Original: Expected Frequency Matrices of Elections: Computation, Geometry, and Preference Learning
Revised: Expected Frequency Matrices of Elections: Computation, Geometry, and Preference Learning

Original: Abstract
Revised: Abstract

Original: We use the “map of elections” approach of Szufa et al.
Revised: We use the “map of elections” approach of Szufa et al.

Original: (AAMAS-2020) to analyzeseveral well-known vote distributions.
Revised: (AAMAS-2020) to analyze several well-known vote distributions.

Original: For each of them, we give an explicit formula or an efﬁcient algorithm for computing its frequency matrix, which capturesthe probability that a given candidate appears in a given position in a sampledvote.
Revised: For each of them, we give an explicit formula or an efﬁcient algorithm for computing its frequency matrix, which captures the probability that a given candidate appears in a given position in a sampled vote.

Original: We use these matrices to draw the “skeleton map” of distributions, evaluateits robustness, and analyze its properties.
Revised: We use these matrices to draw the “skeleton map” of distributions, evaluate its robustness, and analyze its properties.

Original: We further develop a general and uniﬁed framework for learning the distribution of real-world preferences using thefrequency matrices of established vote distributions
Revised: Finally, we develop a general and uniﬁed framework for learning the distribution of real-world preferences using the frequency matrices of established vote distributions.

Original:  Introduction
Revised:  Introduction

Original: Computational social choice is a research area at the intersection of social choice (the science ofcollective decision-making) and computer science, which focuses on the algorithmic analysis ofproblems related to preference aggregation and elicitation (Brandt et al., 2013).
Revised: Computational social choice is a research area at the intersection of social choice (the science of collective decision-making) and computer science, which focuses on the algorithmic analysis of problems related to preference aggregation and elicitation (Brandt et al., 2013).

Original: Many of the earlypapers in this ﬁeld were primarily theoretical, focusing on establishing the worst-case complexityof winner determination and strategic behavior under various voting rules—see, e.g., the papers of
Revised: Many of the early papers in this ﬁeld were primarily theoretical, focusing on establishing the worst-case complexity of winner determination and strategic behavior under various voting rules—see, e g., the papers of

Original: Hemaspaandra et al.
Revised: Hemaspaandra et al.

Original: (1997), Dwork et al.
Revised: (1997), Dwork et al.

Original: (2001), and Conitzer et al.
Revised: (2001), and Conitzer et al.

Original: (2007)—but more recent workoften combines theoretical investigations with empirical analysis.
Revised: (2007)—but more recent work often combines theoretical investigations with empirical analysis.

Original: For example, formal bounds on therunning time and/or approximation ratio of a winner determination algorithm can be complementedby experiments that evaluate its performance on realistic instances; see, e.g., the works of Conitzer(2006), Betzler et al.
Revised: For example, formal bounds on the running time and/or approximation ratio of a winner determination algorithm can be complemented by experiments that evaluate its performance on realistic instances; see, e.g., the works of Conitzer (2006), Betzler et al.

Original: (2014), Faliszewski et al.
Revised: (2014), Faliszewski et al.

Original: (2018) and Wang et al.
Revised: (2018) and Wang et al.

Original: However, performing high-quality experiments requires the ability to organize and understand theavailable data.
Revised: However, performing high-quality experiments requires the ability to organize and understand the available data.

Original: One way to achieve this is to form a so-called “map of elections,” recently introducedby eu and extended by Boehmer et al.
Revised: One way to achieve this is to form a so-called “map of elections,” recently introduced by Szufa et al. and extended by Boehmer et al.

Original: (2021b).
Revised: (2021b).

Original: The idea is as follows.
Revised: The idea is as follows.

Original: First, we ﬁx a distance measurebetween elections.
Revised: First, we ﬁx a distance measure between elections.

Original: Second, we sample a number of elections from various distributions and real-lifedatasets—e.g., those collected in PrefLib (Mattei & Walsh, 2013)—and measure the distance betweeneach pair.
Revised: Second, we sample a number of elections from various distributions and real-life datasets—e.g., those collected in PrefLib (Mattei & Walsh, 2013)—and measure the pairwise distances between them.

Original: Third, we embed these elections into the 2D plane, mapping each election to a point sothat the Euclidean distances between points are approximately equal to the distances between therespective elections.
Revised: Third, we embed these elections into the 2D plane, mapping each election to a point so that the Euclidean distances between points are approximately equal to the distances between the respective elections.

Original: Finally, we plot these points, usually coloring them to reﬂect some information(e.g., the distribution from which a given election was sampled); see Figure 2 on p. 7 for an exampleof such a map.
Revised: Finally, we plot these points, usually coloring them to indicate their origin (e.g., the distribution from which a given election was sampled); see Figure 2 later in the paper for an example of such a map.

Original: A location of an election on a map can say quite a lot about its properties.
Revised: A location of an election on a map provides useful information about its properties.

Original: For example, Szufa et al.
Revised: For example, Szufa et al.

Original: and Boehmer et al.
Revised: and Boehmer et al.

Original: (2021a,b) have shown that it is strongly correlated with (a) the Borda score of the winner of the election, (b) the running time of ILP solvers computing the winnersunder the Harmonic-Borda multiwinner voting rule, or (c) the robustness of Plurality and Bordawinners.
Revised: (2021a,b) have shown that it can be used to predict (a) the Borda score of the winner of the election, (b) the running time of ILP solvers computing the winners under the Harmonic-Borda multiwinner voting rule, or (c) the robustness of Plurality and Borda winners.

Original: Moreover, real-world elections of the same type (such as the ones from politics, sports, orsurveys) tend to cluster in the same areas of the map.
Revised: Moreover, real-world elections of the same type (such as the ones from politics, sports, or surveys) tend to cluster in the same areas of the map; see also the positions on the map of the datasets collected by Boehmer & Schaar (2022).

Revised: As such, the map has proven to be a useful framework to analyze the nature of elections and to visualize experimental results in a non-aggregate fashion.

Original: Unfortunately, extending the map to incorporate additional examples and distributions is a challengingtask as the visual representation becomes cluttered and, more importantly, the embedding algorithms, which map elections to points in 2D, ﬁnd it more difﬁcult to preserve pairwise distances betweenpoints as the number of points increases.
Revised: Unfortunately, extending the map to incorporate additional examples and distributions is a challenging task, as the visual representation becomes cluttered and, more importantly, the embedding algorithms, which map elections to points in 2D, ﬁnd it more difﬁcult to preserve pairwise distances between points as the number of points increases.

Original: It is therefore desirable to reduce the number of points in away that preserves the key features of the framework.
Revised: It is therefore desirable to reduce the number of points in a way that preserves the key features of the framework.

Original: We address this challenge by drawing a map of distributions rather than individual elections, whichwe call the skeleton map .
Revised: We address this challenge by drawing a map of distributions rather than individual elections, which we call the skeleton map .

Original: That is, instead of sampling 20–30 points from each distribution and placingthem all on the map, as Szufa et al.
Revised: That is, instead of sampling 20–30 points from each distribution and placing them all on the map, as Szufa et al.

Original: and Boehmer et al.
Revised: and Boehmer et al.

Original: (2021b) do (obtaining around 800points in total), we create a single point for each distribution.
Revised: (2021b) do (obtaining aroundpoints in total), we create a single point for each distribution.

Original: This approach is facilitated by thefact that prior work on the “map of elections” framework represented elections by their frequencymatrices , which capture their essential features.
Revised: This approach is facilitated by the fact that prior work on the “map of elections” framework represented elections by their frequency matrices , which capture their essential features.

Original: The starting point of our work is the observation thatthis representation extends to distributions in a natural way.
Revised: The starting point of our work is the observation that this representation extends to distributions in a natural way.

Original: Thus, if we can compute the frequencymatrix of some distribution D , then, instead of sampling elections from D and creating a point on themap for each sample, we can create a single point for D itself.
Revised: Thus, if we can compute the frequency matrix of some distribution D , then, instead of sampling elections from D and creating a point on the map for each sample, we can create a single point for D itself.

Revised: Our Contribution.

Original: It turns out that our results can also be useful from the perspective of preference learning.
Revised: We provide three sets of results. First, for a number of prominent vote distributions, we show how to compute their frequency matrices, by providing an explicit formula or an efﬁcient algorithm. Second, we draw the map of distributions (the skeleton map ) and argue for its credibility and robustness. Finally, we use our results to estimate the parameters of the distributions that are closest to the real-world elections considered by Boehmer et al.

Revised: (2021b).

Original: In thissetting we are given an election and we want to learn the parameters of some distribution, sothatthis distribution would provide votes assimilar to those from  the election as possible.
Revised: In more detail, we work in the setting of preference learning, where we are given an election and we want to learn the parameters of some distribution, so as to maximize the similarity of the votes sampled from this distribution and the input election.

Original: For example,we may be interested in ﬁtting the classic model of Mallows (1957).
Revised: For example, we may be interested in ﬁtting the classic model of Mallows (1957).

