . .+ x[t]) to denote its mean. Forp∈[0,1],x∼ Bernoulli(p) indicates thatxgets value 1 with probability p and value 0 with probability 1− p . Elections.An (approval) election is a pairE =(C , V), whereC = c 1 , . . . , c m is a set of candidates andV = (v 1 , . . . , v n ) is a collection of voters. Each voter has an approval ballot (or, a vote) where he or she indicates which candidates he or she approves (and he or she disapproves the other ones). Formally, we represent a vote as a binary vector, whose entries correspond to the candidates and have value 1 if a given candidate is approved and 0 otherwise. To simplify notation, we writev i to refer both to the respective voter and his or her vote; the exact meaning will always be clear from the context. Additionally, for a voterv i we writeA(v i ) to denote the set of candidates that this voter approves, and for a candidatec j we writeA(c j ) to mean the set of voters approvingc j . Given an electionE=(C , V), byE −1 we mean an election obtained fromEby reversing all entries of all votes (i.e., for each voterv i and candidatec j , inE −1 , forc j this voter reports 1− v i [j]). We refer to E −1 as a reverse of E . Pearson Correlation Coefficient.Given twot-dimensional vectors,x , y ∈R t , their Pearson correlation coefficient (PCC) is: pcc(x , y )= P t i=1 ¡ x [i ]−x ¢¡ y [i ]−y ¢ q P t i=1 ¡ x [i ]−x ¢ 2 q P t i=1 ¡ y [i ]−y ¢ 2 .(1) pcc (x , y) takes values between 1 and−1, where 1 means perfect correlation (positive linear dependence between the entries ofxandy),−1 means perfect anti-correlation (negative linear dependence between the entries ofxandy), and 0 means a complete lack of linear correlation. In rare cases in which inxorycontains only 1 or only 0, the denominator would be equal to 0, so we fix the value of pcc(x , y ) as 1. (Dis)similarity Measures Between Votes.Given two votes,uandv, overmcandidates, their Hamming distance is ham(u, v )= P i∈[m] |u[i ]− v [i ]| and their Jaccard distances is jacc(u, v )= 1− P i∈[m] min(u[i ], v [i ]) P i∈[m] max(u[i ], v [i ]) . Hamming distance takes values between 0 andm, whereas Jaccard takes values between 0 and 1. We also sometimes measure similarity between two votes using Pearson correlation coefficient, which in the case of binary vectors is known as the phi coefficient Yule [1912], or Matthews coefficient (MCC) Baldi et al. [2000]. 2 There are many other (dis)similarity measures between binary vectors (see, e.g., the discussion of Duan et al. [2014]) but we restrict our attention to these three as they are common in voting literature. 3Election Data We test our election indices on three kinds of data: Special, fixed elections of particular, well-understood structure, synthetically generated random elections, and elections obtained from various real-life data (ranging from actual approval elections to those derived from other types of preferences). In the discussion below we consider elections with candidate set C= c 1 , . . . , c m and voter collection V = (v 1 , . . . , v n ). Special Elections. Our special elections are: p -Identity (p -ID).In ap-Identity election, all voters approve the same set of⌊p m⌋candidates (e.g.,c 1 , . . . , c ⌊p m⌋ ) and disapprove the remaining ones. k -Party.In ak-Party election there arekdisjoint, equal-sized groups of candidates (±1, depending on divisibility) andkdisjoint, equal-sized groups of voters (±1, depending on divisibility) . For eachi∈[k], the voters in thei-th group approve exactly the candidates in thei-th group. Forx , y∈[0,1], by (x , y)- 2-Party we mean a 2-Party election where the first group includes anxfraction of candidates andy fraction of voters, and the second group includes the remaining ones. For the next three elections we require n= m: Diagonal. This is an m-Party election. Triangle. For each i∈ [n], the i -th voter approves the first i candidates, i.e., c 1 , . . . , c i . Cyclic. The first voter approves the first candidate and last⌊m/2⌋−1 candidates. For eachi ∈ 2, . . . n, we obtain thei-th vote by taking the (i−1)-st one and shifting it cyclically by one position to the right, i.e., replacing each candidate c j by c j+1 , with c m mapped to c 1 . Synthetic Elections. Next let us describe our models of generating random elections. If we only describe a pro- cess of sampling a single vote, it means that all the votes in the election are generated this way, independently from each other: p -Impartial Culture (p -IC).Forp∈[0,1], we sample votevso that for eachi∈[m], we havev[i]∼ Bernoulli(p). (p 1 , . . . ,p m )-Independent Approval Model (IAM).For (p 1 , . . . , p m )∈[0,1] m , we sample votevso that for each i ∈[m],v[i]∼ Bernoulli(p i ). IAMs were discussed, e.g., by Faliszewski et al. [2025a], Lackner and Maly [2025], and Xia [2025]. (p ,φ)-Resampling. Givenp,φ∈[0,1] and a fixed central voteuwith| A(u)|=⌊p m⌋, we sample votevas follows: For eachi ∈[m], with probability 1−φwe letv[i]= u[i] and with probabilityφwe letv[i]∼ Bernoulli (p). This model is due to Szufa et al. [2022]. Some authors, such as Faliszewski et al. [2025a], allow arbitrary central votes, but we use the original variant as it givesp mapproved candidates in expectation. Euclidean.Here, we assume that each voter and candidate is associated with a point in ad-dimensional Euclidean space, and each voterv i approves a candidatecif the distance between their corresponding points is smaller than some threshold valuer i (which can be different for each voter). Such a model was studied, e.g., by Godziszewski et al. [2021]. See Section D for details. We also consider a few special distributions, to generate elections with particular features: p -ID/IC.The first half of the voters form ap-ID election and the second half is generated using thep-IC model. Lin-IC.In this model,v 1 is generated using 1-IC,v 2 using (1− 1 /n)-IC,v 3 using (1− 2 /n)-IC, and so on, until v n , generated using 1 /n-IC. 3 p φ Figure 1: Results of the resampling experiment for Approval Agreement index. Rows correspond to fixedp values and columns to fixedφ values. N(D ,φ). D is a distribution over elections. We first generate an election usingDand then we replace each vote v with a new one as follows: We let p be the fraction of candidates approved in v and we generate a vote using (p,φ)-resampling model with central vote equal tov. In other words, N(D,φ) introduces noise on top of elections generated usingD (N stands for noisy). Real-Life Data. See Section 7 and Section D. 4Indices and Initial Experiments By an election index, we mean a functionfthat given an electionEoutputs a value between 0 and 1, quantifying the extent to whichEhas a feature captured byf. In the following sections we define agreement, diversity, and polarization indices. For now, we discuss indices on a high level and describe our approach to evaluating them. Normalization.Given an electionE=(C , V), we consider its average vote length (avl; the average number of candidates approved by a voter), reverse avl (the average number of candidates disapproved by a voter), and its saturation (satr; the fraction of candidates that an average voter approves): avl(E )= 1 |V| P v∈V | A(v )|,rev-avl(E )= avl(E −1 ),satr(E )= avl(E )/|C|. While saturation is a natural election index by itself, for the other indices we are mostly interested in nor- malizing them in a way that would be independent of saturation. In other words, if two elections are similar in nature except that their saturations differ (as would be the case, e.g., for 0.5-IC and 0.25-IC elections), we would prefer an election index to give them similar values. Naturally, this is not a precise mathematical statement, but rather an intuitive desideratum. Yet, in some cases it is easy to argue that a particular index is not saturation-independent, and we often view it as an argument against this index. Resampling Experiment.For each of our indices, we perform the following resampling experiment: For each p∈ 0.1, . . . ,0.9and eachφ∈ 0,0.1, . . . ,1, we generate 10 (p,φ)-resampling elections with 60 candidates and 60 voters, and compute for them the average value of the index in question. We arrange the results as a matrix, where each row corresponds to a fixed value ofpand each column corresponds to a fixed value ofφ. Color brightness represents the average value of the index. We present such matrices in the first row of Table 1, and we provide an enlarged example (including axis labels and the color-bar scale) in Figure 1. The important feature of these plots is that each column corresponds to a fixed value ofφ, while saturation (i.e.,p) varies across rows. Hence, if the columns of the matrix are not (approximately) uniformly colored, then we claim that a given index is not saturation-independent (as, arguably, resampling elections with different values ofp but the same value ofφare of the same nature). For such an index, we say that it fails the resampling test of saturation independence. 4 av-agrcntr-agrpair-agrpcc-agrjacc-agrpcc + -agr cntr-divpcc-divout-divcntr-polpcc-polpair-pol resampling experiment: saturation independence:✗✓✗✓✗✓– 1. 1 /3-ID 1.00 ±0.00 1.00 ±0.00 1.00 ±0.00 1.00 ±0.00 1.00 ±0.00 1.00 ±0.00 0.00 ±0.00 0.00 ±0.00 0.00 ±0.00 0.00 ±0.00 0.00 ±0.00 0.00 ±0.00 2.2-Party 0.00 ±0.00 0.00 ±0.00 0.00 ±0.00 0.00 ±0.00 0.50 ±0.00 0.50 ±0.00 0.20 ±0.00 0.25 ±0.02 0.10 ±0.00 1.00 ±0.00 1.00 ±0.00 1.00 ±0.00 3.N(2-Party, 0.6) 0.09 ±0.01 0.08 ±0.01 0.01 ±0.00 0.02 ±0.00 0.35 ±0.01 0.10 ±0.01 0.66 ±0.02 0.82 ±0.01 0.29 ±0.00 0.25 ±0.10 0.17 ±0.01 0.24 ±0.01 4.3-Party 0.33 ±0.00 0.00 ±0.00 0.00 ±0.00 0.00 ±0.00 0.33 ±0.00 0.33 ±0.00 0.33 ±0.00 0.31 ±0.01 0.13 ±0.00 0.33 ±0.00 0.50 ±0.00 0.63 ±0.00 5.4-Party 0.50 ±0.00 0.00 ±0.00 0.00 ±0.00 0.00 ±0.00 0.25 ±0.00 0.25 ±0.00 0.47 ±0.04 0.40 ±0.00 0.15 ±0.00 0.25 ±0.00 0.33 ±0.00 0.43 ±0.00 6.( 1 /3, 1 /3)-2-Party 0.33 ±0.00 0.24 ±0.00 0.10 ±0.00 0.11 ±0.00 0.56 ±0.00 0.56 ±0.00 0.15 ±0.00 0.18 ±0.00 0.09 ±0.00 0.76 ±0.00 0.89 ±0.00 0.99 ±0.00 7.Cyclic 0.00 ±0.00 0.00 ±0.00 0.00 ±0.00 0.00 ±0.00 0.39 ±0.00 0.25 ±0.00 0.46 ±0.00 0.57 ±0.01 0.22 ±0.00 0.50 ±0.00 0.33 ±0.00 0.58 ±0.00 8.Diagonal 0.97 ±0.00 0.00 ±0.00 0.00 ±0.00 0.00 ±0.00 0.02 ±0.00 0.02 ±0.00 0.97 ±0.00 0.97 ±0.00 0.63 ±0.01 0.02 ±0.00 0.02 ±0.00 0.01 ±0.00 9.Triangle 0.50 ±0.00 0.49 ±0.00 0.33 ±0.00 0.50 ±0.00 0.51 ±0.00 0.50 ±0.00 0.41 ±0.00 0.32 ±0.00 0.18 ±0.00 0.01 ±0.00 0.14 ±0.00 0.47 ±0.00 10.N(Tri,0.6) 0.21 ±0.01 0.20 ±0.02 0.06 ±0.00 0.15 ±0.02 0.33 ±0.01 0.17 ±0.02 0.89 ±0.01 0.77 ±0.03 0.26 ±0.00 0.00 ±0.02 0.05 ±0.00 0.39 ±0.01 11. 1 /2-ID/IC 0.50 ±0.01 0.49 ±0.01 0.26 ±0.01 0.25 ±0.01 0.51 ±0.00 0.30 ±0.00 0.41 ±0.01 0.50 ±0.00 0.19 ±0.00 0.07 ±0.01 0.26 ±0.02 0.45 ±0.01 12. 1 /2-IC 0.10 ±0.01 0.10 ±0.01 0.02 ±0.00 0.02 ±0.00 0.34 ±0.01 0.07 ±0.00 0.81 ±0.01 0.91 ±0.00 0.29 ±0.00 0.07 ±0.01 0.05 ±0.00 0.18 ±0.00 13. 1 /4-IC 0.49 ±0.01 0.00 ±0.00 0.02 ±0.00 0.02 ±0.00 0.16 ±0.00 0.07 ±0.00 0.97 ±0.00 0.90 ±0.00 0.31 ±0.00 0.01 ±0.01 0.05 ±0.00 0.16 ±0.00 14.Lin-IC 0.08 ±0.01 0.06 ±0.01 0.01 ±0.00 0.09 ±0.02 0.31 ±0.01 0.15 ±0.03 0.96 ±0.01 0.83 ±0.03 0.27 ±0.00 0.00 ±0.01 0.04 ±0.00 0.37 ±0.01 Table 1: Values of our indices for elections from various distributions. The elections have 60 candidates and 60 voters. Index values are averaged over 10 samples and the standard deviation is reported in smaller font. Similar table with more elections can be found in Section D. Index Values on Example Elections.In Table 1 we also give values of our indices on a number of characteristic (distributions over) elections. Each row gives index values for a given family of elections. All elections have 60 candidates and 60 voters and the results for each family are averaged over sampling 10 elections (we report standard deviation in smaller font). For each family we give its name and an icon presenting an example election; in this icon, columns are the candidates, rows are the voters, and darker spots indicate that a given voter approves a given candidate. We discuss this table throughout the following sections. 5Agreement Indices An agreement index should identify whether the voters have similar views, i.e., tend to approve and disapprove the same candidates. We consider two types of such indices: Global ones look at the extent to which all voters agree with each other, whereas local ones also detect large groups of voters that are internally in agreement, but that may disagree with each other. We propose six agreement indices that we describe below. All our indices give value 1 exactly forp-ID elections—which, indeed, represent perfect agreements—and for most of them we also characterize when they give value 0. 