Original: This model is parameterizedby a central vote v and a dispersion parameter φ , which speciﬁes how likely it is to generate avote at some distance from the central one (alternatively, one may use the Placket-Luce model).
Revised: This model is parameterized by a central vote v and a dispersion parameter φ , which speciﬁes how likely it is to generate a vote at some distance from the central one (alternatively, one may use, e.g., the Plackett–Luce model).

Original: Previous works on preference learning typically propose algorithms to learn the parameters of onespeciﬁc (parameterized) vote distribution (see, e.g., the works of Lu & Boutilier (2014); Mandhani & Meil ˇ a (2009); Meila & Chen (2010); Vitelli et al.
Revised: Previous works on preference learning typically proposed algorithms to learn the parameters of one speciﬁc (parameterized) vote distribution (see, e.g., the works of Lu & Boutilier (2014); Mandhani & Meilˇa (2009); Meila & Chen (2010); Vitelli et al.

Original: (2017); Murphy & Martin (2003); Awasthi et al.(2014) for (mixtures of) the Mallows model and the works of Guiver & Snelson (2009); Hunter(2004); Minka (2004); Gormley & Murphy (2008) for (mixtures of) the Placket-Luce model).
Revised: Murphy & Martin (2003); Awasthi et al. (2014) for (mixtures of) the Mallows model and the works of Guiver & Snelson (2009); Hunter (2004); Minka (2004); Gormley & Murphy (2008) for (mixtures of) the Plackett–Luce model).

Original: Usingfrequency matrices, we offer a more generic approach.
Revised: Using frequency matrices, we offer a more general approach.

Original: Indeed, given an election and a parameterizedvote distribution whose frequency matrix we can compute, the task of learning the distribution’sparameters boils down to ﬁnding parameters that minimize the distance between the  matrices ofthedistribution and the election.
Revised: Indeed, given an election and a parameterized vote distribution whose frequency matrix we can compute, the task of learning the distribution’s parameters boils down to ﬁnding parameters that minimize the distance between the election and the matrices of the distribution.

Original: While this minimization process may be quite challenging, our approachoffers a uniform framework for dealing with multiple kinds of distributions at the same time.Our Contribution.
Revised: While this minimization problem may be quite challenging, our approach offers a uniform framework for dealing with multiple kinds of distributions at the same time.

Original: We provide three sets of results.

Original: First, for a number of prominent votedistributions, we show how to compute their frequency matrices, by providing an explicit formula oran efﬁcient algorithm.

Original: Second, we draw the map of distributions (the skeleton map ) and argue for itscredibility and robustness.

Original: Finally, we use our results to estimate the parameters of the distributionsthat are closest to the real-world elections considered by Boehmer et al.

Original: (2021b).

Original: We ﬁnd that for thecase of the Mallows distribution, our approach learns parameters very similar to those establishedusing maximum likelihood-based approaches.
Revised: We ﬁnd that for the case of the Mallows distribution, our approach learns parameters very similar to those established using maximum likelihood-based approaches.

Original: Omitted proofs and discussions are in the appendix.
Revised: Omitted proofs and discussions are in the appendix.

Revised: The source code used for the experiments is available in a GitHub repository 1 .

Original:  Preliminaries
Revised:  Preliminaries

Original: Given an integer t , we write [ t ] to mean the set { 1 , . . .
Revised: Given an integer t , we write [ t ] to denote the set { 1 , . . .

Original: , t } .
Revised: , t } .

Original: We interpret a vector x ∈ R m as an m × 1matrix (i.e., we use column vectors as the default).
Revised: We interpret a vector x ∈ R m as an m × 1 matrix (i.e., we use column vectors as the default).

Original: Preference Orders and Elections.
Revised: Preference Orders and Elections.

Original: Let C be a ﬁnite, nonempty set of candidates.
Revised: Let C be a ﬁnite, nonempty set of candidates.

Original: We refer tototal orders over C as preference orders (or, equivalently, votes ), and denote the set of all preferenceorders over C by L ( C ) .
Revised: We refer to total orders over C as preference orders (or, equivalently, votes ), and denote the set of all preference orders over C by L ( C ) .

Original: Given a vote v and a candidate c , by pos v ( c ) we mean the position of c in v(the top-ranked candidate has position 1 , the next one has position 2 , and so on).
Revised: Given a vote v and a candidate c , by pos v ( c ) we mean the position of c in v (the top-ranked candidate has position 1 , the next one has position 2 , and so on).

Original: If a candidate ais ranked above another candidate b in vote v , we write v : a (cid:31) b .
Revised: If a candidate a is ranked above another candidate b in vote v , we write v : a (cid:31) b .

Original: Let rev( v ) denote the reverse ofvote v .
Revised: Let rev( v ) denote the reverse of vote v .

Original: An election E =
Revised: An election E =

Original: ( C, V ) consists of a set C
Revised: ( C, V ) consists of a set C

Original: = { c 1 , . .
Revised: = { c 1 , . .

Original: , c m } of candidates and a collection
Revised: , c m } of candidates and a collection

Original: V =
Revised: V =

Original: ( v 1 , . .
Revised: ( v 1 , . .

Original: . , v n ) of votes.
Revised: . , v n ) of votes.

Original: Occasionally we refer to the elements of V as voters rather than votes.
Revised: Occasionally we refer to the elements of V as voters rather than votes.

Original: Frequency Matrices.
Revised: Frequency Matrices.

Original: Consider an election E =
Revised: Consider an election E =

Original: ( C, V ) with C
Revised: ( C, V ) with C

Original: = { c 1 , . .
Revised: = { c 1 , . .

Original: , c m } and
Revised: , c m } and

Original: V =
Revised: V =

Original: ( v 1 , . .
Revised: ( v 1 , . .

Original: . , v n ) .
Revised: . , v n ) .

Original: For each candidate c j and position i ∈ [ m ] , we deﬁne #freq E ( c j , i ) tobe the fraction of the votes from V who rank c j in position i .
Revised: For each candidate c j and position i ∈ [ m ] , we deﬁne #freq E ( c j , i ) to be the fraction of the votes from V that rank c j in position i .

Original: We deﬁne the column vector#freq E ( c j ) to be (#freq E ( c j , 1) , . . .
Revised: We deﬁne the column vector #freq E ( c j ) to be (#freq E ( c j , 1) , . . .

Original: , #freq E ( c j , m )) and matrix #freq( E ) to consist of vectors#freq E ( c 1 ) , . . .
Revised: , #freq E ( c j , m )) and matrix #freq( E ) to consist of vectors #freq E ( c 1 ) , . . .

Original: , #freq E ( c m ) .
Revised: , #freq E ( c m ) .

Original: We refer to #freq( E ) as the frequency matrix of election E .
Revised: We refer to #freq( E ) as the frequency matrix of election E .

Original: Frequency matrices are bistochastic, i.e., their entries are nonnegative and each of their rows and columnssums up to one (see Example A.1 in Appendix A).
Revised: Frequency matrices are bistochastic, i.e., their entries are nonnegative and each of their rows and columns sums up to one. Example 2.1. Let E =

Original: Given a vote v  ,we write #freq( v ) to denote thefrequency matrix of the election containing this vote only; #freq( v) is a permutation matrix, with a single 1 in each row and in each column.
Revised: ( C, V ) be an election with candidate set C = { a, b, c, d, e } and four voters, v 1 , v 2 , v 3 , and v 4 . Below, we show the voters’ preference orders (on the left) and the election’s frequency matrix (on the right). v 1 : a (cid:31) b (cid:31)

Revised: c

Revised: (cid:31) d (cid:31) e, v 2 : c (cid:31) b (cid:31) d

Revised: (cid:31) a (cid:31) e, v 3 : d (cid:31) e (cid:31) c

Revised: (cid:31) b (cid:31) a, v 4 : b (cid:31) c

Revised: (cid:31) a (cid:31) d (cid:31) e.

Revised:   a b c d e 1 / 4 1 / 4 1 / 4 1 /   1 / 2 1 / 1 / 1 / 1 / 2 1 /  1 / 4 1 / 1 /  1 /   3 / 

Revised: Given a vote v , we write #freq( v ) to denote the frequency matrix of the election containing this vote only; #freq( v ) is a permutation matrix, with a single 1 in each row and in each column.

Original: Thus, for an election E =
Revised: Thus, for an election E =

Original: ( C, V ) with V =
Revised: ( C, V ) with V =

Original: ( v 1 , . .
Revised: ( v 1 , . .

Original: . , v n )we have #freq( E ) =
Revised: . , v n ) we have #freq( E ) =

Original: 1 n · (cid:80) ni =1 #freq( v i ) .
Revised: 1 n · (cid:80) ni =1 #freq( v i ) .

Original: Compass Matrices.
Revised: Compass Matrices.

Original: For even m , Boehmer et al.
Revised: For even m , Boehmer et al.

Original: (2021b) deﬁned the following four m × m“compass” matrices, which appear to be extreme on the “map of elections”:
Revised: (2021b) deﬁned the following four m × m “compass” matrices, which appear to be extreme on the “map of elections”:

Original: The identity matrix , ID m , has ones on the diagonal and zeroes everywhere else (it corre- sponds to an election where all voters agree on a single preference order).
Revised: The identity matrix , ID m , has ones on the diagonal and zeroes everywhere else (it corresponds to an election where all voters agree on a single preference order).

Original: The uniformity matrix , UN m , has all entries equal to 1 / m (it corresponds to lack of agreement; each candidate is ranked at each position equally often).
Revised: The uniformity matrix , UN m , has all entries equal to 1 / m (it corresponds to lack of agreement; each candidate is ranked at each position equally often).

Original: The stratiﬁcation matrix , ST m , is partitioned into four quadrangles, where all entries in thetop-left and bottom-right quadrangles are equal to 2 / m , and all other entries are equal tozero (it corresponds to partial agreement; the voters agree which half of the candidates issuperior, but disagree on everything else).
Revised: The stratiﬁcation matrix , ST m , is partitioned into four quadrangles, where all entries in the top-left and bottom-right quadrangles are equal to 2 / m , and all other entries are equal to zero (it corresponds to partial agreement; the voters agree which half of the candidates is superior, but disagree on everything else).

Original: The antagonism matrix , AN m , has values 1 / 2 on both diagonals and zeroes elsewhere (itcaptures a conﬂict:  half of the voters rank the candidates in one way, and half of the votersrank them in the opposite way).
Revised: The antagonism matrix , AN m , has values 1 / 2 on both diagonals and zeroes elsewhere (it captures a conﬂict: it is a matrix of an election where half of the voters rank the candidates in one way and half of the voters rank them in the opposite way).

Original: Below, we show examples of these matrices for m = 4 :
Revised: Below, we show examples of these matrices for m = 4 :

Original: [Equation]
Revised: [Equation]

Original: [Equation] [Equation] [Equation]
Revised: / 4 1 / 4 1 / 4 1 / 4 1 /1 / 4 1 / 4 1 / 4 1 / 4 1 / 4 1 / 4 1 / 4 1 /

Original: [Equation]{+ , ID 4 =   1 0 0 0 0 1 00 0 1 0 0 
+}
Revised: {+ , ID 4 =   1 0 0 0 0 1 00 0 1 0 0 
+}

Original: 1 /
Revised: 0

Original: [Equation] AN 4 =   1 / 20 1 /0 1 / 2 1 / 20 1 / 2 1 / 2 0 1 / 20 1 /  .
Revised:  , ST 4 =   1 / 2 1 / 20 1 / 2 1 / 200 1 / 2 1 / 0 1 / 2 1 /  , AN 4 =   1 / 20 1 /0 1 / 2 1 / 20 1 / 2 1 / 2 0 1 / 20 1 /  .

Original: We omit the subscript in the names of these matrices if its value is clear from the context or irrelevant.
Revised: We omit the subscript in the names of these matrices if its value is clear from the context or irrelevant.

Original: EMD.
Revised: EMD.

Original: Let x = ( x 1 , . . .
Revised: Let x = ( x 1 , . . .

Original: , x n ) and y =
Revised: , x n ) and y =

Original: ( y 1 , . . .
Revised: ( y 1 , . . .

Original: , y n ) be two vectors with nonnegative real entriesthat sum up to 1 .
Revised: , y n ) be two vectors with nonnegative real entries that sum up to 1 .

Original: Their Earth mover’s distance , denoted EMD( x, y ) , is the cost of transforming xinto y using operations of the form:
Revised: Their Earth mover’s distance , denoted EMD( x, y ) , is the cost of transforming x into y using operations of the form:

Original: Given indices i, j ∈ [ n ] and a positive value δ such that x i ≥ δ ,at the cost of δ · | i − j | , replace x i with x
Revised: Given indices i, j ∈ [ n ] and a positive value δ such that x i ≥ δ , at the cost of δ · | i − j | , replace x i with x

Original: i − δ and x j with x j + δ
Revised: i − δ and x j with x j + δ

Original: (this corresponds to moving δamount of “earth” from position i to position j ).
Revised: (this corresponds to moving δ amount of “earth” from position i to position j ).

Original: EMD( x, y ) can be computed in polynomial time bya standard greedy algorithm.
Revised: EMD( x, y ) can be computed in polynomial time by a standard greedy algorithm.

Original: Positionwise Distance (Szufa et al., 2020).
Revised: Positionwise Distance (Szufa et al., 2020).

Original: Let A =
Revised: Let A =

Original: ( a 1 , . .
Revised: ( a 1 , . .

Original: . , a m ) and B = ( b 1 , . . .
Revised: . , a m ) and B = ( b 1 , . . .

Original: , b m )be two m × m frequency matrices.
Revised: , b m ) be two m × m frequency matrices.

Original: Their raw positionwise distance is rawPOS( A, B ) =
Revised: Their raw positionwise distance is rawPOS( A, B ) =

Original: min σ ∈ S m (cid:80) mi =1 EMD( a i , b
Revised: min σ ∈ S m (cid:80) mi =1 EMD( a i , b

Original: σ ( i ) ) , where S m denotes the set of all permutations over [ m ] .
Revised: σ ( i ) ) , where S m denotes the set of all permutations over [ m ] .

Original: We willnormalize these distances by 13 ( m 2 − 1) , which Boehmer et al.
Revised: We will normalize these distances by 13 ( m 2 − 1) , which Boehmer et al.

Original: (2021b, 2022) proved to be themaximum distance between two m × m frequency matrices and the distance between ID m and 
Revised: (2021b, 2022) proved to be the maximum distance between two m × m frequency matrices and the distance between ID m and za b c d (a) Tyyya b d c (b) Txa xb xc d (c) T

Original: UN m : nPOS( A, B )
Revised: UN m : nPOS( A, B )

Original: =
Revised: =

Original: rawPOS( A,B ) 13 ( m 2 − 1) .
Revised: rawPOS( A,B ) 13 ( m 2 − 1) .

Original: For two elections E and F with equal-sized candidate sets,their positionwise distance, raw or normalized, is deﬁned as the positionwise distance between theirfrequency matrices.
Revised: For two elections E and F with equal-sized candidate sets, their positionwise distance, raw or normalized, is deﬁned as the positionwise distance between their frequency matrices.

Original: Paths Between the Compass Matrices.
Revised: Paths Between the Compass Matrices.

Original: Let X and Y be two compass matrices.
Revised: Let X and Y be two compass matrices.

Original: Boehmer et al.(2021b) showed that if we take their convex combination Z = αX + (1 − α ) Y ( 0 ≤ α ≤ 1 )then nPOS
Revised: Boehmer et al. (2021b) showed that if we take their afﬁne combination Z = αX + (1 − α ) Y ( 0 ≤ α ≤ 1 ) then nPOS

Original: ( X, Z ) =
Revised: ( X, Z ) =

Original: (1 − α )nPOS( X, Y ) and nPOS( Z, Y )
Revised: (1 − α )nPOS( X, Y ) and nPOS( Z, Y )

Original: =
Revised: =

Original: α nPOS( X, Y ) .
Revised: α nPOS( X, Y ) .

Original: Such convexcombinations form direct paths between the compass matrices.
Revised: Such afﬁne combinations form direct paths between the compass matrices; they are also possible between any two frequency matrices of a given size, not just the compass ones, but may require shufﬂing the matrices’ columns (Boehmer et al., 2021b).

Original: Structured Domains.
Revised: Structured Domains.

Original: We consider two classes of structured elections, single-peaked ones (Black,1958), and group-separable ones (Inada, 1964).
Revised: We consider two classes of structured elections, single-peaked elections (Black, 1958), and group-separable elections (Inada, 1964).

Original: For a discussion of these domains and the motivationbehind them, see the original papers and the overview by Elkind et al.
Revised: For a discussion of these domains and the motivation behind them, see the original papers and the overviews by Elkind et al.

Original: Intuitively, an electionis single-peaked if we can order the candidates so that, as each voter considers the candidates in thisorder (referred to as the societal axis ), his or her appreciation ﬁrst increases and then decreases.
Revised: Intuitively, an election is single-peaked if we can order the candidates so that, as each voter considers the candidates in this order (referred to as the societal axis ), his or her appreciation ﬁrst increases and then decreases.

Original: Theaxis may, e.g., correspond to the left-right political spectrum.
Revised: The axis may, e.g., correspond to the left-right political spectrum.

Original: Deﬁnition 2.1.
Revised: Deﬁnition 2.2.

Original: Let v be a vote over C and let (cid:67) be the societal axis over C .
Revised: Let v be a vote over C and let (cid:67) be the societal axis over C .

Original: We say that v is singlepeaked with respect to (cid:67) if for every t ∈ [ | C | ] its t top-ranked candidates form an interval within (cid:67) .
Revised: We say that v is singlepeaked with respect to (cid:67) if for every t ∈ [ | C | ] its t top-ranked candidates form an interval within (cid:67) .

Original: An election is single-peaked with respect to (cid:67) if all its votes are.
Revised: An election is single-peaked with respect to (cid:67) if all its votes are.

Original: An election is single-peaked (SP) ifit is single-peaked with respect to some axis.
Revised: An election is single-peaked (SP) if it is single-peaked with respect to some axis.

Revised: Note that the election from Example 2.1 is single-peaked with respect to the axis a (cid:67) b (cid:67) c (cid:67) d (cid:67) e .