5.1Approval Agreement Our first index, Approval Agreement, adapts an idea from the world of ordinal elections (where the voters rank the candidates). There, for every pair of candidatesaandbwe compute the absolute difference between the fractions of voters that rank them one way or the other, and output the average of these values [Alcalde-Unzu and Vorsatz, 2013; Hashemi and Endriss, 2014; Can et al., 2015; Faliszewski et al., 2023b]. In the approval 5 setting, we compare the numbers of voters approving and disapproving each given candidate. Definition 5.1. Given an election E= (C , V ), its Approval Agreement is av-agr(E )= 1 |C| X c∈C ̄ ̄ ̄ ̄ | A(c )| |V| − μ 1− | A(c )| |V| ¶ ̄ ̄ ̄ ̄ = 1 |C| X c∈C ̄ ̄ ̄ ̄ 1− 2 | A(c )| |V| ̄ ̄ ̄ ̄ . Unfortunately, this index has several drawbacks. Foremost, as we see in the first row of Table 1, it fails the resampling test of saturation independence (and, indeed, we see that it gives different values for 1 /2-IC and 1 /4-IC). This is also visible in the following proposition, where we show that it assumes value 0 if and only if each candidate is approved by exactly half of the voters and, hence, it is not possible to assume this value for saturation different from 0.5. Proposition 5.1. For an electionE,av-agr(E)=1 exactly ifEis an identity election, andav-agr(E)=0 exactly if each candidate is approved by exactly half of the voters. The index also behaves poorly on Party elections. As we see in Table 1, for 2-Party its value is 0, but it increases as we consider 3-Party and 4-Party, to become close to 1 for Diagonal. Indeed, the index treats the fact that many voters disapprove the same candidates as a sign of agreement among them (while in some cases—such as elections within parliaments—this indeed might be a reasonable view, in others—such as in participatory budgeting—it is ungrounded). 5.2Central Agreement(s) Our second index, Central Agreement, is designed to resemble the Kemeny voting rule for ordinal elections [Ke- meny, 1959]. In that setting, a Kemeny ranking is a ranking that minimizes the sum of swap distances to all the votes in a given election. The larger is this value, the bigger is the disagreement among the voters. We implement the same idea for approval elections using the Hamming distance (while computing a Kemeny ranking is well-known to be intractable [Bartholdi I et al., 1989; Hemaspaandra et al., 2005], in our case the analogous task is very simple). LetE=(C , V) be an election. A voteuover candidate setCis central forEif for each candidatec∈ C,u agrees regardingcwith at least half of the voters. So, if exactly half of the voters approve some candidate and half disapprove it, then there is more than one central vote for the given election. We writecen(E) to denote the set of all central votes for E . Definition 5.2. Let E= (C , V ) be an election and let u be an arbitrary member of cen(E ). We define cntr-agr(E )= 1− P v i ∈V ham(v i , u) |V|· min(avl(E ), rev-avl(E )) . The denominator in the definition ofcntr-agr(E) is chosen to be (nearly) equal to the largest possible value of the numerator for elections of the same size and saturation as E . Proposition 5.2. For an electionE,cntr-agr(E)=1 exactly ifEis an identity election, andcntr-agr(E)=0 exactly if eithercen(E) contains a vote approving all candidates orcen(E) contains a vote disapproving all candidates. Central Agreement has two major drawbacks. First, as indicated by Theorem 5.2, it is quite non-discriminative and often assumes value 0 (e.g., on the majority of Pabulib elections). Second, it fails the resampling test of saturation independence (see Table 1). Interestingly, Central Agreement and Approval Agreement differ mostly by their normalization (replacing the denominator withmin Theorem 5.3 below, we would get Approval Agreement). Proposition 5.3. Let E= (C , V ), then cntr-agr(E )= 1− X c∈C 1− ̄ ̄ ̄ 1− 2| A(c )| |V| ̄ ̄ ̄ 2· min(avl(E ), rev-avl(E )) . We do not consider central agreement based on the Jaccard distance as it would be intractable [Chierichetti et al., 2010] and we find sufficiently many other agreement indices. 6 5.3Pairwise Agreements The final four indices are based on computing expected (dis)similarity between two randomly selected votes. Definition 5.3. For an electionE=(C , V), we define Hamming, Jaccard, PCC, and PCC + Pairwise Agreement Indices: pair-agr(E )= 1− P u∈V P v∈V ham(u,v ) 2|V| 2 |C|·satr(E )(1−satr(E )) , jacc-agr(E )= 1 |V| 2 P u∈V P v∈V (1− jacc(u, v )), pcc-agr(E )= 1 |V| 2 P u∈V P v∈V pcc(u, v ), pcc + -agr(E )= 1 |V| 2 P u∈V P v∈V max(0, pcc(u, v )). The normalization for Hamming Pairwise Agreement follows that for Central Agreement: the denominator is equal to the maximum possible value of the numerator among elections with the same size and saturation (assuming that it is possible to ensure that every candidate has the same approval score). The normalizations for Jaccard and PCC + use the fact that we average over values that are between 0 and 1. For PCC Agreement, we need an additional argument. Proposition 5.4. For each election E , pcc-agr(E )∈ [0, 1]. At first sight, computing each of these indices requiresO(|C||V| 2 ) time: Computing similarity between a pair of votes requiresO(|C|) time and there areO(|V| 2 ) pairs to consider. Yet, by changing the order of summation, we can compute Hamming Pairwise Agreement in O(|C||V|) time. Theorem 5.5. For an election E= (C , V ), we have that pair-agr(E )= 1− P c∈C | A(c )|· (|V|−| A(c )|) |V| 2 |C|· satr(E )(1− satr(E )) . Next we characterize when Hamming and Jaccard Pairwise indices assume values 1 and 0. In case of PCC-based indices we only provide some examples and observations. Theorem 5.6. Each of Pairwise, Jaccard, PCC, and PCC + Agreement Indices assumes value 1 exactly for identity elections. Hamming Pairwise Agreement has value 0 exactly if all candidates have equal approval score, and Jaccard Pairwise Agreement has value 0 exactly if all the voters approve disjoint sets of candidates. Proposition 5.7. For eachk-Party electionE =(C , V) such that both|C|and|V|are divisible byk, we have pcc-agr(E )= 0 and pcc + -agr(E )= 1 /k. Remark 5.1. Letp∈[0,1] be a number and letuandvbe two votes such that ap 2 fraction of candidates is approved in both votes, a (1− p) 2 fraction is disapproved in both, ap(1− p) fraction is only approved inu, and ap(1− p) fraction is only approved inv. Thenpcc(u, v)=0. As votes inp-IC elections are close to satisfying these conditions, we expect PCC and PCC + Agreements to be close to 0 for IC elections. We note that Hamming Pairwise Agreement and PCC Agreement tend to give very similar results for the examples in Table 1. In fact, if all voters approve the same number of candidates, then the two indices are equal (but this is not the case when vote lengths differ; see the Triangle election). Proposition 5.8. LetE=(C , V) be an election such that 0<| A(u)|=| A(v)|<|C|for allu, v∈ V. Then we have that pair-agr(E )= pcc-agr(E ). In the resampling experiment both Hamming and PCC Agreements give results independent of saturation. Overall, we view both of them as good global agreement indices. Jaccard and PCC + Pairwise Agreement indices capture local agreement, by detecting large agreeing groups, even if they disagree with other such groups (indeed, in PCC + we only sum positive PCC values to account for like-minded voters, disregarding negative correlations that could cancel them out). For example, they give high—and nearly identical—values for party elections (the fewer parties, the higher agreement, as there are larger agreeing groups) and detect agreeing groups in 1 /2-ID/IC, and Lin-IC. Yet, we prefer PCC + Agreement as it passes the resampling test of saturation independence and assigns close-to-zero values toICelections. The fact that Jaccard Pairwise Agreement fails the resampling test of saturation is expected, as it treats approvals and disapprovals assymetrically. If such asymmetry is natural in a given setting it still might be a valuable index. 7 6Diversity and Polarization Indices Intuitively, an election is diverse if (a) its votes tend to be different from each other, and (b) they cover the space of all possible votes (modulo saturation). For example, we view Diagonal as a maximally diverse election, because for its saturation (one approval per voter) it contains exactly one copy of each possible vote. Yet, we view Cyclic, which is similar in spirit to Diagonal, as much less diverse than 1 /2-IC; these two types of election have the same saturation, but the votes in the latter are notably more different from each other and, intuitively, cover the preference space more evenly. On the other hand, we say that an election is polarized if it consists of two groups of voters that agree internally but strongly disagree with each other. Consequently, 2-Party is an example of a maximally polarized election. We acknowledge that elections with three, four, or more internally consistent but disagreeing groups could also be seen as maximally polarized—and seeking indices capturing this intuition would be valuable—but we leave this for future work. For a broader view of diversity and polarization, see the special issue edited by Levin et al. [2021]. In voting literature, see, e.g., the works of Nehring and Puppe [2002], Hashemi and Endriss [2014], Can et al. [2015], Karpov et al. [2024], Ammann and Puppe [2025], and Faliszewski et al. [2023b; 2025b; 2026a; 2026b]. 6.1Clustering-Based Diversity and Polarization Our first approach to designing diversity indices follows the view of Faliszewski et al. [2023b, 2026a], that an election is diverse if even after clustering its voters into like-minded groups, these groups disagree internally. As opposed to them, we parameterize our definition with an agreement index. For a given integerkand an electionE=(C , V), letP k (V) be a set of all complete partitions of voters into kdisjoint subsetsV 1 , . . . , V k , and letabe some agreement index. We aim to find a partition that maximizes average agreement within the subsets, weighted by the size of each cluster, i.e., P k i=1 |V i | |V| a(C , V i ). In principle, there are many ways in which the values for differentkcan be aggregated to give a single index. We follow a simple approach of Faliszewski et al. [2025b] and take the average of values forkfrom 1 to 5. This results in the following definition. Definition 6.1. For an election E= (C , V ), and agreement index a, we define the a-Diversity index as: a-div(E )= 1− 1 5 P 5 k=1 max V 1 ,...,V k ∈P k (V ) P k i=1 |V i | |V| a(C , V i ). Taking Central Agreement as the underlying index, we obtain Central Diversity, denotedcntr-div, and taking PCC Pairwise Agreement we obtain PCC Diversity, denotedpcc-div. We do not consider Hamming Pairwise Agreement for the sake of focus and due to its similarity topcc-agr. In both cases, obtaining the optimal partition of votes is not computationally feasible, thus we rely on clustering heuristics. We approxi- mate cntr-div with an algorithm based on thek-means clustering: We employ thek-medoids version using Hamming distance, where cluster centers are restricted to observed votes and are iteratively updated to minimize the sum of intra-cluster distances. For pcc-div we use spectral clustering [Ng et al., 2001]; our input for spectral clustering is the numberkof clusters and the affinity matrix, where for each pair of votesu, vwe provide value 1+ 1 2 pcc(u, v ) (value 0 means total dissimilarity, and 1 means the votes are equal). To quantify polarization, we measure the increase of agreement arising from clustering the election into two groups instead of viewing it as a single group (again, the general idea is due to Faliszewski et al. [2023b, 2026a]). Definition 6.2. For an election E= (C , V ), and agreement index a, we define the a-Polarization index as: a-pol(E )=max V 1 ,V 2 ∈P 2 (V ) P 2 i=1 |V i | |V| a(C , V i )− a(E ). We consider Central Polarization (cntr-pol) and PCC Polarization (pcc-pol), based on the Central and PCC Pairwise Agreement indices, and we use the same clustering heuristics as in the case of diversity indices. Additionally, we consider the Pairwise Polarization index, which outputs the standard deviation of Ham- ming distances between all pairs of votes. The intuition is that in a polarized electorate each two voters either have very similar or close-to-opposite preferences (factor of 2/|C| ensures values in [0, 1]). 8 Definition 6.3. For an election E= (C , V ), the Pairwise Polarization index is defined as pair-pol(E )= 2 r P u,v∈V (ham(u,v )− P x ,y∈V ham(x ,y )/|V| 2 ) 2 |V| 2 |C| 2 . 