Original: We also consider group-separable elections , introduced by Inada (1964).
Revised: We also consider group-separable elections , introduced by Inada (1964).

Original: For our purposes, it willbe convenient to use the tree-based deﬁnition of Karpov (2019).
Revised: For our purposes, it will be convenient to use the tree-based deﬁnition of Karpov (2019).

Original: Let C
Revised: Let C

Original: = { c 1 , . .
Revised: = { c 1 , . .

Original: , c m } be a set of candidates, and consider a rooted, ordered tree T whose leaves are elements of C .
Revised: , c m } be a set of candidates, and consider a rooted, ordered tree T whose leaves are elements of C .

Original: The frontier ofsuch a tree is the preference order that ranks the candidates in the order in which they appear in thetree from left to right.
Revised: The frontier of such a tree is the preference order that ranks the candidates in the order in which they appear in the tree from left to right.

Original: A preference order is consistent with a given tree if it can be obtained as itsfrontier by reversing the order in which the children of some nodes appear.
Revised: A preference order is consistent with a given tree if it can be obtained as its frontier by reversing the order in which the children of some nodes appear.

Original: Deﬁnition 2.2.
Revised: Deﬁnition 2.3.

Original: An election E =
Revised: An election E =

Original: ( C, V ) is group-separable if there is a rooted, ordered tree T whoseleaves are members of C , such that each vote in V is consistent with T .
Revised: ( C, V ) is group-separable if there is a rooted, ordered tree T whose leaves are members of C , such that each vote in V is consistent with T .

Original: The trees from Deﬁnition 2.2 form a subclass of clone decomposition trees , which are examples of
Revised: The trees from Deﬁnition 2.3 form a subclass of clone decomposition trees , which are examples of

Original: PQ-trees (Elkind et al., 2012; Booth & Lueker, 1976).
Revised: PQ-trees (Elkind et al., 2012; Booth & Lueker, 1976).

Original: Example 2.3.
Revised: Example 2.4.

Original: Consider candidate set C = { a, b, c, d } , trees T 1 , T 2 , and T 3 from Figure 1, and votesv 1 : a (cid:31) b (cid:31) c (cid:31) d , v 2 : c (cid:31) d (cid:31) b (cid:31) a , and v 3 : b (cid:31) d (cid:31) c (cid:31) a .
Revised: Consider candidate set C = { a, b, c, d } , trees T 1 , T 2 , and T 3 from Figure 1, and votes v 1 : a (cid:31) b (cid:31) c (cid:31) d , v 2 : c (cid:31) d (cid:31) b (cid:31) a , and v 3 : b (cid:31) d (cid:31) c (cid:31) a .

Original: Vote v 1 is consistent with each ofthe trees, v 2 is consistent with T 2 (reverse the children of y 1 and y 2 ), and v 3 is consistent with T 3(reverse the children of x 1 and x 3 ).
Revised: Vote v 1 is consistent with each of the trees, v 2 is consistent with T 2 (reverse the children of y 1 and y 2 ), and v 3 is consistent with T(reverse the children of x 1 and x 3 ).

Original:  Frequency Matrices for Vote Distributions
Revised:  Frequency Matrices for Vote Distributions

Original: We show how to compute frequency matrices for several well-known distributions over votes.
Revised: We show how to compute frequency matrices for several well-known distributions over votes.

Original: 3.1 Setup and Interpretation
Revised: 3.1 Setup and Interpretation

Original: A vote distribution for a candidate set C is a function D that assigns a probability to each preference order over C .
Revised: A vote distribution for a candidate set C is a function D that assigns a probability to each preference order over C .

Original: Formally, we require that for each v ∈ L ( C ) it holds that D ( v ) ≥ 0 and153 (cid:80) v ∈L ( C ) D ( v )
Revised: Formally, we require that for each v ∈ L ( C ) it holds that D ( v ) ≥ 0 and (cid:80) v ∈L ( C ) D ( v )

Original: = 1 .
Revised: = 1 .

Original: We say that a vote v is in the support of D if D ( v ) > 0 .
Revised: We say that a vote v is in the support of D if D ( v ) > 0 .

Original: Given such a distribution, we can form an election by repeatedly drawing votes according to the speciﬁed probabilities.
Revised: Given such a distribution, we can form an election by repeatedly drawing votes according to the speciﬁed probabilities.

Original: For example, we can sample each element of L ( C ) with equal probability; this distribution, which isknown as impartial culture (IC) , is denoted by D IC (we omit the candidate set from our notation as itwill always be clear from the context).
Revised: For example, we can sample each element of L ( C ) with equal probability; this distribution, which is known as impartial culture (IC) , is denoted by D IC (we omit the candidate set from our notation as it will always be clear from the context).

Original: The frequency matrix of a vote distribution D over a candidateset C is #freq( D ) =
Revised: The frequency matrix of a vote distribution D over a candidate set C is #freq( D ) =

Original: (cid:80) v ∈L ( C ) D
Revised: (cid:80) v ∈L ( C ) D

Original: ( v ) · #freq( v ) .
Revised: ( v ) · #freq( v ) .

Original: For example, we have #freq( D IC ) = UN .
Revised: For example, we have #freq( D IC ) = UN .

Original: Oneinterpretation of #freq( D ) is that the entry for a candidate c j and a position i is the probabilitythat a vote v sampled from D has c j in position i (which we denote as P [pos v ( c j )
Revised: One interpretation of #freq( D ) is that the entry for a candidate c j and a position i is the probability that a vote v sampled from D has c j in position i (which we denote as P [pos v ( c j )

Original: =
Revised: =

Original: i ] ).
Revised: i ] ).

Original: Anotherinterpretation is that if we sample numerous votes then the resulting election’s frequency matrixwould be close to #freq( D ) with high probability.
Revised: Another interpretation is that if we sample a large number of votes then the resulting election’s frequency matrix would be close to #freq( D ) with high probability.

Original: More formally, if we let M n be a random variableequal to the frequency matrix of an n -voter election generated according to D , then it holds thatlim n →∞ E ( M n ) =
Revised: More formally, if we let M n be a random variable equal to the frequency matrix of an n -voter election generated according to D , then it holds that lim n →∞ E ( M n ) =

Original: #freq( D ) .
Revised: #freq( D ) .

Original: 3.2 Group-Separable Elections
Revised: 3.2 Group-Separable Elections

Original: We ﬁrst consider sampling group-separable votes.
Revised: We ﬁrst consider sampling group-separable votes.

Original: Given a rooted tree T whose leaves are labeled byelements of C = { c 1 , . . .
Revised: Given a rooted tree T whose leaves are labeled by elements of C = { c 1 , . . .

Original: , c m } , let D T GS be the distribution assigning equal probability to all votesconsistent with T , and zero probability to all other votes; one can think of D T GS as impartial culturerestricted to the group-separable subdomain deﬁned by T .
Revised: , c m } , let D T GS be the distribution assigning equal probability to all votes consistent with T , and zero probability to all other votes; one can think of D T GS as impartial culture restricted to the group-separable subdomain deﬁned by T .

Original: To sample from D T GS , we can toss a faircoin for each internal node of T , reversing the order of its children if this coin comes up heads, andoutput the frontier of the resulting tree.
Revised: To sample from D T GS , we can toss a fair coin for each internal node of T , reversing the order of its children if this coin comes up heads, and output the frontier of the resulting tree.

Original: We focus on the following types of trees:
Revised: We focus on the following types of trees:

Original: Flat( c 1 , . . .
Revised: Flat( c 1 , . . .

Original: , c m ) is a tree with a single internal node, whose children, from left to right, are c 1 , c 2 , . . 
Revised: , c m ) is a tree with a single internal node, whose children, from left to right, are c 1 , c 2 , . . .

Original: , c m .
Revised: , c m .

Original: There are only two preference orders consistent with this tree, c 1 (cid:31) · · · (cid:31) c m and its reverse. 
Revised: There are only two preference orders consistent with this tree, c 1 (cid:31) · · · (cid:31) c m and its reverse. 2.

Original: Bal( c 1 , . . .
Revised: Bal( c 1 , . . .

Original: , c m ) is a perfectly balanced binary tree with frontier c 1 , . . .
Revised: , c m ) is a perfectly balanced binary tree with frontier c 1 , . . .

Original: , c m (hence we assume the number m of candidates to be a power of two).
Revised: , c m (hence we assume the number m of candidates to be a power of two).

Original: CP( c 1 , . . .
Revised: CP( c 1 , . . .

Original: , c m ) is a binary caterpillar tree: it has internal nodes x 1 , . . .
Revised: , c m ) is a binary caterpillar tree: it has internal nodes x 1 , . . .

Original: , x m − 1 ; for eachj ∈ [ m − 2] , x j has c j as the left child and x j +1 as the right one, whereas x m − 1 has both c m − 1 and c m as children.
Revised: , x m − 1 ; for each j ∈ [ m − 2] , x j has c j as the left child and x j +1 as the right one, whereas x m − 1 has both c m − 1 and c m as children.

Original: The ﬁrst tree in Figure 1 is ﬂat, the second one is balanced, and the third one is a caterpillar tree.
Revised: The ﬁrst tree in Figure 1 is ﬂat, the second one is balanced, and the third one is a caterpillar tree.

Original: If T is a caterpillar tree, then we refer to D T GS as the GS/caterpillar distribution .
Revised: If T is a caterpillar tree, then we refer to D T GS as the GS/caterpillar distribution .

Original: We use a similarterminology for the other trees.
Revised: We use a similar terminology for the other trees.

Original: Theorem 3.1.
Revised: Theorem 3.1.

Original: Let F be the frequency matrix of distribution D T GS .
Revised: Let F be the frequency matrix of distribution D T GS .

Original: If T is ﬂat then F = AN , andif it is balanced then F = UN .
Revised: If T is ﬂat then F = AN , and if it is balanced then F = UN .

Original: If T is a caterpillar tree CP( c 1 , . .
Revised: If T is a caterpillar tree CP( c 1 , . .

Original: . , c m ) , then for each candidatec j the probability that c j appears in a position i ∈ [ m ] in a random vote v sampled from D T GS is:
Revised: . , c m ) , then for each candidate c j the probability that c j appears in a position i ∈ [ m ] in a random vote v sampled from D T GS is:

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: The cases of ﬂat and balanced trees are immediate, so we focus on caterpillar trees.
Revised: The cases of ﬂat and balanced trees are immediate, so we focus on caterpillar trees.

Original: Let
Revised: Let

Original: T = CP( c 1 , . . .
Revised: T = CP( c 1 , . . .

Original: , c m ) with internal nodes x 1 , . .
Revised: , c m ) with internal nodes x 1 , . .

Original: , x m − 1 , and consider a candidate c j and a positioni ∈ [ m ] .
Revised: , x m − 1 , and consider a candidate c j and a position i ∈ [ m ] .

Original: Let v be a random variable equal to a vote sampled from D T GS .
Revised: Let v be a random variable equal to a vote sampled from D T GS .

Original: We say that a node x (cid:96) ,(cid:96) ∈ [ m − 1] , is reversed if the order of its children is reversed.
Revised: We say that a node x (cid:96) , (cid:96) ∈ [ m − 1] , is reversed if the order of its children is reversed.

Original: Note that for (cid:96) <
Revised: Note that for (cid:96) <

Original: r it holds that c rprecedes c (cid:96) in the frontier if and only if x (cid:96) is reversed.
Revised: r it holds that c r precedes c (cid:96) in the frontier if and only if x (cid:96) is reversed.

Original: Suppose that x j is not reversed.
Revised: Suppose that x j is not reversed.

Original: Then v ranksc j above each of c j +1 , . . .
Revised: Then v ranks c j above each of c j +1 , . . .

Original: , c m .
Revised: , c m .

Original: This means that for c j to be ranked exactly in position i , it must bethat j ≥
Revised: This means that for c j to be ranked exactly in position i , it must be that j ≥

Original: i and exactly i − 1 nodes among x 1 , . . .
Revised: i and exactly i − 1 nodes among x 1 , . . .

Original: , x j − 1 are not reversed.
Revised: , x j − 1 are not reversed.

Original: If j ≥ i , the probability thatx j and i − 1 nodes among x 1 , . . .
Revised: If j ≥ i , the probability that x j and i − 1 nodes among x 1 , . . .

Original: , x j − 1 are not reversed is 12 j · (cid:0) j − 1 i − 1 (cid:1) .
Revised: , x j − 1 are not reversed is 12 j · (cid:0) j − 1 i − 1 (cid:1) .

Original: On the other hand, if x j isreversed, then v ranks candidates c j +1 , . . .
Revised: On the other hand, if x j is reversed, then v ranks candidates c j +1 , . . .

Original: , c m above c j .
Revised: , c m above c j .

Original: As there are m − j of them, for c j to beranked exactly in position i it must hold that i > m − j and exactly ( i − 1) − ( m − j ) nodes amongx 1 , . . .
Revised: As there are m − j of them, for c j to be ranked exactly in position i it must hold that i > m − j and exactly ( i − 1) − ( m − j ) nodes among x 1 , . . .

Original: , x j − 1 are not reversed.
Revised: , x j − 1 are not reversed.

Original: This happens with probability 12 j · (cid:0) j − 1 ( i − 1) − ( m − j ) (cid:1) .za b c d (a) Tyyya b d c (b) Txa xb xc d (c) T
Revised: This happens with probability 12 j · (cid:0) j − 1 ( i − 1) − ( m − j ) (cid:1) .

Original: Regarding distributions D T GS not handled in Theorem 3.1, we still can compute their frequence matrices efﬁciently.
Revised: Regarding distributions D T GS not handled in Theorem 3.1, we still can compute their frequence matrices efﬁciently.

Original: Theorem 3.2.
Revised: Theorem 3.2.

Original: Given a tree T , one can compute #freq( D T GS ) using polynomially many arithmetic operations with respect to the number of nodesin T .
Revised: There is an algorithm that given a tree T computes #freq( D T GS ) using polynomially many arithmetic operations with respect to the number of nodes in T .

Original:  From Caterpillars to Single-Peaked Preferences.
Revised: 3.3 From Caterpillars to Single-Peaked Preferences.

Original: There is a relationshipbetween GS/caterpillar votes and single-peaked ones, which will be veryuseful when computing one of the frequency matrices in the next section.
Revised: There is a relationship between GS/caterpillar votes and single-peaked ones, which will be very useful when computing one of the frequency matrices in the next section.

Original: Theorem 3.3.
Revised: Theorem 3.3.

Original: Given a ranking v over C = { c 1 , . . .
Revised: Given a ranking v over C = { c 1 , . . .

Original: , c m } , let (cid:98) v be anotherranking over C such that, for each j ∈ [ m ] , if c j is ranked in position i in vthen
Revised: , c m } , let (cid:98) v be another ranking over C such that, for each j ∈ [ m ] , if c j is ranked in position i in v then

Original: c i is ranked in position m − j + 1 in (cid:98) v .
Revised: c i is ranked in position m − j + 1 in (cid:98) v .

Original: Suppose that v is in the supportof D T GS , where T = CP( c 1 , . ..
Revised: Suppose that v is in the support of D T GS , where T = CP( c 1 , . .

Original: , c m ) .
Revised: c m ) .

Original: Then (cid:98) v is single-peaked with respectto c 1 (cid:67) · · · (cid:67) c m .
Revised: Then (cid:98) v is single-peaked with respect to c 1 (cid:67) · · · (cid:67) c m .

Original: There are exactly 2 m − 1 votes in the support of D T GS (this follows by simplecounting) and there are 2 m − 1 votes that are single-peaked with respect toc 1 (cid:67) · · · (cid:67) c m .
Revised: There are exactly 2 m − 1 votes in the support of D T GS (this follows by simple counting) and there are 2 m − 1 votes that are single-peaked with respect to c 1 (cid:67) · · · (cid:67) c m .

Original: As u (cid:54) = v implies (cid:98) u (cid:54) =
Revised: As u (cid:54) = v implies (cid:98) u (cid:54) =

Original: (cid:98) v , it follows that the mapping v (cid:55)→ (cid:98) vis a bijection between all votes in the support of D T GS and all votes that aresingle-peaked with respect to c 1 (cid:67) · · · (cid:67) c m .
Revised: (cid:98) v , it follows that the mapping v (cid:55)→ (cid:98) v is a bijection between all votes in the support of D T GS and all votes that are single-peaked with respect to c 1 (cid:67) · · · (cid:67) c m .

Original: 3.3 Single-Peaked Elections
Revised: 3.4 Single-Peaked Elections

Original: We consider two models of generating single-peaked elections, one due to Walsh (2015) and one dueto Conitzer (2009).
Revised: We consider two models of generating single-peaked elections, one due to Walsh (2015) and one due to Conitzer (2009).

Original: Let us ﬁx a candidate set C
Revised: Let us ﬁx a candidate set C

Original: = { c 1 , . . .
Revised: = { c 1 , . . .

Original: , c m } and a societal axis
Revised: , c m } and a societal axis

Original: c 1 (cid:67) · · · (cid:67) c m .
Revised: c 1 (cid:67) · · · (cid:67) c m .

Original: Under the Walsh distribution, denoted D WalSP , each vote that is single-peaked with respect to (cid:67) hasequal probability (namely, 1 2 m − 1 ), and all other votes have probability zero.
Revised: Under the Walsh distribution, denoted D WalSP , each vote that is single-peaked with respect to (cid:67) has equal probability (namely, 1 2 m − 1 ), and all other votes have probability zero.

Original: By Theorems 3.1 and 3.3,we immediately obtain the frequency matrix for the Walsh distribution (in short, it is the transposedmatrix of the GS/caterpillar distribution).
Revised: By Theorems 3.1 and 3.3, we immediately obtain the frequency matrix for the Walsh distribution (in short, it is the transposed matrix of the GS/caterpillar distribution).

Original: Corollary 3.4.
Revised: Corollary 3.4.