6.2Outer Diversity Recently, Faliszewski et al. [2026b] proposed an alternative diversity index, based on the average distance between the votes in the election and all possible votes. If the average distance is small, the votes are well spread in the whole space, i.e., diverse. It was originally proposed in the ordinal setting and for domains (sets of votes without specified multiplicity) rather than elections, but we adapt it to our setting as follows. For two collections of votes over the same set of candidates,U =(u 1 , . . . , u k ) andV =(v 1 , . . . , v n ), by ham(U , V) we mean the average Hamming distance betweenncopies of votes inUandkcopies of votes inV, matched so as to minimize this distance, i.e.: ham(U , V )= 1 k n min π:[k n]→[k n] P k n i=1 ham(u i , v π(i ) ), whereπ is a bijection and indices from U are taken modulo k and from V modulo n. For a candidate setC, letU C consist of a single copy of each vote from0,1 |C| . Then, in principle, we could measure diversity of an electionE=(C , V) based onham(V ,U C ), i.e., the average distance between votes inV and all possible votes. However, if the saturation ofEis very small or large, the value ofham(V ,U C ) is always substantial (as most votes inU C have moderate saturation). Thus, no election with small/large saturation could be considered diverse according to such a measure. To circumvent this issue, instead ofU C we use p -U C , defined as a collection in which the number of copies of each vote is proportional to the probability of drawing such a vote from p -IC (so, 1 /2-U C =U C ). We would like the index to be maximum and equal to 1 ifV = p -U C , and minimum and equal to 0 if Vis a singleton. From the following lemma we get that for a single votevwith saturationp, it holds that ham(v , p -U C )= 2p (1− p ). Lemma 6.1. For candidate set C , q∈ 0, 1 |C| , . . . , 1 and v∈ 0, 1 |C| with q|C| ones it holds that: ham(v , p -U C )= p (1− q )+ q (1− p ). Definition 6.4. For an election E= (C , V ) with satr(E )= p , the Outer Diversity index is defined as out-div(E )= 1− 1 2p (1−p ) ham(V , p -U C ). As discussed, the value 1 obtained forV = p -U C , and a single votevalways yields the value of 0. It remains to prove that this is indeed the minimum and maximum value. Proposition 6.2. For each election E , out-div(E )∈ [0, 1]. 6.3Comparison of the Indices Let us now analyze the values of the indices given in Table 1. First, we see that Central Diversity fails the resampling test of saturation independence, whereas the other diversity indices pass it. For polarization indices the test is not meaningful, as we expect low polarization for all values ofpandφparameters. We also see that clustering-based diversity indices give close-to-1 values for Diagonal and IC elections (including Lin-IC). Outer diversity performs surprisingly poorly on these elections, giving them low values, but we have verified that its values are much closer to 1 for analogous elections with a notably larger number of voters. Still, this dependence on election size seems a major drawback of the index. This is quite disappointing, given how well the outer diversity idea worked for ordinal preference domains [Faliszewski et al., 2026b]. Further, all diversity indices give higher values for the noisy variants of respective elections (see the values of 2-Party versus N(2-Party, 0.6) and Triangle versus N(Triangle, 0.6)). All polarization indices give value 1 for 2-Party and lower values for its noisy variant, as well as for 3- and 4-Party, as expected. Pairwise Polarization still gives nearly-1 for ( 1 /3, 1 /3)-2-Party whereas Central and PCC Polarization give lower ones, but both approaches could be justified. Overall, Pairwise Polarization tends to give highest values. For example, for Triangle it gives nearly 1 /2, while the other indices give values much closer to 0. Both approaches could be justified, either by Triangle’s similarity to 2-Party, or by saying that in Triangle all voters agree on best-to-worst ranking of the candidates, but have different approval thresholds. 9 Figure 2: Map of synthetic elections.Figure 3: Map of real-life elections. 7Experiments To present experimental results for our indices, we use the map of elections framework [Szufa et al., 2025]. We construct the map as follows. First, we assemble a collection of elections. Next, we compute pairwise distances between elections using the feature distance, as in the work of Faliszewski et al. [2025b]. Specifically, letf(E)=(f 1 (E), . . . , f k (E)) denote the feature vector of electionE, wheref 1 , . . . , f k are our indices. For two electionsEandF, we define their distance as the (standard) Euclidean distance between their feature vectors, d f (E , F)=∥ f(E)− f(F)∥ 2 . Finally, we embed the resulting distance matrix using multidimensional scaling (MDS [Kruskal, 1964]). For the map, we use the following three indices:pcc-agr,pcc-div, andpcc-pol. 2 In Figure 2, we show a map based on synthetic elections, while Figure 3 presents a map based on real-life ones (all our datasets will be publicly available). Each point corresponds to a single election. We describe the selected/generated elections in the Section D. Interestingly, the synthetic map forms a triangle whose vertices correspond to three extreme cultures: ID (maximizing agreement), IC (maximizing diversity), and 2-party (maximizing polarization). The edge between IC and ID is populated by Resampling elections; the edge between 2-party and IC is filled by N(2-Party) elections; and the edge between 2-party and ID is filled by (x , y)-2-Party elections. Both 2D-Euclidean (brown) and IAM (red) elections lie close to IC, reflecting high diversity together with low polarization and agreement. Finally, the partylist (green) elections occupy the upper region of the triangle, indicating high polarization. Turning to the real-life map (Figure 3), we observe that most elections exhibit very low polarization and, on average, substantially higher diversity than agreement. Compared with other real-world datasets, the Polkadot blockchain data (stakeholders voting over validators) shows relatively high agreement. Both Polkadot and Pol.is (users approving comments on an online deliberation platform) exhibit particularly low polarization. Intuitively, this may reflect a shared “search for ground truth” dynamic in both settings. Notably, the Eurovision (song contest) data looks similar to the Chopin (piano competition) one, which is interesting given that both involve people evaluating musical performances. In comparison to other real-life data, the French (presidential) elections seem relatively polarized. Regarding Pabulib (participatory budgeting voting), with few exceptions, the data shows high diversity, with low agreement and low polarization. (This may stem from the fact that, in most cities, people can approve only a very limited number of projects, so the saturation is extremely low). To conclude, it is worth noting that our map of real-life elections resembles the one from the work of Faliszewski et al. [2025b], in which (while focusing on ordinal elections), most real-life instances appeared in a similar region of the map, i.e., low polarization, mid to high diversity, and low to mid agreement. Finally, we note that MDS embeddings of maps in Figures 2 and 3 achieve mean multiplicative distortions of 1.014 and 1.010, respectively, indicating extremely good representation. In particular, this is far more accurate than the original maps presented in [Szufa et al., 2025, Section 4.3.2], where overall distortion is between 1.21 and 1.26. 2 We chose these indices as their sum is the most consistently close to 1 among those studied (see Section C.1 for details), which shows that they are complementary and, hence, together provide a good high-level summary of the nature of a given election. For elections with more than 200 candidates or more than 1000 voters, we computed the indices using sampled subsets of the elections. 10 8Future Work We foresee three directions for future work: First, a deeper study of agreement indices in ordinal elections, going beyond the index we mention in Section 5.1 and Kemeny rule. Second, finding more principled diversity and polarization indices, also for polarization with more than two disagreeing groups. 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AAdditional Tables Table 2 shows the results of the resampling experiment for agreement indices and elections of different sizes. BMissing Proofs Given two votes,uandv, letn 11 , n 10 , n 01 , andn 00 be the number of candidates approved in bothuandv, only u, only v , and neither of them. Then we have: pcc(u, v )= n 00 ·n 11 −n 01 ·n 10 p (n 10 +n 11 )·(n 00 +n 01 )·(n 01 +n 11 )·(n 00 +n 10 ) .(2) B.1Proof of Theorem 5.1 Let E= (C , V ) be an election. By definition of the Approval Agreement, we have that: av-agr(E )= 1 |C| P c∈C ̄ ̄ ̄ 1− 2 | A(c )| |V| ̄ ̄ ̄ . Hence, the value ofav-agr(E) is 1 if and only if for each candidatec∈ Cit holds that ̄ ̄ ̄ 1− 2 | A(c )| |V| ̄ ̄ ̄ = 1, which happens exactly if| A(c)|=0 or| A(c)|=|V|. This means thatav-agr(E)=1 if and only ifEis an identity election. On the other had,av-agr(E)=0 if and only if for each candidatec∈ Cwe have ̄ ̄ ̄ 1− 2 | A(c )| |V| ̄ ̄ ̄ =0, which happens exactly when 2 | A(c )| |V| = 1, i.e.,| A(c )|= |V| /2. 14 B.2Proof of Theorem 5.2 Let E= (C , V ) be an election. By definition of Central Agreement, we have that: cntr-agr(E )= 1− P v i ∈V ham(v i , u) |V|· min(avl(E ), rev-avl(E )) . Hence,cntr-agr(E) is equal to 1 exactly if every summand is equal to 0, which happens when all the votes are identical. Hence, cntr-agr(E )= 1 if and only if E is an identity election. To characterize whencntr-agr(E)=0, let us first introduce an auxiliary function. By central Hamming distance (CHD) of election E= (C , V ) we mean: chd(E )= P v i ∈V ham(v i , u). Naturally, we have: cntr-agr(E )= 1− chd(E ) |V|· min(avl(E ), rev-avl(E )) . Let us rephrasechd(E) as follows: Given an electionE=(C , V) and one of its central votesu∈ cen(E), for each candidatec j , we definechd(c j )= P v i ∈V |v i [j]− u[j]|. Then, by changing the order of summation, we can have: chd(E )= P v i ∈V ham(v i , u) = P v i ∈V P c j ∈C |v i [ j ]− u[ j ]|(3) = P c j ∈C P v i ∈V |v i [ j ]− u[ j ]|= P c j ∈C chd(c j ). Using this formulation, we characterize the elections with a given size and saturation, for whichchd(E) is largest (specifically, it is easier to verify the claims in the proof of Lemma B.1 by using Equation (3)). Lemma B.1. The valuechd(E) is largest among elections withmcandidates,nvoters, and saturationαif and only if (a) all candidates are approved in some vote incen(E), or (b) all candidates are disapproved in some vote in cen(E ), or (c) for each candidate c j we have that| A(c j )|∈ ⌊ n /2⌋,⌈ n /2⌉. Proof. Let us first consider case (b). We know that there is an electionEwithmcandidates,nvoters, and saturationsatr(E)=α, such that all candidates are disapproved in some central vote incen(E). We arbitrarily partition the candidates into those that can receive approvals and those that can provide them. Then we perform a sequence of operations where, in each of them, we move a single approval from a providing candidate to a receiving one. We observe that theCHDvalues remains constant, until the first operation where for the receiving candidatec k all the central votes for the current election approve it. When this happens, the value ofCHDdrops by one. From this point on, each operation either maintains the current value ofCHDor decreases it further. Since, with an appropriate sequence of operations and an appropriate choice of providing and receiving candidates, we can reach an arbitrary elections with the given size and saturation, it must be the case that the starting election had the maximum value ofCHD. The same reasoning applies to case (a), but for moving disapprovals rather than approvals. Let us consider case (c), assuming thatnis odd (if it were even, then cases (a) and (b) would have held as well and we have handled them already). Consider an analogous sequence of operations as in the preceding paragraph, except that the candidates who are originally approved by⌈ n /2⌉voters (call them big candidates) cannot give approvals to those, who were originally approved by⌊ n /2⌋voters (call them small candidates; this is w.l.o.g., as if we wanted to make such swaps, then we could have simply permuted the candidates). Now we observe that whenever an approval is moved from a small candidate or to a big candidate, then theCHD value of the election decreases. Yet, all our moves are of this form, so the original CHD value must have been maximal. Note that every election can be reached by moving approvals from one in cases (a), (b), or (c). This completes the proof. Next, let us consider some electionE=(C , V), such that there is a voteu∈ cen(E) where all candidates are disapproved. By the above lemma, we know that itschd(E) value is largest for its size and saturation. Further, 15 we note that: chd(E )= P v i ∈V ham(v i , u) = P v i ∈V | A(v i )|=|V|· avl(E ). It is also immediate