Original: Consider a candidate set C
Revised: Consider a candidate set C

Original: = { c 1 , . . .
Revised: = { c 1 , . . .

Original: , c m } and an axis c 1 (cid:67) · · · (cid:67) c m .
Revised: , c m } and an axis c 1 (cid:67) · · · (cid:67) c m .

Original: The probabilitythat candidate c j appears in position i in a vote sampled from D WalSP is:
Revised: The probability that candidate c j appears in position i in a vote sampled from D WalSP is:

Original: [Equation]
Revised: [Equation]

Original: [Equation] To sample a vote from the Conitzer distribution, D ConSP (also known as  random peak distribution ), wepick some candidate c j uniformly at random and rank him or her on top.
Revised: To sample a vote from the Conitzer distribution, D ConSP (also known as the random peak distribution ), we pick some candidate c j uniformly at random and rank him or her on top.

Original: Then we perform m − 1iterations, where in each we choose (uniformly at random) a candidate directly to the right or the leftof the already selected ones, and place him or her in the highest available position in the vote.
Revised: Then we perform m −iterations, where in each we choose (uniformly at random) a candidate directly to the right or the left of the already selected ones, and place him or her in the highest available position in the vote.

Original: Theorem 3.5.
Revised: Theorem 3.5.

Original: Let c 1 (cid:67) · · · (cid:67)
Revised: Let c 1 (cid:67) · · · (cid:67)

Original: c m be the societal axis, where m is an even number, and let v be arandom vote sampled from D ConSP for this axis.
Revised: c m be the societal axis, where m is an even number, and let v be a random vote sampled from D ConSP for this axis.

Original: For j ∈ [ m 2 ] and i ∈ [ m ] we have:
Revised: For j ∈ [ m 2 ] and i ∈ [ m ] we have:

Original: P [pos v ( c j ) = i ] =
Revised: P [pos v ( c j ) = i ] =

Original:   2 / 2 m if i < j , ( j +1) / 2 m if i = j , 1 / 2 m if j < i < m − j + 1 , ( m − j +1) / 2 m if i = m − j + 1 , if i + j > m .
Revised:   2 / 2 m if i < j , ( j +1) / 2 m if i = j , 1 / 2 m if j < i < m − j + 1 , ( m − j +1) / 2 m if i = m − j + 1 , if i + j > m .

Original: Further, for each candidate c j ∈ C and each position i ∈ [ m ] we have P [pos v ( c j ) =
Revised: Further, for each candidate c j ∈ C and each position i ∈ [ m ] we have P [pos v ( c j ) =

Original: i ] 
Revised: i ] =

Original: = 233 P [pos v ( c m − j +1 )
Revised: P [pos v ( c m − j +1 )

Original: = i ] .
Revised: = i ] .

Original: 3.4 Mallows Model
Revised: 3.5 Mallows Model

Original: Finally, we consider the classic Mallows distribution.
Revised: Finally, we consider the classic Mallows distribution.

Original: It has two parameters, a central vote v ∗ overm candidates and a dispersion parameter φ ∈ [0 , 1] .
Revised: It has two parameters, a central vote v ∗ over m candidates and a dispersion parameter φ ∈ [0 , 1] .

Original: The probability of sampling a vote v from thisdistribution (denoted D v ∗ ,φ Mal ) is: D v ∗ ,φ Mal ( v ) =
Revised: The probability of sampling a vote v from this distribution (denoted D v ∗ ,φ Mal ) is: D v ∗ ,φ Mal ( v ) =

Original: 1 Z φ κ ( v,v ∗ ) , where Z = 1 · (1 + φ ) · (1 + φ + φ 2 ) ·· · · · (1 + · · · + φ m − 1 ) is a normalizing constant and κ ( v, v ∗ ) is the swap distance between v and v ∗(i.e. , the number of swaps of adjacent candidates needed to transform v into v ∗ ).
Revised: 1 Z φ κ ( v,v ∗ ) , where Z = 1 · (1 + φ ) · (1 + φ + φ 2 ) · · · · · (1 + · · · + φ m − 1 ) is a normalizing constant and κ ( v, v ∗ ) is the swap distance between v and v ∗ (i.e., the number of swaps of adjacent candidates needed to transform v into v ∗ ).

Original: In our experiments,we consider a new parameterization, introduced by Boehmer et al.
Revised: In our experiments, we consider a new parameterization, introduced by Boehmer et al.

Original: (2021b).
Revised: (2021b).

Original: It uses a normalizeddispersion parameter norm - φ , which is converted to a value of φ so that the expected swap distancebetween the central vote v ∗ and a sampled vote v is norm - φtimes the maximum swap distancebetween two votes (so, norm - φ = 1 is equivalent to IC and for norm - φ = 0 .
Revised: It uses a normalized dispersion parameter norm - φ , which is converted to a value of φ so that the expected swap distance between the central vote v ∗ and a sampled vote v is norm - φtimes the maximum swap distance between two votes (so, norm - φ = 1 is equivalent to IC and for norm - φ = 0 .

Original: 5 we get elections thatlie close to the middle of the UN – ID path).
Revised: 5 we get elections that lie close to the middle of the UN – ID path).

Original: Our goal is now to compute the frequency matrix of D v ∗ ,φ Mal .
Revised: Our goal is now to compute the frequency matrix of D v ∗ ,φ Mal .

Original: That is, given the candidate rankedin position j in the central vote, we want to compute the probability that he or she appears in agiven position i ∈
Revised: That is, given the candidate ranked in position j in the central vote, we want to compute the probability that he or she appears in a given position i ∈

Original: [ m ] in the sampled vote.
Revised: [ m ] in the sampled vote.

Original: Given a positive integer m , consider the candidate set
Revised: Given a positive integer m , consider the candidate set

Original: C ( m ) =
Revised: C ( m ) =

Original: { c 1 , . . .
Revised: { c 1 , . . .

Original: , c m } and the central vote v ∗ m : c 1 (cid:31) · · · (cid:31) c m .
Revised: , c m } and the central vote v ∗ m : c 1 (cid:31) · · · (cid:31) c m .

Original: Fix a candidate c j ∈ C ( m ) , and aposition i ∈ [ m ] .
Revised: Fix a candidate c j ∈ C ( m ) , and a position i ∈ [ m ] .

Original: For every integer k between 0 and m ( m − 1) / 2 , let S ( m, k ) be the number of votesin L ( C ( m )) that are at swap distance k from v ∗ m , and deﬁne T ( m, k, j, i ) to be the number of suchvotes that have c j in position i .
Revised: For every integer k between 0 and m ( m − 1) / 2 , let S ( m, k ) be the number of votes in L ( C ( m )) that are at swap distance k from v ∗ m , and deﬁne T ( m, k, j, i ) to be the number of such votes that have c j in position i .

Original: One can compute S ( m, k ) in time polynomial in m (OEIS Foundation
Revised: One can compute S ( m, k ) in time polynomial in m (OEIS Foundation

Original: Inc., 2020); using S ( m, k ) , we show that the same holds for T ( m, k, j, i ) .
Revised: Inc., 2020); using S ( m, k ) , we show that the same holds for T ( m, k, j, i ) .

Original: Lemma 3.6.
Revised: Lemma 3.6.

Original: There is an algorithm that computes T ( m, k, j, i ) in polynomial time with respect to m .
Revised: There is an algorithm that computes T ( m, k, j, i ) in polynomial time with respect to m .

Original: We can now express the probability of sampling a vote v , where the candidate ranked in position j inthe central vote v ∗ ends up in position i under D v ∗ ,φ Mal , as:
Revised: We can now express the probability of sampling a vote v , where the candidate ranked in position j in the central vote v ∗ ends up in position i under D v ∗ ,φ Mal , as:

Original: [Equation]
Revised: [Equation]

Original: The correctness follows from the deﬁnitions of T and D v ∗ ,φ Mal .
Revised: The correctness follows from the deﬁnitions of T and D v ∗ ,φ Mal .

Original: By Lemma 3.6, we have the following.
Revised: By Lemma 3.6, we have the following.

Original: Theorem 3.7.
Revised: Theorem 3.7.

Original: There exists an algorithm that, given a number m of candidates, a vote v ∗ , and aparameter φ , computes the frequency matrix of D v ∗ ,φ Mal using polynomially many operations in m .
Revised: There exists an algorithm that, given a number m of candidates, a vote v ∗ , and a parameter φ , computes the frequency matrix of D v ∗ ,φ Mal using polynomially many operations in m .

Original: Note that Equation (1) only depends on φ , j and i (and, naturally, on m ).
Revised: Note that Equation (1) only depends on φ , j and i (and, naturally, on m ).

Original: Using this fact, we can alsocompute frequency matrices for several variants of the Mallows distribution.
Revised: Using this fact, we can also compute frequency matrices for several variants of the Mallows distribution.

Original: Remark 3.8.
Revised: Remark 3.8.