that|V|· avl(E)≤|V|· rev-avl(E) in this case. Similarly, if there isu∈ cen(E) that approves all the candidates, then we havechd(E)=|V|· rev-avl(E) andrev-avl(E)≤ avl(E). This means that if either all candidates inEare approved by at least half of the voters or all candidates inEare disapproved by at least half of the voters, then cntr-agr(E )= 0. To finish the proof, it remains to consider an electionE=(C , V) where some candidates are approved by fewer than half of the voters (denote the set of these candidates asC < ), some candidates are approved by more than half of the voters (denote the set of these candidates asC > ), and some (possibly zero) are approved by exactly half of the voters (denote the set of these candidates asC = ). Let us formE < =(C < , V),E > =(C > , V) and E = = (C = , V ). The following holds (note that chd(E = )=|V|avl(E = )=|V|rev-avl(E = )): chd(E )= chd(E < )+ chd(E > )+ chd(E = ) =|V|(avl(E < )+ rev-avl(E > )+ avl(E = )) One can easily verify that this value is strictly smaller than both|V|avl(E) and|V|rev-avl(E) (the former inequality holds becauseavl(E)= avl(E < )+ avl(E > )+ avl(E = ) whilerev-avl(E > )< avl(E < ); the latter inequality holds for analogous reasons). Consequently, cntr-agr(E )> 0. B.3Proof of Theorem 5.3 We use the notation as in the proof of Theorem 5.2, in Equation(3). Consequently, we have electionE=(C , V) and chd(E )= P c j ∈C chd(c j ), where: chd(c j )= X v i ∈V |v i [ j ]− u[ j ]| for some u∈ cen(E ). We note that: chd(c j )= ( | A(c j )|if u[ j ]= 0, |V|−| A(c j )|if u[ j ]= 1. Further, we have thatu[j]=0 whenever| A(c j )|≤|V|/2 andu[j]=1 otherwise (strictly speaking, if| A(c j )| were equal to|V|/2 then we could also takeu[j]=1 in this case, but this would not change the logic of the argument). Consequently, we have that: chd(c j )= |V| 2 − ̄ ̄ ̄ |V| 2 −| A(c j )| ̄ ̄ ̄ . After substituting this to the formula for cntr-agr(E ), we get: cntr-agr(E )= 1− chd(E ) |V|·min(avl(E ),rev-avl(E )) = 1− P c j ∈C ¡ |V| 2 − ̄ ̄ |V| 2 −| A(c j )| ̄ ̄ ¢ |V|·min(avl(E ),rev-avl(E )) = 1− |V| 2 P c j ∈C ³ 1− ̄ ̄ ̄ 1− 2| A(c j )| |V| ̄ ̄ ̄ ́ |V|·min(avl(E ),rev-avl(E )) = 1− P c j ∈C ³ 1− ̄ ̄ ̄ 1− 2| A(c j )| |V| ̄ ̄ ̄ ́ 2·min(avl(E ),rev-avl(E )) This completes the proof. B.4Proof of Theorem 5.4 This result is, in essence, folk knowledge. We provide an argument for the sake of completeness. Letxandy be two vectors of dimension m. We form two new vectors, x ′ and y ′ , such that for each i∈ [m] we have: x ′ [i ]= x [i ]− x [i ] q P m j=1 (x [i ]−x [i ]) 2 and y ′ [i ]= y [i ]−y [i ] q P m j=1 (y [i ]−y [i ]) 2 . 16 We observe that pcc(x , y ) is equal to the dot product of x ′ and y ′ . Now let us consider some electionE=(C , V), whereV =(v 1 , . . . , v n ). We form ann× nmatrixM, where entry in thei-th row andj-th column ispcc(v i , v j ). By the argument from the previous paragraph, we know thatMis a Gram matrix (i.e., a matrix of dot products of a sequence of vectors). Since all Gram matrices are semi-definite positive, and sums of entries of all semi-definite positive matrices are nonnegative, we have that: X v i ∈V X v j ∈V pcc(v i , v j )≥ 0. On the other hand, since for each two votes v i and v j we have that pcc(v i , v j )≤ 1, it is immediate that: X v i ∈V X v j ∈V pcc(v i , v j )≤ n 2 . Together, this means that pcc-agr(E )∈ [0, 1]. B.5Proof of Theorem 5.5 LetE =(C , V) be an election with candidate setC = c 1 , . . . , c m . We define its expected Hamming distance (EHD) as follows: ehd(E )= P u∈V P v∈V ham(u, v ). Naturally, we have: pair-agr(E )= ehd(E ) /2|V| 2 |C|·satr(E )(1−satr(E )). By definition of ehd(E ) and changing the order of summation, we have: ehd(E )= P u∈V P v∈V ham(u, v ) = P u∈V P v∈V P c j ∈C |u[ j ]− v [ j ]| = P c j ∈C P u∈V P v∈V |u[ j ]− v [ j ]|. Now we note that for each candidatec j , there are| A(c j )|voters who approve it andn−| A(c j )|who do not. Hence, for a given candidate c j ∈ C , we have that: P u∈V P v∈V |u[ j ]− v [ j ]| =| A(c j )|· P v∈V |1− v [ j ]| + (n−| A(c j )|)· P v∈V |0− v [ j ]| =| A(c j )|(n−| A(c j )|)+ (n−| A(c j )|)| A(c j )| = 2| A(c j )|(n−| A(c j )|). By substituting this equality into the definition of ehd(E ) we obtain: ehd(E )= P c j ∈C 2| A(c j )|(n−| A(c j )|), which substituted to the formula for pair-agr(E ) gives: pair-agr(E )= P c j ∈C | A(c j )|(n−| A(c j )|) |V| 2 |C|·satr(E )(1−satr(E )) . This completes the proof. B.6Proof of Theorem 5.6 It is easy to verify that, indeed, each of the four indices takes value 1 if and only if the input election is an identity one. Further, closer inspection of the definition of Jaccard Pairwise Agreement indicates that it assumes value 0 if and only if all voters approve disjoint sets of candidates (otherwise one of the summands in its definition would be greater than 0). It remains to consider when Hamming Pairwise Agreement assumes value 0. We use the same notation as in the proof of Theorem 5.5 and we first prove the next lemma, which characterizes for which elections E of a given size and saturation value ehd(E ) is largest. 17 Lemma B.2. Letmandnbe two positive integers, and letα∈[0,1] be a rational number such thatα· nmis an integer. LetE=(C , V) be an election withmcandidates,nvoters, andsatr(E)=α. Valueehd(E) is largest among elections of its size and saturation if for each two candidates c j , c k ∈ C , | | A(c j )|−| A(c k )| | ≤ 1. Proof.LetE =(C , V) be an arbitrary election such thatC = c 1 , . . . , c m ,V =(v 1 , . . . , v n ), andsatr(E)=α. We claim that if there are two candidatesc j , c k ∈ Csuch thatc k has at least two more approvals thanc j then by swapping two approvals within E we can obtain a new election E ′ such that ehd(E ′ )> ehd(E ). Let us first expressehd(E) in a convenient way. Let us define functionf(x)= x(1−x) and for each candidate c t ∈ C, letp t be its approval probability, i.e.,p t = | A(c t )| /n. Now, using the formula forehd(E) given in the proof of Theorem 5.5, we have: ehd(E )= 2 n 2 P c t ∈C n· f (p t ) = 2 n P c t ∈C f (p t ) = 2 n ¡ f (p j )+ f (p k ) ¢ + X (E ), whereX(E)= 2 n P c t ∈C \c j ,c k f (p t ). Next, we form electionE ′ that is identical toEexcept that we remove a single approval from c k and add a single approval to c j . We have that: ehd(E ′ )= 2 n ¡ f (p j + 1 /n)+ f (p k − 1 /n) ¢ + X (E ), To establish that ehd(E ′ )> ehd(E ), we need to show that f (p j + 1 /n)− f (p j )> f (p k )− f (p k − 1 /n).(4) If we have that bothp j , p k ≤0.5 or bothp j , p k ≥0.5 then it is clear that this holds (as the derivative off(x) is strictly decreasing and assumes value 0 forx=0.5). On the other hand, ifp j <0.5 andp k >0.5 then either the left-hand side of(4)is positive and the right-hand side is at most zero, or the left-hand side if zero and the right-hand size is negative (this follows from the shape of function f (x ), which grows for x∈ [0, 0.5] and then symmetrically decreases for x∈ [0.5, 1], and the fact that c k has at least two more approvals than c j in E ). By repeating the above reasoning, we find that elections where the numbers of approvals received by some two candidates differ by more than one cannot have the largestEHDvalue. By Theorem 5.5 we know that an election’sEHDvalue—for a given number of voters—depends only on the candidates’ approval scores. Consequently, an election has the highestEHDvalue among those with a given numbers of candidates and voters, and a given saturation, if and only if approval scores of its candidates differ at most by one. Now let us consider an electionE=(C , V), where each candidate has the same approval score equal toT. In such an election, we havesatr(E)= T/|V|. By Theorem 5.5 and substituting candidates’ approval scores into the formula, we have that: pair-agr(E )= 1− |C|T (|V|−T ) |V| 2 |C|·satr(E )(1−satr(E )) = 1− |C|T (|V|−T ) |V| 2 |C|·T /|V|(1−T /|V|) = 1− |C|T (|V|−T ) T (|V|−T ) = 0. Next, we observe that for an election where candidates’ approval scores differ by at most one (and, indeed, there are two candidates with different scores) it holds thatpair-agr(E)>0. To this end, let us consider election E=(C , V) withmcandidates andnvoters, whereTvoters approve all the candidates, 1 voter approvesX < m candidates, and remaining n− T− 1 voters do not approve any candidates. We have: ehd(E )= T (0+ (m− X )+ m(n− T− 1)) + T (m− X )+ 0+ X (n− T− 1) + (n− T− 1)(mT+ X+ 0) = 2T (m− X )+ 2mT (n− T− 1)+ 2X (n− T− 1) = 2mT− 2T X+ 2nmT− 2mT 2 − 2mT+ 2X n− 2T X− 2X = 2nmT− 4T X− 2mT 2 + 2X n− 2X = n 2 m( 2T n − 4T X n 2 m − 2T 2 n 2 + 2X nm − 2X n 2 m ). 18 For election E we have that satr(E )= mT+X nm , so: pair-agr(E )= 1− n 2 m( 2T n − 4T X n 2 m − 2T 2 n 2 + 2X nm − 2X n 2 m ) 2n 2 m( mT+X nm )(1− mT+X nm ) = 1− 2T n − 4T X n 2 m − 2T 2 n 2 + 2X nm − 2X n 2 m 2( mT+X nm )(1− mT+X nm ) = 1− 2T n − 4T X n 2 m − 2T 2 n 2 + 2X nm − 2X n 2 m 2T n − 4T X n 2 m − 2T 2 n 2 + 2X nm − 2X 2 n 2 m 2 > 0. The final inequality follows by computing the numerator and denominator. This concludes the proof as, by Theorem B.2, for every other election of this size and saturation, pair-agr(E ) is even larger. B.7Proof of Theorem 5.7 LetE=(C , V) be ak-Party election. We writemandnto denote|C|and|V|, and we assume that bothmand nare divisible byk. Given two votes,uandvinV, that approve the same groups of m /kcandidates, we have that: pcc(u, v )= 1. On the other hand, if u and v approve disjoint sets of m /k candidates each, then, by Equation (2) we have: pcc(u, v )= − m k · m k p m k ·(m− m k )· m k ·(m− m k ) =− m 2 k 2 m 2 k (1− 1 k ) =− 1 k 1 1− 1 k . Then, we have that (the second equality follows by substituting the above values for PCCs): pcc-agr(E )= 1 n 2 P v i ∈V P v j ∈V pcc(v i , v j ) = 1 n 2 · n· ( n k − (n− n k ) 1 k 1 1− 1 k ) = ( 1 k − 1 k )= 0. Similarly, we have: pcc + -agr(E )= 1 n 2 P v i ∈V P v j ∈V max(0, pcc(v i , v j )) = 1 n 2 · n· n k = 1 k . This completes the proof. B.8Proof of Theorem 5.8 Let E= (C , V ) be an election such that 0<| A(u)|=| A(v )|< 1 for all u, v∈ V . First, we derive an explicit dependence between thepcc(u, v) andham(u, v) for any pair ofu, v∈ V. We note that an implicit dependence has been known in the literature on association coefficients (see, e.g., Warrens [2008]); for completeness we provide the exact formula in our notation. Specifically, Warrens [2008] considers Yule’sφcoefficient (denoted there byS Yule1 ) Yule [1912], which coincides with PCC for binary vectors, and also the simple matching coefficientS SM Sokal and Michener [1958], which in our notation equals to the number of matches, i.e.,S SM (u, v)= m− ham(u, v). Warrens [2008] shows that, under fixed marginals,S Yule1 can be expressed as an affine transformation ofS SM , but do not provide an explicit formula, which is as follows. Proposition B.3. If 0<| A(u)|=| A(v )|= p m< m holds then we have pcc(u, v )= 1− ham(u,v ) 2m(1−p )p . Proof.Letn 11 , n 10 , n 01 , n 00 denote the numbers of candidates approved in bothuandv, only inu, only inv, and in neither of them, respectively. By these definitions we have n 11 + n 10 = n 11 + n 01 = p m.(5) 19 This implies: n 00 = m− n 11 − n 10 − n 01 (5) = m− n 11 − (p m− n 11 )− (p m− n 11 ) = m− 2p m+ n 11 . Now consider the numerator of pcc(u, v ) (defined in Equation (2)). n 11 · n 00 − n 01 · n 10 (5) = n 11 · (m− 2p m+ n 11 )− (p m− n 11 ) 2 = n 11 · m− (p m) 2 . The denominator of pcc(u, v ) equals to: p (n 10 + n 11 )· (n 00 + n 01 )· (n 01 + n 11 )· (n 00 + n 10 ) = p p m· (m− p m)· p m· (m− p m) = p m· (m− p m)= m 2 · p (1− p ). Putting both together we obtain: pcc(u, v )= n 11 · m− (m p ) 2 m 2 · p (1− p ) = n 11 − m p 2 m p (1− p ) .(6) By the definition of the Hamming distance we haveham(u, v)= n 10 +n 01 =2(p m−n 11 ) which is equivalent to n 11 = p m− ham(u, v ) 2 .(7) Putting these together we obtain the final equality: pcc(u, v ) (6),(7) = m p− ham(u,v ) 2 − m p 2 m p (1− p ) = m p− m p 2 m p (1− p ) − ham(u,v ) 2 m p (1− p ) = 1− ham(u, v ) 2m p (1− p ) . Since| A(v)|= p mfor allv ∈ V, we haveavl(E)= p m, and thussatr(E)= avl(E ) m = pand together with Theorem B.3 we can prove the final equality: pcc-agr(E ) Def. 5.3 = 1 n 2 X u∈V X v∈V pcc(u, v ) Prop. B .3 = 1 n 2 X u∈V X v∈V μ 1− ham(u, v ) 2m p (1− p ) ¶ =1− X u∈V X v∈V ham(u, v ) 2n 2 m p (1− p ) Def. 5.3 =pair-agr(E ). This completes the proof. 20 B.9Proof of Theorem 6.1 Proof.We will prove that both sides of the equation can be linked to the ratios of the sizes of two combinatorial sets defined as follows. Note that inp -U C every vote appearsa r · b |C|−r times, whereaandbare some integers such thatp= a/(a+ b) andris the number of ones in the vote. LetFbe a set of functions from the set ofmnumbers to the set ofa+ bnumbers,f: [m]→[a+ b]. Then, letXbe set of pairs (x , f)∈[m]× F, such thatf(x)∈[a], ifx∈[(1− q)m], orf(x)∈ a+1, . . . , a+ b, ifx∈ (1− q)m+1, . . . , m. We claim that (1) ham(v , p -U C )=|X|/(m·|F|) and (2)|X|/(m·|F|)= p (1− q )+ q (1− p ). (1) First, observe that |F|= (a+ b) m = m X k=0 à m k ! a k b m−k =|p -U C |.