Original: Given a vote v , two dispersion parameters φ and ψ , and a probability p ∈ [0 , 1] , wedeﬁne the distribution p - D v,φ,ψ
Revised: Given a vote v , two dispersion parameters φ and ψ , and a probability p ∈ [0 , 1] , we deﬁne the distribution p - D v,φ,ψ

Original: Mal as p · D v,φ Mal + (1 − p ) · D rev( v ) ,ψ Mal , i.e., with probability p we samplea vote from D v,φ Mal and with probability 1 − p
Revised: Mal as p · D v,φ Mal + (1 − p ) · D rev( v ) ,ψ Mal , i.e., with probability p we sample a vote from D v,φ Mal and with probability 1 − p

Original: we sample a vote from D rev( v ) ,ψ Mal .
Revised: we sample a vote from D rev( v ) ,ψ Mal .

Original: The probability thatcandidate c j appears in position i in the resulting vote is p · f m ( φ, j, i )+(1 − p ) · f m ( ψ, m − j +1 , i ) .
Revised: The probability that candidate c j appears in position i in the resulting vote is p · f m ( φ, j, i )+(1 − p ) · f m ( ψ, m − j +1 , i ) .

Original: Remark 3.9.
Revised: Remark 3.9.

Original: Consider a candidate set C
Revised: Consider a candidate set C

Original: = { c 1 , . . .
Revised: = { c 1 , . . .

Original: , c m } .
Revised: , c m } .

Original: Given a vote distribution D over L ( C )and a parameter φ , deﬁne a new distribution D (cid:48) as follows:
Revised: Given a vote distribution D over L ( C ) and a parameter φ , deﬁne a new distribution D (cid:48) as follows:

Original: Draw a vote ˆ v according to D andthen output a vote v sampled from D ˆ v,φ Mal ; indeed, such models are quite natural, see, e.g., the workof Kenig & Kimelfeld (2019).
Revised: Draw a vote ˆ v according to D and then output a vote v sampled from D ˆ v,φ Mal ; indeed, such models are quite natural, see, e.g., the work of Kenig & Kimelfeld (2019).

Original: For each t ∈ [ m ] , let g ( j, t ) be the probability that c j appears inposition t in a vote sampled from D .
Revised: For each t ∈ [ m ] , let g ( j, t ) be the probability that c j appears in position t in a vote sampled from D .

Original: The probability that c j appears in position i ∈ [ m ] in avote sampled from D (cid:48) is (cid:80) mt
Revised: The probability that c j appears in position i ∈ [ m ] in a vote sampled from D (cid:48) is (cid:80) mt

Original: =1 g ( j, t ) · f ( φ, t, i ) .
Revised: =1 g ( j, t ) · f ( φ, t, i ) .

Original: In terms of matrix multiplication, this means that et al. (2021b). ID UN AN
Revised: In terms of matrix multiplication, this means that #freq( D (cid:48) ) =

Original: ST

Original: WAL

Original: CON

Original: [-CAT MID   
 #freq( D (cid:48) ) =-] #freq( D v ∗ ,φ Mal ) · #freq( D ) , where v ∗ is c 1 (cid:31) · · · (cid:31)
Revised: [-CAT MID   
 #freq( D (cid:48) ) =-] #freq( D v ∗ ,φ Mal ) · #freq( D ) , where v ∗ is c 1 (cid:31) · · · (cid:31)

Original: c m .
Revised: c m .

Original: We write φ -Conitzer ( φ -Walsh)to refer to this model where we use the Conitzer (Walsh) distribution as the underlying one andnormalized dispersion parameter φ .
Revised: We write φ -Conitzer ( φ -Walsh) to refer to this model where we use the Conitzer (Walsh) distribution as the underlying one and normalized dispersion parameter φ .

Original:  Skeleton Map
Revised:  Skeleton Map

Original: Our goal in this section is to form what we call a skeleton map of vote distributions (skeleton map,for short), evaluate its quality and robustness, and compare it to the map of Boehmer et al.
Revised: Our goal in this section is to form what we call a skeleton map of vote distributions (skeleton map, for short), evaluate its quality and robustness, and compare it to the map of Boehmer et al.

Original: (2021b).
Revised: (2021b).

Original: Throughout this section, whenever we speak of a distance between elections or matrices, we meanthe positionwise distance (occasionally we will also refer to the Euclidean distances on our maps, butwe will always make this explicit).
Revised: Throughout this section, whenever we speak of a distance between elections or matrices, we mean the positionwise distance (occasionally we will also refer to the Euclidean distances on our maps, but we will always make this explicit).

Original: Let Φ = { 0 , 0 . 05 , 0 . 1 , . .
Revised: Let Φ = { 0 , 0 . 05 , 0 . 1 , . .

Original: , 1 } be a set of normalized dispersionparameters that we will be using for Mallows-based distributions in this section.
Revised: , 1 } be a set of normalized dispersion parameters that we will be using for Mallows-based distributions in this section.

Original: We form the skeleton map following the general approach of Szufa et al.
Revised: We form the skeleton map following the general approach of Szufa et al.

Original: and Boehmeret al.
Revised: and Boehmer et al.

Original: (2021b).
Revised: (2021b).

Original: For a given number of candidates, we consider the four compass matrices (UN,
Revised: For a given number of candidates, we consider the four compass matrices (UN,

Original: ID, AN, ST) and paths between each matrix pair consisting of their convex combinations (graydots), the frequency matrices of the Mallows distribution with normalized dispersion parametersfrom Φ (blue triangles), and the frequency matrices of the Conitzer (CON), Walsh (WAL), and
Revised: ID, AN, ST) and paths between each matrix pair consisting of their convex combinations (gray dots), the frequency matrices of the Mallows distribution with normalized dispersion parameters from Φ (blue triangles), and the frequency matrices of the Conitzer (CON), Walsh (WAL), and

Original: GS/caterpillar distribution (CAT).
Revised: GS/caterpillar distribution (CAT).

Original: Moreover, we add the frequency matrices of the following votedistributions (we again use the dispersion parameters from Φ ): (i) The distribution 1 / 2 - D v,φ,φ Mal asdeﬁned in Remark 3.8 (red triangles), (ii) the distribution where with equal probability we mix thestandard Mallows distribution and 1 / 2 - D v,φ,φ Mal (green triangles), and (iii) the φ -Cointzer and φ -Walshdistributions as deﬁned in Remark 3.9 (magenta and orange crosses).
Revised: Moreover, we add the frequency matrices of the following vote distributions (we again use the dispersion parameters from Φ ): (i) The distribution 1 / 2 - D v,φ,φ Mal as deﬁned in Remark 3.8 (red triangles), (ii) the distribution where with equal probability we mix the standard Mallows distribution and 1 / 2 - D v,φ,φ Mal (green triangles), and (iii) the φ -Conitzer and φ -Walsh distributions as deﬁned in Remark 3.9 (magenta and orange crosses).

Original: For each pair of these matriceswe compute their positionwise distance.
Revised: For each pair of these matrices we compute their positionwise distance.

Original: Then we ﬁnd an embedding of the matrices into a 2D plane,so that each matrix is a point and the Euclidean distances between these points are as similar tothe positionwise distances as possible (we use the MDS algorithm, as implemented in the Pythonsklearn.manifold.
Revised: Then we ﬁnd an embedding of the matrices into a 2D plane, so that each matrix is a point and the Euclidean distances between these points are as similar to the positionwise distances as possible (we use the MDS algorithm, as implemented in the Python sklearn.manifold.

Original: MDS package).
Revised: MDS package).

Revised: In Figure 3 we show our map for the case of 10 candidates (the

Revised: ID UN

Revised: AN

Revised: ST

Original: In Figure 3 we show our map for the case of 10 candidates (thelines between some points/matrices show their positionwise distances; to maintain clarity, we onlyprovide some of them).
Revised: WAL CON CAT MID   5  lines between some points/matrices show their positionwise distances; to maintain clarity, we only provide some of them).

Original: We now verify the credibility of the skeleton map.
Revised: We now verify the credibility of the skeleton map.

Original: As the map does not have many points, we expectits embedding to truly reﬂect the positionwise distances between the matrices.
Revised: As the map does not have many points, we expect its embedding to truly reﬂect the positionwise distances between the matrices.

Original: This, indeed, seemsto be the case, although some distances are represented (much) more accurately than the others.
Revised: This, indeed, seems to be the case, although some distances are represented (much) more accurately than the others.

Original: In Figure 4 we provide the following data for a number of matrices (for m = 10 ; matrix M2W isthe Mallows matrix in our data set that is closest to the Walsh matrix).
Revised: In Figure 4 we provide the following data for a number of matrices (for m = 10 ; matrix M2W is the Mallows matrix in our data set that is closest to the Walsh matrix).

Original: In the top-right part (thewhite-orange area), we give positionwise distances between the matrices, and in the bottom-left part(the blue area), for each pair of matrices X and Y we report the misrepresentation ratio Euc( X,Y ) nPOS( X,Y ) ,where Euc( X, Y ) is the Euclidean distance between X and Y in the embedding, normalized by the
Revised: In the top-right part (the white-orange area), we give positionwise distances between the matrices, and in the bottom-left part (the blue area), for each pair of matrices X and Y we report the misrepresentation ratio Euc( X,Y ) nPOS( X,Y ) , where Euc( X, Y ) is the Euclidean distance between X and Y in the embedding, normalized by the

Original: Euclidean distance between ID and UN .
Revised: Euclidean distance between ID and UN .