(8) Now, without loss of generality, let us assume that the first (1− q)mpositions ofvare 0s and the lastq m positions are 1s. According to the definition ofp -U C , each voteu∈ 0,1 m appears therea α b β times, whereα is the number of ones inuandβis the number of zeros. This corresponds to the number of functionsfthat give values in [a] to exactly those arguments in [m] that belong toA(u). Now, for each such functionf, we can choosexin the number of different ways that is equal to the number of indices inA(u) that are also in [(1− q)m] plus the number of indices that are not inA(u) but are in(1− q)m+1, . . . , m, i.e., the symmetric difference betweenA(u) and(1− q)m+1, . . . , m. But this is exactly the Hamming distance betweenuand v. Hence,|X|is equal to the sum of the (unnormalized) Hamming distances betweenvand each vote from p -U C . Combining this with eq. (8), we get that indeed ham(v , p -U C )=|X|/(m·|F|). (2) To express the size ofXin a different way, we first fixx∈[m], and then select matchingf. We split Xinto two disjoint subsets depending on the value ofx, in a way thatX 1 = (x , f)∈ X:x∈[(1− q)m]and X 2 = (x , f )∈ X : x∈ (1− q )m+ 1, . . . , m. InX 1 we can selectxin (1− q)mdifferent ways, thenf(x) can haveadifferent values, andf(y) for each y∈[m]\x can have (a+b) different values. Thus,|X 1 |=(1−q)m·a·(a+b) m−1 . Analogously,|X 2 |= q·m·b·(a+ b ) m−1 . SinceX 1 andX 2 are disjoint andX = X 1 ∪ X 2 , we get that|X|=|X 1 |+|X 2 |=((1− q)a+ q b)m(a+ b) m−1 . Thus, by eq. (8) |X| m|F| = (1− q )a+ q b a+ b = (1− q )p+ q (1− p ). B.10Proof of Theorem 6.2 Proof.Clearly, for every electionE=(C , V) andp= satr(E), it holds thatham(V , p -U C )≥0,henceout-div satr h (E)≤ 1. For the lower bound, by a series of (in)equalities, we will show thatham(V , p -U C )≤2(p− p 2 ),which will imply that out-div satr h (E )≥ 0. DenoteV = v 1 , . . . , v n . Let us start by considering copyingntimes each vote inp -U C , we denote the resulting collection of votes asn· p -U C . Observe that introducing these additional copies does not affect the Hamming distance fromV(as otherwise it would mean that there is a fractional matching giving less total weight than an integral matching, which is not possible Lovász and Plummer [2009]). Hence, ham(V , p -U C )= ham(V , n· p -U C ). To computeham(V , n· p -U C ) we match the votes fromVto those inn· p -U C in an optimal way. Another way of matching the votes would be to match each separate set of copies of votes inp -U C to a single, each time different vote in V . Thus, we get ham(V , p -U C )= ham(V , n· p -U C ) ≤ 1 n n X i=1 ham(v i , p -U C ). 21 Now, let us denote a fraction of ones in v i by q i =| A(v i )|/|C|. From Theorem 6.1, we get that ham(V , p -U C )≤ 1 n n X i=1 ham(v i , p -U C ) = 1 n n X i=1 ¡ p· (1− q i )+ (1− p )· q i ¢ = p à 1− 1 n n X i=1 q i ! + (1− p )· 1 n n X i=1 q i = 2p (1− p ), where the last equality we can obtain from the fact that P n i=1 q i = satr(E )= p . This concludes the proof. CFurther Data Analysis In this appendix, we elaborate on the measure of complementarity of computed indices, and provide correla- tion matrix of their values. C.1Complementarity Faliszewski et al. [2023b] suggest that the elections they consider approximately fit into a simplex in the space of their agreement, diversity, and polarization indices. We would like to measure how much this is the case for all triples of our indices. To this end, we use the following measure of complementarity (we note that similar measures have been used to measure a diversification of a portfolio in financial sciences, see e.g., Choueifaty and Coignard [2008]; Han et al. [2025]). Definition C.1. Let x=(x 1 , . . . , x n ), y=(y 1 , . . . , y n ), and z=(z 1 , . . . , z n ), be three vectors of index values. We define their complementarity as one minus the standard deviation of the sum of the indices divided by the sum of the standard deviation of the indices, i.e., cmpl(x, y, z)= 1− std(x 1 + y 1 + z 1 , . . . , x n + y n + z n ) std(x)+ std(y)+ std(z) , where std(a 1 , . . . , a n )= q 1 n P n i=1 (a i − 1 n P n j=1 a j ) 2 . Intuitively, complementarity measures how close is the sum of indices to being constant as compared to the variability of the indices alone. If the sum is constant, complementarity would give value equal to 1. On the other hand, for any vector x∈R n , we have cmpl(x, x, x)= 0. We checked the values of the complementarity measure for all triples of our indices. The four that gave the highest values where: • pcc-agr, pcc-div, pcc-pol: the value of 0.9016, • cntr-agr, cntr-div, pcc-pol: the value of 0.8793, • cntr-agr, cntr-div, cntr-pol: the value of 0.8775, and • pair-agr, cntr-div, pcc-pol: the value of 0.8748. This motivated us to focus on the pcc-agr, pcc-div, pcc-pol map in Section 7. For comparison, in the original paper by Faliszewski et al. [2023b] that introduced diversity-agreement- polarization framework, the complementarity of their indices for the ordinal elections in their extended dataset is equal to 0.8818 (we have normalized the values of the indices to have the maximum value of 1, as it is not the case there). In the follow-up paper by Faliszewski et al. [2025b], the complementarity of the indices of the election in the main dataset is equal to 0.9086. 22 satrav-agrcntr-agrpair-agrpcc-agrpcc + -agr jacc-agrcntr-divpcc-divout-divcntr-polpcc-polpair-pol satr– -0.4960.4250.1630.0320.1150.644-0.345-0.121-0.1110.1320.1490.294 app_agr -0.461 – -0.1430.1770.3200.100-0.2020.211-0.0570.049-0.247-0.302-0.490 cntr_agr 0.3740.095 – 0.5540.3990.3500.580-0.567-0.328-0.2510.0940.1000.196 pair_agr 0.2430.3280.909 – 0.7090.4530.397-0.437-0.413-0.1570.0800.0420.044 pcc_agr 0.1060.4800.6900.816 – 0.6500.281-0.288-0.531-0.163 0.009 -0.052 -0.007 pcc+_agr 0.1730.2420.6130.6910.855 – 0.422-0.555-0.827-0.4210.2790.2700.302 jacc_agr 0.812-0.1270.6890.6560.5160.603 – -0.592-0.421-0.3180.2190.1950.287 cntr_div -0.3920.198-0.668-0.626-0.420-0.699-0.707 – 0.6360.469-0.477-0.490-0.464 pcc_div -0.174-0.115-0.495-0.576-0.667-0.913-0.5770.832 – 0.498-0.366-0.407-0.363 out_div -0.1340.040-0.374-0.371-0.295-0.503-0.3980.6420.609 – -0.222-0.252-0.280 cntr_pol 0.181-0.379-0.059-0.059-0.1100.3530.284-0.649-0.567-0.461 – 0.6130.497 pcc_pol 0.180-0.399-0.028-0.081-0.1510.3540.276-0.669-0.604-0.5040.929 – 0.619 pair_pol 0.257-0.5160.030-0.081-0.1250.3600.295-0.640-0.569-0.4440.8170.912 – Table 3: Correlations between our features based on the values for elections in the main dataset. The values in the top-right triangle are Kendall’sτcorrelation coefficients, while those in the bottom-left triangle are Pearson’sρ correlation coefficients. C.2Correlations In Table 3 we report the Pearson’sρ and Kendall’sτ correlation coefficients for each pair of our indices. We observe that agreement indices (except for Approval Agreement), diversity indices, and polarization indices all form relatively well correlated clusters which confirms our intuitions that they are measuring more or less the same features. Across these clusters there is usually either negative correlation or a lack of correlation. The only exception is the fact that Jaccard and PCC + Pairwise Agreement indices are slightly correlated with the polarization indices. This is due to the fact that Jaccard and PCC + Pairwise Agreement indices capture the local agreement, which means that the presence of voter groups that have similar preference inside each group, but possibly very dissimilar across the groups, increases the value of these two agreement indices. However, the same structures also increase the polarization indices. Finally, we note that the Saturation index is not significantly correlated with any other index, except for Jaccard Pairwise Agreement. This can be explained by noting that these are the only two indices that we consider that do not treat approvals and disapprovals symmetrically. DDetailed Datasets Description In this appendix we provide a complete description of the data used to produce Figures 2 and 3. We begin by providing details on exact distributions used to generate synthetic elections as well as sources of real-world data and the methods of its preprocessing. We conclude with Table 4, in which for each election in the dataset we provide the values of all considered features as well as its icon akin to those in Table 1. D.1Synthetic Data Let us start by describing elections generated by us using synthetic distributions. CompassCompass elections include exactly those that are listed in Table 1, one copy each. Impartial CultureWe generated ten IC elections with 100 candidates and 1000 voters, two for each value of p= 0.1, 0.3, 0.5, 0.7, 0.9. Lin-ICWe generated five Lin-IC elections with 100 candidates and 1000 voters each, as described in Section 3. 23 ResamplingWe generated 50 resampling elections drawingφandpuniformly at random from [0,1], each with 100 candidates and 1000 voters. N(2-Party)We generated 25 noisy 2-party elections, each with 100 voters and 100 candidates. We sampledφ uniformly at random from [0, 1]. (x,y)-2-PartyWe generated 25 (x,y)-2-Party elections, with 100 voters and 100 candidates each. For each we chose x and y uniformly at random and independently from [0, 1]. Party-ListWe generated 25 uneven party-list elections, each with 100 voters and 100 candidates, in the following way. First we sampled the number of partieskfrom a Poisson distribution withλ=1 and added 3, so that we have at least 3 parties. Next, we uniformly at random chose two partitions of 100 intoknumbers. One of the partitions was then used as the number of voters in each party, while the other was used as the number of candidates. ID MixtureForp∈[0,1] and integerk, ink , p-ID Mixture model, we first split the voters randomly intok groups. Then, each group has identical approvals drawn fromp-IC. We generated 20 ID Mixture elections, each with 100 candidates and 1000 voters, one for each combination of k∈ 2, 3, 4, 5 and p∈ 0.1, 0.3, 0.5, 0.7, 0.9. IAM MixtureFor a givenk, we generate an IAM Mixture election by first splitting the voters randomly intok groups, then for each group and candidatecdrawingp c uniformly at random from [0,1], and finally sampling a sub-election for each group using the Independent Approval Model described in Section 3. We generated 20 IAM Mixture elections, each with 100 candidates and 1000 voters, five for each k∈ 2, 3, 4, 5. 2D-EuclideanWe sampled 50 election from the 2D Euclidean model, replicating first five distributions from Godziszewski et al. [2021] with 10 elections in each (each with 100 candidates and 1000 voters). Specifically, we assumed that both voters and candidates are sampled uniformly from [0, 1]× [0, 1] square, and then: • each voter i approves all candidates in radius r = 0.117 from i ’s position, • each voter i approves all candidates in radius r = 0.167 from i ’s position, • each voteriapproves all candidates in radiusr i fromi’s position, wherer i is drawn uniformly from [0, 0.5], • each voter i approves 10 closest candidates, or • each voter i approves some number of closest candidates, that is drawn uniformly from 1, 2, . . . , 100. D.2Real-World Elections Let us now cover elections that are based on real-world data. We include elections from Preflib Mattei and Walsh [2013], Pabulib Faliszewski et al. [2023a], as well as a number of datasets that we have gathered or converted ourselves. ChopinThese elections are based on the XIX International Chopin Piano Competition. Each election is based on one of four stages of the competition. In each, every member of the jury gives a score from 1 to 25 to every participant of a given stage (except for a relatively rare case in which the participant is a student of the member of the jury). The jury is fixed and consists of 17 members, which are voters in our election, and the number of participants, i.e., candidates, decreases with each stage (starting from 84 in the first stage and ending at 11 in the final stage) as only the best candidates pass to the next stage. For each stage, we converted a score above 20 to an approval. French ElectionsEach of these datasets consists of exit-poll surveys conducted during the French presiden- tial elections of 2007 Baujard et al. [2025a], 2012 Baujard et al. [2025b], 2017 Bouveret et al. [2018], and 2022 Delemazure and Bouveret [2024]. 24 JesterThe Jester approval elections were derived from a Kaggle dataset 3 in which users assigned numerical scores to jokes Goldberg et al. [2001]. In order to convert the data into approval profile, jokes were ranked according to their scores, with unranked jokes placed randomly at the bottom of the ranking. A fixed top percentage of jokes was then labeled as approved candidates, while the remaining jokes were considered unapproved. MovieLensThis dataset is a subset of MovieLens 20M Dataset that can be found on Kaggle Harper and Konstan [2015]. 4 The subset consists of 50 films (candidates) and 423 users (voters) each of which rates each film with a score from1.0,1.5,2.0, . . . ,5.0. We have converted this subset into three elections by setting an approval for every score that is equal or above 4.0, 4.5, or 5.0. NetworksWe have included seven elections that have been converted from canonical network datasets. In each of them we have treated the set of nodes as both voters and candidates with an approval corresponding to an edge from a voter to a candidate. Additionally, each node approves itself. Three networks have been taken from the Mark Newman’s collection of network datasets. 5 These in- clude American College Football Network Girvan and Newman [2002], Dolphin Social Network Lusseau et al. [2003], and Political Blogs Network Adamic and Glance [2005]. The next four network has been taken from the predefined datasets in the Editor4Centralities online tool. 6 These include September 11 Attack Terror- ist Network Krebs [2002], Wood-Processing Facility Strike Network Michael [1997], Zachary’s Karate Club Network Zachary [1977], and Les Miserables Character Network Knuth [1993]. PolkadotWe sampled 10 elections from the Polkadot dataset on Preflib. Each election represents votes of the stakeholders in the Polkadot blockchain over the set of validators in the Nominated Proof-of-Stake (NPoS) protocol Boehmer et al. [2024]. Our SurveysWe collected these datasets in an online survey filled predominantly by students. Voters were asked to mark their like or dislike for each of the candidates which then translated to approvals and disapprovals. No restrictions regarding the vote length were imposed. The topics of the surveys include colors, fruits, pets, pizza types, and school subjects (each topic corresponds to one election). Bidding DataWe include six elections based on bids of reviewers over the papers which they want to review in computer science conferences. All of the data comes from Preflib. Three elections comes from AAMAS 2015, 2016, and 2021 editions and three come from unspecified conferences. In each case, we have interpreted "yes" bids as an approval, and all other responses (and lack thereof ) as a lack of approval. CampsongsThese four elections are surveys in which the participants had to state their preferences over CCM-genre music pieces. The data is available on Preflib. CTUThis is an approval election over the time of tutorial at the Czech Technical University in Prague. EurovisionWe include a sample of 20 election based on the Eurovision Song Contest from the dataset available on Preflib Boehmer [2023]. Originally the data consists of rankings over most preferred participants as seen by each member of the jury (the number of ranked participants is fixed each time). We have converted these top-truncated ordinal elections to approval elections, by setting an approval each for each ranked participant. PostersThese are two approval elections from the San Sebastian Poster Competition available at Preflib. Two elections were held over two groups of posters (A group) and (B group). 3 https://w.kaggle.com/datasets/aakaashjois/jester-collaborative-filtering-dataset 4 https://w.kaggle.com/datasets/grouplens/movielens-20m-dataset 5 https://websites.umich.edu/~mejn/netdata/ 6 https://centrality.mimuw.edu.pl/editor/ 25 SushiThese are two elections based on two surveys on the favorite kinds of sushi Kamishima [2003]. In one of them the participants had to indicate most preferred 10 types from the total of 100 types and provide their strict ranking. We have converted this data into an approval election by setting an approval for each of the selected 10 types. In the other survey the participants were allowed to provide a weak ordering of the 10 chosen types (with ties). Here, we included only the kinds that were tied at the first place. Pol.isThe data comes from online deliberation platform Pol.is, in which the users can express their support, neutrality, or disapprovement over the comments posted by other users. The data is incomplete, as not every user has to react to every comment. We have transformed all datasets of this type available at Preflib into approval elections by recognizing the support as an approval. PabulibFinally, we include a sample of participatory budgeting elections from Pabulib Faliszewski et al. [2023a]. In each such election, citizens of a given municipality or district might express their preferences over the projects that can be implemented in their neighborhood. Most of the elections that we included are natively approval elections (the only exception is Toulouse 2019 election in which the voters could distribute 7 points between the projects—we treated each positive amount of points as an approval). However, the projects are associated with costs that could impact the voters decisions, and which are ignored for our purposes. We include five elections from each of the following five cities: Amsterdam, Budapest, Łód ́z, Toulouse, Warsaw. The elections included are the ones that have the highest “quality” score on the Pabulib webpage. Apart from that, we included all elections from the districts of Warsaw in 2026. D.3Index Values The following is the table with values of our indices for all elections in our main dataset. For each election we present an icon representing its approval matrix, akin to those presented in Table 1. If the election had more than 500 candidates (or voters), we have sampled 500 candidates (or voters) for the purpose of drawing the icon. For most of the elections we ordered the rows (and columns) in the icon based on the Jaccard distances between the votes (and candidates). Based on the distances we computed the tree of hierarchical clustering and ordered the rows (and columns) according to the ordering of the leaves in the the tree Eisen et al. [1998]. ElectionIcon |C| |V|satrav-agrcntr-agrpair-agrpcc-agrpcc + -agr jacc-agrcntr-divpcc-divout-divcntr-polpcc-polpair-pol 1 /3-ID6060 0.331.001.001.001.001.001.000.000.000.000.000.000.00 2-Party6060 0.500.000.000.000.000.500.500.200.230.101.001.001.00 N(2-Party, 0.6)6060 0.490.080.050.010.010.100.330.660.820.290.320.180.24 3-Party6060 0.330.330.000.000.000.330.330.330.300.130.330.500.63 4-Party6060 0.250.500.000.000.000.250.250.450.400.150.250.330.43 2-Party( 1 /3)6060 0.560.330.240.100.110.560.560.150.180.090.760.890.99 Cyclic6060 0.500.000.000.000.000.250.390.460.570.220.500.330.58 Diagonal6060 0.020.970.000.000.000.020.020.970.970.620.020.020.01 Triangle6060 0.510.500.490.330.500.500.510.400.320.180.010.140.47 N(Tri,0.6)6060 0.510.210.190.060.150.180.340.900.770.270.000.050.38 1 /2-ID/IC6060 0.500.510.510.270.270.300.510.400.500.190.090.250.45 26 ElectionIcon |C| |V|satrav-agrcntr-agrpair-agrpcc-agrpcc + -agr jacc-agrcntr-divpcc-divout-divcntr-polpcc-polpair-pol 1 /2-IC6060 0.500.120.110.020.020.070.340.800.900.290.080.040.18 1 /4-IC6060 0.250.510.000.020.020.070.150.980.910.300.000.040.15 Lin-IC 6060 0.500.080.070.010.050.100.300.960.880.270.000.040.37 IC (p= 0.1), 1 100 1000 0.100.800.000.000.000.040.051.000.980.390.000.010.08 IC (p= 0.1), 2 100 1000 0.100.800.000.000.000.040.051.000.980.390.000.010.08 IC (p= 0.3), 1 100 1000 0.300.410.000.000.000.040.181.000.980.330.000.010.10 IC (p= 0.3), 2 100 1000 0.300.400.000.000.000.040.181.000.980.330.000.010.10 IC (p= 0.5), 1 100 1000 0.500.030.020.000.000.040.340.920.980.320.050.010.10 IC (p= 0.5), 2 100 1000 0.500.030.020.000.000.040.340.920.980.320.050.010.10 IC (p= 0.7), 1 100 1000 0.700.390.000.000.000.040.541.000.980.330.000.010.10 IC (p= 0.7), 2100 1000 0.700.400.000.000.000.040.541.000.980.330.000.010.10 IC (p= 0.9), 1100 1000 0.900.800.000.000.000.040.821.000.980.390.000.010.08 IC (p= 0.9), 2100 1000 0.900.800.000.000.000.040.821.000.980.390.000.010.08 Lin-IC, 1100 1000 0.500.020.020.000.040.080.290.990.940.280.000.010.35 Lin-IC, 2100 1000 0.500.020.020.000.040.080.290.990.940.280.000.010.35 Lin-IC, 3100 1000 0.500.020.020.000.030.070.290.990.950.280.000.010.35 Lin-IC, 4100 1000 0.500.020.020.000.040.080.290.990.940.280.000.010.34 Lin-IC, 5100 1000 0.500.020.020.000.040.080.290.990.950.280.000.010.35 Resampling (φ= 0.0045) 100 1000 0.501.001.000.990.990.990.990.000.010.010.000.000.01 Resampling (φ= 0.0095) 100 1000 0.931.000.980.980.980.981.000.020.010.010.000.000.01 Resampling (φ= 0.0155) 100 1000 0.120.990.970.970.970.970.940.030.030.020.000.000.02 Resampling (φ= 0.0761) 100 1000 0.550.930.920.860.860.860.880.080.140.080.000.000.05 Resampling (φ= 0.0954) 100 1000 0.040.990.810.810.820.820.720.160.140.100.000.030.02 Resampling (φ= 0.1139) 100 1000 0.300.900.840.790.790.790.740.160.210.110.000.000.06 Resampling (φ= 0.1307) 100 1000 0.410.870.840.750.760.760.750.160.240.130.000.000.06 Resampling (φ= 0.1444) 100 1000 0.190.910.770.730.740.740.650.230.250.140.000.010.05 27 ElectionIcon |C| |V|satrav-agrcntr-agrpair-agrpcc-agrpcc + -agr jacc-agrcntr-divpcc-divout-divcntr-polpcc-polpair-pol Resampling (φ= 0.1829) 100 1000 0.640.830.770.670.670.670.790.230.320.170.000.000.07 Resampling (φ= 0.2044) 100 1000 0.340.820.730.630.640.640.610.270.350.180.000.000.07 Resampling (φ= 0.2051) 100 1000 0.480.790.790.630.630.630.680.210.360.180.000.000.08 Resampling (φ= 0.2381) 100 1000 0.720.810.650.580.580.580.790.350.410.200.000.010.07 Resampling (φ= 0.2605) 100 1000 0.670.770.650.550.550.550.740.350.440.210.000.010.08 Resampling (φ= 0.3078) 100 1000 0.410.700.640.480.480.480.530.360.510.230.000.010.09 Resampling (φ= 0.3157) 100 1000 0.220.780.510.470.470.470.420.490.510.240.000.010.07 Resampling (φ= 0.3281) 100 1000 0.990.990.420.580.940.940.990.430.020.120.000.050.01 Resampling (φ= 0.3476) 100 1000 0.190.780.440.420.430.430.370.550.550.250.010.010.07 Resampling (φ= 0.3492) 100 1000 0.580.660.600.430.430.430.610.400.560.240.000.010.09 Resampling (φ= 0.3925) 100 1000 0.590.620.540.370.370.370.590.460.620.250.000.010.10 Resampling (φ= 0.4575) 100 1000 0.320.600.380.290.300.300.350.620.690.270.000.010.09 Resampling (φ= 0.4626) 100 1000 0.570.540.470.290.290.290.530.530.700.270.000.010.10 Resampling (φ= 0.4887) 100 1000 0.860.770.150.270.270.270.820.800.700.300.030.020.07 Resampling (φ= 0.4889) 100 1000 0.470.510.480.260.260.260.440.510.720.270.000.010.10 Resampling (φ= 0.5196) 100 1000 0.870.770.090.230.230.240.820.850.740.310.070.020.07 Resampling (φ= 0.5344) 100 1000 0.440.480.410.220.220.220.390.590.760.280.000.010.10 Resampling (φ= 0.5850) 100 1000 0.030.940.000.170.230.240.120.890.660.430.170.070.04 Resampling (φ= 0.6007) 100 1000 0.600.420.290.160.160.170.490.710.820.290.000.010.10 Resampling (φ= 0.6028) 100 1000 0.580.410.300.160.160.160.480.700.830.290.000.010.10 Resampling (φ= 0.6386) 100 1000 0.660.430.150.130.130.140.550.830.850.300.000.010.10 Resampling (φ= 0.6599) 100 1000 0.370.380.170.120.120.120.290.820.870.300.000.010.10 Resampling (φ= 0.6785) 100 1000 0.930.860.000.110.110.130.880.940.840.400.050.030.06 Resampling (φ= 0.6964) 100 1000 0.970.930.000.110.150.160.940.940.760.450.010.060.04 Resampling (φ= 0.7170) 100 1000 0.200.600.000.080.080.090.150.980.900.330.010.010.09 Resampling (φ= 0.7187) 100 1000 0.890.780.000.080.080.100.820.970.890.360.000.020.08 28 ElectionIcon |C| |V|satrav-agrcntr-agrpair-agrpcc-agrpcc + -agr jacc-agrcntr-divpcc-divout-divcntr-polpcc-polpair-pol Resampling (φ= 0.7261) 100 1000 0.900.800.000.080.080.100.830.980.890.370.010.020.07 Resampling (φ= 0.7706) 100 1000 0.920.840.000.050.050.080.860.970.910.400.020.020.07 Resampling (φ= 0.7714) 100 1000 0.540.230.160.050.050.070.400.830.930.310.000.010.10 Resampling (φ= 0.7846) 100 1000 0.860.720.000.050.050.070.770.990.920.360.000.020.08 Resampling (φ= 0.7869) 100 1000 0.510.210.190.050.050.070.370.800.940.310.000.010.10 Resampling (φ= 0.8090) 100 1000 0.280.430.000.040.040.060.180.990.940.320.010.010.10 Resampling (φ= 0.8420) 100 1000 0.070.870.000.020.020.060.050.990.930.430.010.030.06 Resampling (φ= 0.8536) 100 1000 0.850.700.000.020.020.050.741.000.950.360.000.010.09 Resampling (φ= 0.8616) 100 1000 0.130.730.000.020.020.050.081.000.960.360.000.010.08 Resampling (φ= 0.8667) 100 1000 0.190.630.000.020.020.050.111.000.960.340.000.010.09 Resampling (φ= 0.9096) 100 1000 0.160.680.000.010.010.050.091.000.970.360.000.010.09 Resampling (φ= 0.9149) 100 1000 0.520.090.060.010.010.050.350.910.970.320.020.010.10 Resampling (φ= 0.9387) 100 1000 0.540.090.010.000.000.040.370.950.980.320.020.010.10 Resampling (φ= 0.9467) 100 1000 0.260.470.000.000.000.040.151.000.980.330.000.010.10 Resampling (φ= 0.9676) 100 1000 0.560.110.000.000.000.040.390.970.980.320.010.010.10 Resampling (φ= 0.9960) 100 1000 1.001.000.000.000.980.981.001.000.020.630.000.000.01 Noisy AN (φ= 0.0292) 100100 0.500.020.020.000.000.470.480.220.260.110.950.940.94 Noisy AN (φ= 0.0952) 100100 0.500.030.020.000.000.410.440.270.380.160.880.830.83 Noisy AN (φ= 0.0988) 100100 0.510.040.030.000.000.410.440.280.390.160.870.810.81 Noisy AN (φ= 0.1458) 100100 0.500.040.040.000.000.370.420.300.450.180.820.740.74 Noisy AN (φ= 0.1735) 100100 0.500.040.040.000.000.340.410.330.460.190.780.690.69 Noisy AN (φ= 0.2060) 100100 0.500.040.040.000.000.320.400.360.490.200.750.630.64 Noisy AN (φ= 0.2238) 100100 0.500.050.050.000.000.310.400.360.540.200.740.620.63 Noisy AN (φ= 0.2538) 100100 0.500.050.040.000.000.290.380.390.550.210.700.570.57 Noisy AN (φ= 0.3319) 100100 0.510.060.050.010.010.230.370.460.640.220.610.450.46 Noisy AN (φ= 0.3395) 100100 0.500.060.050.010.010.220.370.470.650.220.600.440.45 29 ElectionIcon |C| |V|satrav-agrcntr-agrpair-agrpcc-agrpcc + -agr jacc-agrcntr-divpcc-divout-divcntr-polpcc-polpair-pol Noisy AN (φ= 0.4155) 100100 0.500.070.070.010.010.180.360.520.710.230.510.340.36 Noisy AN (φ= 0.4475) 100100 0.500.060.060.010.010.160.350.550.760.230.490.320.34 Noisy AN (φ= 0.5728) 100100 0.510.070.060.010.010.100.350.650.840.240.360.190.23 Noisy AN (φ= 0.5966) 100100 0.500.070.060.010.010.090.340.670.840.240.330.170.21 Noisy AN (φ= 0.6129) 100100 0.500.080.070.010.010.090.340.660.860.240.320.170.21 Noisy AN (φ= 0.6266) 100100 0.500.070.060.010.010.080.350.690.850.250.300.150.20 Noisy AN (φ= 0.6957) 100100 0.500.080.080.010.010.070.350.770.890.240.210.100.17 Noisy AN (φ= 0.7028) 100100 0.500.070.070.010.010.060.340.740.900.250.200.100.17 Noisy AN (φ= 0.7337) 100100 0.490.070.060.010.010.060.330.820.910.250.180.080.16 Noisy AN (φ= 0.7346) 100100 0.490.080.060.010.010.060.330.780.910.240.180.080.16 Noisy AN (φ= 0.7541) 100100 0.500.090.090.010.010.060.340.800.910.250.150.070.15 Noisy AN (φ= 0.8031) 100100 0.500.080.080.010.010.050.340.830.930.250.100.050.15 Noisy AN (φ= 0.8212) 100100 0.500.070.070.010.010.050.340.840.940.250.040.040.14 Noisy AN (φ= 0.9024) 100100 0.500.070.070.010.010.050.340.860.940.250.060.030.14 Noisy AN (φ= 0.9666) 100100 0.490.070.060.010.010.050.330.860.940.250.060.030.14 2-Party (0.09, 0.04)100100 0.880.920.670.640.850.920.920.070.030.030.330.150.53 2-Party (0.11, 0.61)100100 0.410.220.060.020.050.520.520.190.190.070.940.951.00 2-Party (0.13, 0.11)100100 0.790.780.480.410.610.800.800.100.080.050.520.390.79 2-Party (0.14, 0.61)100100 0.420.220.070.020.050.520.520.190.190.080.930.951.00 2-Party (0.16, 0.82) 100100 0.280.640.360.270.410.700.700.130.120.060.640.590.91 2-Party (0.27, 0.58) 100100 0.460.160.090.020.030.510.510.180.190.070.910.971.00 2-Party (0.28, 0.36) 100100 0.560.280.180.060.080.540.540.160.180.070.820.921.00 2-Party (0.28, 0.45) 100100 0.520.100.060.010.010.500.510.190.200.080.940.991.00 2-Party (0.32, 0.46) 100100 0.510.080.050.010.010.500.500.190.200.080.950.991.00 2-Party (0.43, 0.47) 100100 0.500.060.050.000.000.500.500.190.200.080.951.001.00 2-Party (0.46, 0.92)100100 0.470.840.830.700.710.850.850.030.060.030.170.290.71 30 ElectionIcon |C| |V|satrav-agrcntr-agrpair-agrpcc-agrpcc + -agr jacc-agrcntr-divpcc-divout-divcntr-polpcc-polpair-pol 2-Party (0.47, 0.34)100100 0.510.320.310.100.100.550.550.140.180.070.690.900.99 2-Party (0.52, 0.68)100100 0.510.360.350.130.130.560.560.130.170.070.650.870.99 2-Party (0.56, 0.98) 100100 0.560.960.950.920.920.960.960.010.020.010.050.080.39 2-Party (0.57, 0.16) 100100 0.460.680.650.460.460.730.730.070.110.050.350.540.89 2-Party (0.58, 0.63) 100100 0.520.260.230.070.070.530.530.150.190.070.770.931.00 2-Party (0.62, 0.23) 100100 0.440.540.470.280.290.650.650.110.140.060.530.710.96 2-Party (0.74, 0.28) 100100 0.390.440.290.160.190.600.600.140.160.070.710.810.98 2-Party (0.79, 0.52) 100100 0.510.040.020.000.000.500.500.200.200.080.981.001.00 2-Party (0.85, 0.23) 100100 0.310.540.260.170.290.650.650.150.140.070.740.710.96 2-Party (0.90, 0.47) 100100 0.480.060.010.000.000.500.500.200.200.080.991.001.00 2-Party (0.91, 0.31)100100 0.340.380.100.050.140.570.570.180.170.080.900.860.99 2-Party (0.92, 0.75)100100 0.710.500.140.090.250.620.620.170.150.070.860.750.97 2-Party (0.92, 0.19)100100 0.240.620.210.160.380.690.690.160.120.060.790.620.92 2-Party (0.95, 0.39)100100 0.400.220.030.010.050.520.520.190.190.080.970.951.00 Party-List 1100100 0.330.340.000.120.100.380.380.260.240.090.760.610.65 Party-List 2100100 0.360.500.300.220.200.480.480.190.210.080.420.550.69 Party-List 3100100 0.270.670.380.360.330.490.490.210.210.080.310.420.57 Party-List 4100100 0.570.360.240.130.190.500.500.190.190.080.580.670.91 Party-List 5100100 0.120.770.000.260.260.340.340.390.250.090.560.400.27 Party-List 6 100100 0.310.440.110.140.170.440.440.220.210.090.180.590.66 Party-List 7 100100 0.390.220.000.000.030.340.340.290.260.090.530.670.88 Party-List 8 100100 0.140.720.000.070.080.210.210.580.440.130.180.290.25 Party-List 9 100100 0.200.710.290.290.360.500.500.250.220.080.380.300.51 Party-List 10 100100 0.220.560.000.010.060.250.250.480.380.120.170.350.55 Party-List 11 100100 0.370.270.020.090.080.410.410.240.240.090.770.640.72 Party-List 12100100 0.310.380.000.050.080.340.340.320.280.100.450.480.70 31 ElectionIcon |C| |V|satrav-agrcntr-agrpair-agrpcc-agrpcc + -agr jacc-agrcntr-divpcc-divout-divcntr-polpcc-polpair-pol Party-List 13100100 0.170.660.000.160.160.300.300.480.320.110.040.400.34 Party-List 14100100 0.240.610.190.200.360.550.550.200.150.080.640.520.65 Party-List 15 100100 0.220.580.030.120.160.350.350.340.310.100.490.300.61 Party-List 16 100100 0.190.640.060.090.250.430.430.320.230.090.290.330.69 Party-List 17 100100 0.350.300.000.020.050.340.340.320.260.100.410.600.73 Party-List 18 100100 0.300.400.000.030.080.370.370.310.260.100.470.530.69 Party-List 19 100100 0.280.440.000.030.020.280.280.390.340.110.360.470.53 Party-List 20 100100 0.270.460.000.030.140.370.370.320.250.100.400.490.65 Party-List 21 100100 0.140.710.000.110.170.290.290.550.370.120.060.350.36 Party-List 22 100100 0.220.560.000.110.140.330.330.330.290.100.520.450.55 Party-List 23100100 0.290.420.010.050.190.410.410.290.210.100.550.570.74 Party-List 24100100 0.320.360.000.130.110.360.360.300.260.090.600.570.62 Party-List 25100100 0.270.490.060.080.140.400.400.280.250.090.500.470.75 ID Mixture (k= 2, p= 0.1) 100 1000 0.120.990.940.940.920.920.920.010.020.010.060.080.08 ID Mixture (k= 2, p= 0.3) 100 1000 0.320.580.350.440.440.510.570.130.110.060.650.560.50 ID Mixture (k= 2, p= 0.5) 100 1000 0.500.660.660.570.570.570.700.070.090.050.340.430.47 ID Mixture (k= 2, p= 0.7) 100 1000 0.740.830.660.640.650.660.850.070.070.040.340.350.38 ID Mixture (k= 2, p= 0.9) 100 1000 0.900.920.560.610.690.690.930.090.060.040.440.310.20 ID Mixture (k= 3, p= 0.1) 100 1000 0.100.830.140.460.460.470.490.190.120.060.780.460.18 ID Mixture (k= 3, p= 0.3) 100 1000 0.320.550.300.370.370.390.490.190.170.080.450.420.43 ID Mixture (k= 3, p= 0.5) 100 1000 0.470.780.760.630.630.640.750.060.090.050.180.280.50 ID Mixture (k= 3, p= 0.7) 100 1000 0.710.800.660.620.610.630.830.080.090.050.290.330.41 ID Mixture (k= 3, p= 0.9) 100 1000 0.880.860.450.600.590.590.910.130.100.050.090.330.17 ID Mixture (k= 4, p= 0.1) 100 1000 0.120.770.060.340.320.380.390.270.200.080.560.420.23 ID Mixture (k= 4, p= 0.3) 100 1000 0.320.610.400.410.410.450.530.150.150.070.460.420.46 ID Mixture (k= 4, p= 0.5) 100 1000 0.550.370.300.220.220.260.520.260.300.110.310.300.45 32 ElectionIcon |C| |V|satrav-agrcntr-agrpair-agrpcc-agrpcc + -agr jacc-agrcntr-divpcc-divout-divcntr-polpcc-polpair-pol ID Mixture (k= 4, p= 0.7) 100 1000 0.680.550.300.330.330.330.670.260.230.090.280.320.38 ID Mixture (k= 4, p= 0.9) 100 1000 0.930.860.020.290.300.330.900.320.220.090.470.380.14 ID Mixture (k= 5, p= 0.1) 100 1000 0.100.800.020.220.210.260.270.430.320.110.260.290.16 ID Mixture (k= 5, p= 0.3) 100 1000 0.320.720.560.510.510.510.590.130.140.070.250.320.42 ID Mixture (k= 5, p= 0.5) 100 1000 0.490.380.360.220.220.290.490.230.290.110.320.370.48 ID Mixture (k= 5, p= 0.7) 100 1000 0.680.520.250.280.280.310.650.270.270.100.130.320.41 ID Mixture (k= 5, p= 0.9) 100 1000 0.910.840.100.370.390.400.900.390.190.090.520.370.16 IAM Mixture (k= 2), 1 100 1000 0.530.480.450.300.310.310.510.540.680.260.000.010.11 IAM Mixture (k= 2), 2 100 1000 0.490.460.450.260.260.260.460.510.650.270.060.100.16 IAM Mixture (k= 2), 3 100 1000 0.500.340.340.180.180.200.430.540.690.280.150.150.19 IAM Mixture (k= 2), 4 100 1000 0.500.430.430.260.260.270.460.520.650.270.020.100.17 IAM Mixture (k= 2), 5 100 1000 0.500.470.470.290.290.300.480.500.640.260.020.080.18 IAM Mixture (k= 3), 1 100 1000 0.520.360.340.180.180.200.440.560.680.280.010.130.21 IAM Mixture (k= 3), 2 100 1000 0.510.400.380.220.220.230.450.570.690.270.060.080.18 IAM Mixture (k= 3), 3 100 1000 0.440.290.200.110.110.140.340.640.740.290.120.130.18 IAM Mixture (k= 3), 4 100 1000 0.500.320.320.140.150.170.400.600.710.280.080.130.20 IAM Mixture (k= 3), 5 100 1000 0.480.280.260.120.120.150.380.590.720.290.180.150.20 IAM Mixture (k= 4), 1 100 1000 0.490.320.310.140.140.160.400.590.700.280.090.120.19 IAM Mixture (k= 4), 2 100 1000 0.550.420.350.230.240.250.490.620.700.270.020.060.17 IAM Mixture (k= 4), 3 100 1000 0.480.330.300.150.150.170.390.620.730.290.070.110.19 IAM Mixture (k= 4), 4 100 1000 0.510.330.310.160.160.190.430.570.680.280.140.150.23 IAM Mixture (k= 4), 5 100 1000 0.520.400.370.210.210.220.460.570.690.280.010.090.18 IAM Mixture (k= 5), 1 100 1000 0.490.310.300.140.140.160.400.580.720.290.140.140.19 IAM Mixture (k= 5), 2 100 1000 0.470.260.220.100.110.130.360.650.750.290.110.080.18 IAM Mixture (k= 5), 3 100 1000 0.480.230.200.080.080.110.350.680.790.300.120.110.17 IAM Mixture (k= 5), 4 100 1000 0.520.280.250.110.110.140.410.650.750.290.070.100.18 33 ElectionIcon |C| |V|satrav-agrcntr-agrpair-agrpcc-agrpcc + -agr jacc-agrcntr-divpcc-divout-divcntr-polpcc-polpair-pol IAM Mixture (k= 5), 5 100 1000 0.510.260.240.100.100.130.390.650.760.290.100.110.18 2D-Euclidean (r = 0.117), 1 100 1000 0.040.920.000.000.050.090.030.970.880.370.020.040.06 2D-Euclidean (r = 0.117), 2 100 1000 0.040.920.000.000.020.050.030.960.910.370.030.040.05 2D-Euclidean (r = 0.117), 3 100 1000 0.040.920.000.000.050.080.030.960.870.360.030.050.05 2D-Euclidean (r = 0.117), 4 100 1000 0.040.930.000.000.020.060.030.960.900.370.010.040.05 2D-Euclidean (r = 0.117), 5 100 1000 0.040.920.000.000.060.090.030.970.880.370.020.030.06 2D-Euclidean (r = 0.117), 6 100 1000 0.040.920.000.000.030.070.030.960.900.370.020.040.05 2D-Euclidean (r = 0.117), 7 100 1000 0.040.920.000.000.040.080.030.970.890.370.010.040.06 2D-Euclidean (r = 0.117), 8 100 1000 0.040.920.000.000.050.090.030.960.870.370.030.050.06 2D-Euclidean (r = 0.117), 9 100 1000 0.040.920.000.000.050.090.030.960.870.370.030.040.06 2D-Euclidean (r = 0.117), 10 100 1000 0.040.920.000.000.040.070.030.960.890.370.010.040.06 2D-Euclidean (r = 0.167), 1 100 1000 0.080.850.000.000.010.070.050.930.860.300.020.070.10 2D-Euclidean (r = 0.167), 2 100 1000 0.080.850.000.000.000.070.050.940.870.310.040.070.10 2D-Euclidean (r = 0.167), 3 100 1000 0.080.840.000.000.010.080.050.940.860.310.040.070.10 2D-Euclidean (r = 0.167), 4 100 1000 0.080.840.000.000.030.090.050.940.840.310.040.070.10 2D-Euclidean (r = 0.167), 5 100 1000 0.070.850.000.010.000.070.050.940.870.310.050.070.09 2D-Euclidean (r = 0.167), 6 100 1000 0.080.850.000.000.000.070.050.940.870.310.030.070.09 2D-Euclidean (r = 0.167), 7 100 1000 0.080.850.000.000.000.070.050.940.870.310.030.070.08 2D-Euclidean (r = 0.167), 8 100 1000 0.080.850.000.000.010.070.050.940.860.300.030.070.11 2D-Euclidean (r = 0.167), 9 100 1000 0.080.850.000.000.000.070.050.930.860.310.030.070.09 2D-Euclidean (r = 0.167), 10 100 1000 0.080.850.000.000.020.090.050.920.840.310.030.070.10 2D-Euclidean (Uniform r ), 1 100 1000 0.190.610.000.020.190.280.100.870.660.250.070.100.36 2D-Euclidean (Uniform r ), 2 100 1000 0.180.630.000.020.230.310.100.890.620.250.060.100.35 2D-Euclidean (Uniform r ), 3 100 1000 0.180.640.000.010.220.300.100.880.620.250.060.100.36 2D-Euclidean (Uniform r ), 4 100 1000 0.190.620.000.010.210.300.100.870.620.250.100.120.36 2D-Euclidean (Uniform r ), 5 100 1000 0.190.630.000.020.210.290.100.880.630.250.060.100.34 34 ElectionIcon |C| |V|satrav-agrcntr-agrpair-agrpcc-agrpcc + -agr jacc-agrcntr-divpcc-divout-divcntr-polpcc-polpair-pol 2D-Euclidean (Uniform r ), 6 100 1000 0.180.630.000.020.210.290.100.880.640.250.080.100.36 2D-Euclidean (Uniform r ), 7 100 1000 0.190.620.000.020.240.320.100.870.610.250.070.100.37 2D-Euclidean (Uniform r ), 8 100 1000 0.180.640.000.020.220.300.100.890.630.250.070.090.35 2D-Euclidean (Uniform r ), 9 100 1000 0.190.630.000.010.210.290.100.880.630.250.070.100.35 2D-Euclidean (Uniform r ), 10 100 1000 0.190.610.000.010.190.270.100.860.650.250.070.100.37 2D-Euclidean (Top10), 1 100 1000 0.100.800.000.010.010.090.080.970.810.270.020.100.09 2D-Euclidean (Top10), 2 100 1000 0.100.800.000.010.010.090.080.950.810.270.000.100.09 2D-Euclidean (Top10), 3 100 1000 0.100.800.000.010.010.090.080.960.810.280.000.090.09 2D-Euclidean (Top10), 4 100 1000 0.100.800.000.010.010.090.080.940.790.270.060.110.09 2D-Euclidean (Top10), 5 100 1000 0.100.800.000.010.010.090.080.970.820.270.000.090.09 2D-Euclidean (Top10), 6 100 1000 0.100.800.000.010.010.090.080.960.810.270.000.100.09 2D-Euclidean (Top10), 7 100 1000 0.100.800.000.010.010.090.080.960.810.270.000.100.09 2D-Euclidean (Top10), 8 100 1000 0.100.800.000.010.010.090.070.980.830.280.000.090.09 2D-Euclidean (Top10), 9 100 1000 0.100.800.000.010.010.090.080.970.820.270.000.090.09 2D-Euclidean (Top10), 10 100 1000 0.100.800.000.010.010.090.080.970.820.270.000.100.09 2D-Euclidean (Uniform Top), 1 100 1000 0.490.190.170.050.070.170.320.840.730.230.000.160.42 2D-Euclidean (Uniform Top), 2 100 1000 0.510.200.180.060.080.170.340.840.730.240.000.150.42 2D-Euclidean (Uniform Top), 3 100 1000 0.480.190.150.050.060.160.300.840.730.240.000.160.42 2D-Euclidean (Uniform Top), 4 100 1000 0.490.210.200.060.080.170.320.830.720.240.000.150.41 2D-Euclidean (Uniform Top), 5 100 1000 0.510.210.190.060.080.180.340.820.720.240.000.150.41 2D-Euclidean (Uniform Top), 6 100 1000 0.500.200.200.060.090.170.330.830.720.240.000.160.41 2D-Euclidean (Uniform Top), 7 100 1000 0.490.200.180.050.070.170.320.830.730.240.000.160.41 2D-Euclidean (Uniform Top), 8 100 1000 0.510.220.200.070.100.180.340.830.720.230.000.150.42 2D-Euclidean (Uniform Top), 9 100 1000 0.510.230.220.070.100.180.340.820.710.230.000.150.41 2D-Euclidean (Uniform Top), 10 100 1000 0.490.220.210.060.080.180.320.820.720.240.000.160.41 Chopin 18417 0.370.410.220.190.210.220.330.670.640.170.080.080.25 35 ElectionIcon |C| |V|satrav-agrcntr-agrpair-agrpcc-agrpcc + -agr jacc-agrcntr-divpcc-divout-divcntr-polpcc-polpair-pol Chopin 24017 0.470.280.240.110.120.160.370.700.690.260.000.100.29 Chopin 32017 0.530.360.320.170.180.220.440.550.610.380.000.130.30 Chopin Final 1117 0.510.250.240.110.360.410.380.710.470.500.000.100.38 French 2007 All 122836 0.180.630.000.150.260.340.210.730.500.550.320.200.27 French 2012 Louvigny 10930 0.250.530.070.160.220.310.250.670.540.580.080.210.33 French 2012 Saint-Etienne 10387 0.260.500.050.140.200.300.240.670.550.610.260.230.34 French 2012 Strasbourg 101023 0.270.510.070.160.280.370.250.720.510.570.070.190.34 French 2017 All 11 20076 0.240.620.230.280.340.400.320.570.450.460.090.170.30 French 2022 All 121379 0.260.570.180.230.300.370.290.660.510.530.100.150.31 Jester 0.25 100 73421 0.250.540.080.170.170.180.240.790.750.290.120.070.14 Jester 0.5100 73421 0.490.510.510.310.310.320.490.420.580.240.050.090.27 Jester 0.75100 73421 0.740.770.560.530.530.550.800.360.350.150.120.150.32 MovieLens 4.050423 0.570.430.340.220.270.270.490.650.690.340.000.040.25 MovieLens 4.550423 0.320.440.140.190.210.230.270.780.740.360.080.030.22 MovieLens 5.050423 0.190.620.000.130.240.260.150.900.700.370.090.040.28 Dolphins Network 6262 0.100.800.000.030.020.080.070.930.820.320.020.110.13 Football Network 115115 0.100.800.000.000.000.070.070.970.860.260.000.080.08 Karate Club Network 3434 0.160.670.010.090.110.180.160.800.640.360.180.200.28 Les Miserables Network 7777 0.100.800.000.070.060.110.090.850.730.260.070.130.19 Polblogs Network 1490 1490 0.010.990.000.020.800.800.300.980.200.320.010.000.04 September11 Network 1919 0.240.530.000.050.040.170.190.740.610.410.130.220.30 Strike Network 2424 0.170.650.000.020.020.140.140.810.660.380.090.190.22 Polkadot 18 929 20621 0.010.980.000.060.450.460.090.960.460.260.020.060.02 Polkadot 28 931 22252 0.010.980.000.050.540.540.110.970.400.270.050.040.02 Polkadot 137 993 19442 0.010.980.000.080.460.470.090.970.470.240.000.050.02 Polkadot 1581002 20603 0.010.980.000.050.490.500.100.980.450.260.000.040.02 36 ElectionIcon |C| |V|satrav-agrcntr-agrpair-agrpcc-agrpcc + -agr jacc-agrcntr-divpcc-divout-divcntr-polpcc-polpair-pol Polkadot 1931051 23318 0.010.990.000.050.540.550.120.970.380.240.040.050.02 Polkadot 2521013 30608 0.010.980.000.040.520.520.100.970.430.240.010.040.02 Polkadot 302 995 36726 0.010.980.000.050.580.590.130.980.380.260.000.030.02 Polkadot 353 983 41974 0.010.980.000.050.600.600.140.970.320.280.010.080.02 Polkadot 401 924 46340 0.010.980.000.030.620.630.150.980.340.260.000.020.02 Polkadot 496 973 44918 0.010.990.000.040.600.610.140.950.350.270.000.030.02 Survey Colours 8140 0.750.490.000.140.520.580.640.780.350.600.240.090.35 Survey Fruits 8140 0.820.640.000.040.670.720.700.860.250.680.080.060.44 Survey Pets 8140 0.590.240.080.090.420.500.440.750.420.610.070.140.44 Survey Pizzas 8140 0.660.330.020.100.290.380.530.760.540.670.250.130.37 Survey Subjects 8112 0.670.420.130.130.250.350.550.670.540.640.170.150.37 Bidding AAMAS 1 613201 0.010.980.000.010.490.490.090.990.480.360.000.010.03 Bidding AAMAS 2 442161 0.010.980.000.010.450.460.080.990.510.350.010.020.03 Bidding AAMAS 3 526667 0.020.960.000.010.250.260.030.990.710.360.010.030.05 Bidding CS conf 1 5431 0.100.810.000.090.210.240.110.920.660.310.070.090.14 Bidding CS conf 2 5224 0.160.670.000.070.070.120.140.910.760.290.040.110.17 Bidding CS conf 3 176146 0.030.940.000.020.180.200.030.990.770.260.010.030.09 Campsongs 17839 0.290.450.040.100.110.130.220.890.790.230.030.060.24 Campsongs 2839 0.310.390.010.060.300.400.210.810.500.650.170.150.39 Campsongs 3 8256 0.280.470.050.100.110.130.210.900.810.240.030.040.25 Campsongs 4 1056 0.360.300.030.050.060.180.240.800.740.650.110.130.34 CTU 2382 0.300.400.010.080.100.170.200.850.740.440.060.130.36 Eurovision 2 1818 0.560.520.460.320.320.350.550.410.490.360.080.120.31 Eurovision 4 1920 0.530.510.480.320.320.330.530.420.510.340.060.110.26 Eurovision 9 1820 0.560.400.320.210.210.250.500.530.610.400.090.120.29 Eurovision 152122 0.480.320.290.170.170.210.410.540.640.360.150.120.28 37 ElectionIcon |C| |V|satrav-agrcntr-agrpair-agrpcc-agrpcc + -agr jacc-agrcntr-divpcc-divout-divcntr-polpcc-polpair-pol Eurovision 202425 0.420.500.400.310.310.350.450.480.530.310.060.100.30 Eurovision 262424 0.420.520.420.320.320.340.450.470.510.310.110.110.28 Eurovision 32 2439 0.420.480.370.270.270.280.420.530.570.340.110.120.25 Eurovision 36 2442 0.420.450.340.260.260.280.420.520.580.360.100.110.25 Eurovision 37 2842 0.360.500.300.260.260.270.370.610.610.320.050.090.21 Eurovision 39 1921 0.530.530.500.360.360.370.550.370.460.340.110.130.25 Eurovision 41 2542 0.400.450.310.230.230.240.380.590.660.370.090.080.23 Eurovision 44 2539 0.400.410.260.190.190.210.360.640.670.370.070.090.24 Eurovision 48 1921 0.530.310.270.130.130.200.430.540.640.380.130.160.32 Eurovision 52 1821 0.560.380.300.190.190.230.490.560.630.390.100.110.30 Eurovision 571619 0.620.440.250.220.220.260.560.540.550.400.180.140.30 Eurovision 631842 0.560.510.450.280.280.310.530.490.590.420.060.080.29 Eurovision 661842 0.560.380.300.200.200.240.490.580.640.440.110.100.27 Eurovision 682686 0.380.400.220.170.170.210.340.680.700.410.070.110.23 Eurovision 691944 0.530.420.390.220.220.250.470.520.650.430.070.080.27 Eurovision 721740 0.530.380.340.190.190.230.460.550.670.440.070.090.28 Posters A1765 0.320.380.040.080.150.220.230.820.700.520.010.100.32 Posters B1760 0.330.410.110.090.160.230.240.800.700.510.050.100.29 Sushi 10100 5000 0.100.800.000.090.090.100.101.000.880.350.000.020.05 Sushi≤ 10 100 5000 0.040.920.000.060.060.080.050.970.870.420.030.040.05 Pol.is 1 54339 0.070.850.000.080.390.410.090.970.560.400.040.030.19 Pol.is 2 896 2031 0.060.870.000.110.420.430.090.960.550.210.040.020.19 Pol.is 3 1039 1756 0.040.920.000.100.440.440.090.950.520.220.070.040.14 Pol.is 4 522 1116 0.060.870.000.130.370.380.090.920.560.200.060.030.19 Pol.is 5 2138 3142 0.030.950.000.140.500.500.110.980.460.200.020.030.09 Pol.is 6613 1503 0.040.920.000.120.400.410.090.940.550.210.070.030.12 38 ElectionIcon |C| |V|satrav-agrcntr-agrpair-agrpcc-agrpcc + -agr jacc-agrcntr-divpcc-divout-divcntr-polpcc-polpair-pol Pol.is 71452 3616 0.030.940.000.120.510.520.110.970.450.210.020.020.10 Pol.is 8371126 0.210.590.000.090.180.200.120.890.740.150.110.050.38 Pol.is 9 50204 0.260.570.190.220.280.290.260.730.620.320.040.080.25 Pol.is 10 174448 0.090.820.000.080.260.270.060.940.700.200.060.030.27 Pol.is 11 298 1487 0.010.970.000.090.590.590.130.980.370.150.010.030.05 Pol.is 12 3926 0.390.450.290.250.290.300.360.600.560.250.050.090.29 Pol.is 13 2162 6289 0.030.950.100.350.600.600.220.680.330.100.230.050.04 Pol.is 14 165381 0.130.740.000.140.250.260.110.900.680.190.090.060.24 Pol.is 15 316536 0.100.810.030.250.370.380.170.790.540.160.160.070.17 Pol.is 16 80117 0.270.610.270.360.430.440.320.600.480.200.120.070.21 Pol.is 17148334 0.170.700.130.300.390.390.240.740.530.180.130.070.16 Pol.is 1871234 0.280.530.160.270.370.380.260.660.530.230.210.080.27 Pol.is 19297404 0.130.750.060.250.340.340.180.820.590.170.150.060.16 Pol.is 20197 1921 0.080.830.000.140.280.290.080.910.660.200.090.050.18 PB Amsterdam 252 858520 0.220.570.000.070.070.100.150.950.870.340.040.040.16 PB Amsterdam 309 58 14411 0.180.640.000.080.080.110.140.940.860.430.020.050.14 PB Amsterdam 417 399562 0.320.400.060.090.090.130.230.840.830.480.050.050.20 PB Amsterdam 621 60 10081 0.290.440.040.090.100.130.200.880.840.380.080.050.21 PB Amsterdam 631 62 11519 0.240.520.010.100.090.120.180.920.850.390.070.050.17 PB Budapest 2022 Jozsefvaros 33520 0.120.770.000.110.110.160.140.920.770.510.100.090.11 PB Budapest 2023 Jozsefvaros 40918 0.120.770.050.130.120.160.140.910.800.510.020.050.09 PB Budapest 2024 Jozsefvaros 40864 0.120.760.000.080.080.130.120.930.840.530.030.060.09 PB Budapest 2024 175 28943 0.050.900.000.070.070.080.061.000.840.270.000.090.08 PB Budapest 2025 Ujbuda 402040 0.070.850.000.060.060.110.080.940.810.540.010.090.09 PB Lodz 2020 151 51472 0.030.950.000.050.040.060.040.950.870.360.000.050.03 PB Lodz 2022 160 90494 0.020.960.000.040.040.060.040.880.820.340.000.110.03 39 ElectionIcon |C| |V|satrav-agrcntr-agrpair-agrpcc-agrpcc + -agr jacc-agrcntr-divpcc-divout-divcntr-polpcc-polpair-pol PB Lodz 2022 Teofilow Wielkopolska 24 16771 0.130.880.560.560.550.590.560.370.280.220.050.180.22 PB Lodz 2023 139 70747 0.030.950.000.040.040.060.040.960.860.390.040.060.03 PB Lodz 2024 163 66045 0.020.950.000.040.040.050.040.940.870.380.000.050.03 PB Toulouse 2019 301494 0.090.810.000.060.070.130.110.900.720.560.010.160.14 PB Toulouse 2022 5 10423 0.180.640.000.090.100.210.180.790.620.640.160.200.26 PB Toulouse 2022 12 10659 0.180.740.280.320.350.410.350.550.420.470.080.180.24 PB Toulouse 2022 199 4532 0.010.970.000.020.020.030.020.970.910.400.010.050.01 PB Toulouse 2024 183 7260 0.020.970.000.010.010.030.021.000.940.400.010.020.01 PB Warszawa 2021 106 95899 0.080.840.000.110.100.120.100.950.840.360.030.040.07 PB Warszawa 2022 129 81234 0.060.870.000.080.070.090.070.970.870.340.000.040.06 PB Warszawa 2023 138 78904 0.060.880.000.090.090.100.080.960.850.320.010.040.06 PB Warszawa 2024 118 75255 0.070.850.000.100.090.110.090.970.850.340.020.040.06 PB Warszawa 2025 86 67615 0.100.800.000.080.080.100.090.970.850.380.000.040.08 PB Warszawa 2026 Bemowo 354714 0.130.740.000.090.090.140.130.920.810.550.020.070.11 PB Warszawa 2026 Bialoleka 306542 0.150.700.000.100.100.150.140.910.810.570.040.060.13 PB Warszawa 2026 Bielany 274656 0.170.660.000.070.070.130.140.900.820.610.050.070.14 PB Warszawa 2026 Mokotow 176902 0.270.480.030.080.080.160.210.830.790.680.070.080.22 PB Warszawa 2026 Ochota 233060 0.200.590.000.100.100.150.170.860.790.620.100.080.17 PB Warszawa 2026 Praga Polnoc 251746 0.190.630.010.120.120.160.180.870.780.580.110.070.15 PB Warszawa 2026 Praga Poludnie 216202 0.210.580.000.080.080.150.170.850.780.640.070.090.19 PB Warszawa 2026 Rembertow 401555 0.110.780.000.050.050.110.090.930.820.540.060.080.10 PB Warszawa 2026 Srodmiescie 102113 0.410.290.130.090.100.200.300.700.730.710.160.130.32 PB Warszawa 2026 Targowek 243793 0.190.620.000.060.060.120.140.900.830.630.040.080.16 PB Warszawa 2026 Ursus 153415 0.300.490.150.180.180.230.280.700.700.600.110.080.25 PB Warszawa 2026 Ursynow 365782 0.130.740.000.090.090.140.130.910.810.530.080.070.11 PB Warszawa 2026 Wawer 193259 0.220.560.000.110.110.170.190.880.770.640.130.080.20 40 ElectionIcon |C| |V|satrav-agrcntr-agrpair-agrpcc-agrpcc + -agr jacc-agrcntr-divpcc-divout-divcntr-polpcc-polpair-pol PB Warszawa 2026 Wesola 491256 0.100.810.000.050.050.100.080.920.840.500.030.100.08 PB Warszawa 2026 Wilanow 81367 0.510.240.230.080.090.210.380.690.740.750.090.130.36 PB Warszawa 2026 Wlochy 102102 0.440.240.130.080.080.190.330.680.750.740.210.140.33 PB Warszawa 2026 Wola 154210 0.300.460.100.140.140.210.260.750.730.640.180.090.25 PB Warszawa 2026 Zoliborz 202282 0.230.570.040.150.150.210.220.780.720.560.110.100.21 Table 4: Values of our indices for elections in our main dataset. 41