Original: The closer they are to 1 , the more accurate is the embedding.
Revised: The closer they are to 1 , the more accurate is the embedding.

Original: The misrepresentation ratios are typically between 0 . 8 and 1 . 15 , with many of them between 0 . 9 and 
Revised: The misrepresentation ratios are typically between 0 . 8 and 1 . 15 , with many of them between 0 . 9 and 1 . 05 .

Original: Thus, in most cases, the map is quite accurate and offers good intuition about the relationsbetween the matrices.
Revised: Thus, in most cases, the map is quite accurate and offers good intuition about the relations between the matrices.

Original: Yet, some distances are represented particularly badly.
Revised: Yet, some distances are represented particularly badly.

Original: As an extreme example,the Euclidean distance between the Walsh matrix and the closest Mallows matrix, M2W, is off byalmost a factor of 8 (these matrices are close, but not as close as the map suggests).
Revised: As an extreme example, the Euclidean distance between the Walsh matrix and the closest Mallows matrix, M2W, is off by almost a factor of 8 (these matrices are close, but not as close as the map suggests).

Original: Thus, whileone always has to verify claims suggested by the skeleton map, we view it as quite credible.
Revised: Thus, while one always has to verify claims suggested by the skeleton map, we view it as quite credible.

Original: Thisconclusion is particularly valuable when we compare the skeleton map and the map of Boehmeret al.
Revised: This conclusion is particularly valuable when we compare the skeleton map and the map of Boehmer et al.

Original: (2021b), shown in Figure 2.
Revised: (2021b), shown in Figure 2.

Original: The two maps are similar, and analogous points (mostly) appearin analogous positions.
Revised: The two maps are similar, and analogous points (mostly) appear in analogous positions.

Original: Perhaps the biggest difference is the location of the Conitzer matrix on theskeleton map and Conitzer elections in the map of Boehmer et al., but even this difference is nothuge.
Revised: Perhaps the biggest difference is the location of the Conitzer matrix on the skeleton map and Conitzer elections in the map of Boehmer et al., but even this difference is not huge.

Original: We remark that the Conitzer matrix is closer to UN and AN than to ID and ST , whereas forthe Walsh matrix the opposite is true.
Revised: We remark that the Conitzer matrix is closer to UN and AN than to ID and ST , whereas for the Walsh matrix the opposite is true.

Original: Boehmer et al.
Revised: Boehmer et al.

Original: (2021b) make a similar observation; our resultsallow us to make this claim formal.
Revised: (2021b) make a similar observation; our results allow us to make this claim formal.

Original: In Appendix F, we analyze the robustness of the skeleton mapwith respect to varying the number of candidates.
Revised: In Appendix E, we analyze the robustness of the skeleton map with respect to varying the number of candidates.

Original: We ﬁnd that except for pairs including the Walsh or
Revised: We ﬁnd that except for pairs including the Walsh or

Original: GS/caterpillar matrices, which "travel" on the map as the number of candidates increases, the distancebetween each pair of matrices in the skeleton map stays nearly constant.
Revised: GS/caterpillar matrices, which "travel" on the map as the number of candidates increases, the distance between each pair of matrices in the skeleton map stays nearly constant.

Original:  Learning Vote Distributions
Revised:  Learning Vote Distributions

Original: We demonstrate how the positionwise distance and frequency matrices can be used to ﬁt votedistributions to given real-world elections.
Revised: We demonstrate how the positionwise distance and frequency matrices can be used to ﬁt vote distributions to given real-world elections.

Original: Speciﬁcally, we consider the Mallows model ( D v,φ Mal ) andthe φ -Conitzer and φ -Walsh models.
Revised: Speciﬁcally, we consider the Mallows model ( D v,φ Mal ) and the φ -Conitzer and φ -Walsh models.

Original: Naturally, we could use more distributions, but we focus onshowcasing the technique and the general uniﬁed approach.
Revised: Naturally, we could use more distributions, but we focus on showcasing the technique and the general uniﬁed approach.

Original: Concerning our results, among others, weverify that for Mallows model our approach is strongly correlated with existing maximum-likelihoodapproaches.
Revised: Concerning our results, among others, we verify that for Mallows model our approach is strongly correlated with existing maximumlikelihood approaches.

Original: Moreover, unlike in previous works, we compare the capabilities of different distributionsto ﬁt the given elections.
Revised: Moreover, unlike in previous works, we compare the capabilities of different distributions to ﬁt the given elections.

Original: We remark that if we do not have an algorithm for computing a frequencymatrix of a given vote distribution, we can obtain an approximate one by sampling sufﬁciently manyvotes from this distribution.
Revised: We remark that if we do not have an algorithm for computing a frequency matrix of a given vote distribution, we can obtain an approximate matrix by sampling sufﬁciently many votes from this distribution.

Original: In principle, it is also possible to deal with distributions  over electionswhich do not correspond to vote distributions and which do not have expected frequency matrices (asis the case, e.g., for the Euclidean models where candidates do not have ﬁxed positions; see the workof Szufa et al.
Revised: In principle, it is also possible to deal with distributions [Table] over elections that do not correspond to vote distributions and hence are not captured by expected frequency matrices (as is the case, e.g., for the Euclidean models where candidates do not have ﬁxed positions; see the work of Szufa et al.

Original: (2020) for examples of such models in the context of the map of elections):
Revised: (2020) for examples of such models in the context of the map of elections):

Original: If we wantto compute the distance of such a distribution, we samplea sufﬁciently large number of electionsand compute their average distance from the input one.
Revised: If we want to compute the distance of such a distribution, we sample sufﬁciently many elections and compute their average distance from the input one.

Original: However, it remains unclear how robust thisapproach is.
Revised: However, it remains unclear how robust this approach is.

Original: Approach.
Revised: Approach.

Original: To ﬁt our vote distributions to a given election, we compute the election’s distance tothe frequency matrices of D v,φ Mal , φ -Conitzer, and φ -Walsh, for φ ∈ { 0 , 0 .
Revised: To ﬁt our vote distributions to a given election, we compute the election’s distance to the frequency matrices of D v,φ Mal , φ -Conitzer, and φ -Walsh, for φ ∈ { 0 , 0 .

Original: We select thedistribution corresponding to the closest matrix.
Revised: We select the distribution corresponding to the closest matrix.

Original: [Table]

Original: Data.
Revised: Data.

Original: We consider elections from the real-worlddatasets used by Boehmer et al.
Revised: We consider elections from the real-world datasets used by Boehmer et al.

Original: (2021b).
Revised: (2021b).

Original: They generated 15 elections with 10 candidates and 100 voters (with strict preferences) from each of the elevendifferent real-world election datasets (so, altogether,they generated 165 electins, most of them from Preﬂib (Mattei & Walsh, 2013)).
Revised: They generated 15 elections with 10 candidates and 100 voters (with strict preferences) from each of the eleven different real-world election datasets (so, altogether, they generated 165 elections, most of them from Preﬂib (Mattei & Walsh, 2013)).

Original: They used four datasetsof political elections (from North Dublin (Irish), various non-proﬁt and professional organizations (ERS),and city council elections from Glasgow and Aspen),four datasets of sport-based elections (from Tour de France (TDF), Giro d’Italia (GDI), speed skating, andﬁgure skating) and three datasets with survey-basedelections (from preferences over T-shirt designs, sushi, and cities).
Revised: They used four datasets of political elections (from North Dublin (Irish), various non-proﬁt and professional organizations (ERS), and city council elections from Glasgow and Aspen), four datasets of sport-based elections (from Tour de France (TDF), Giro d’Italia (GDI), speed skating, and ﬁgure skating) and three datasets with survey-based elections (from preferences over T-shirt designs, sushi, and cities).

Original: We present the results of ouranalysis for seven illustrative and particularly interesting elections in Table 1 and also include themin our skeleton map from Figure 3.
Revised: We present the results of our analysis for seven illustrative and particularly interesting elections in Table 1 and also include them in our skeleton map from Figure 3.

Original: Basic Test.
Revised: Basic Test.

Original: There is a standard maximum-likelihood estimator (MLE; based on Kemeny voting (Mandhani & Meilˇa, 2009)) that given an election provides the most likely dispersion parameterof the Mallows distribution that might have generated this election.
Revised: There is a standard maximum-likelihood estimator (MLE; based on Kemeny voting (Mandhani & Meil ˇ a, 2009)) that given an election provides the most likely dispersion parameter of the Mallows distribution that might have generated this election.

Original: To test our approach, we compared the parameters provided by our approach and by the MLE for our 165 elections and found them to be highly correlated (with Pearson correlation coefﬁcient around 0 . 97 ).
Revised: To test our approach, we compared the parameters provided by our approach and by the MLE for our 165 elections and found them to be highly correlated (with Pearson correlation coefﬁcient around 0 . 97 ).

Original: In particular, the averageabsolute difference between the dispersion parameter calculated by our approach and the MLE isonly 0 .
Revised: In particular, the average absolute difference between the dispersion parameter calculated by our approach and the MLE is only 0 .

Original: See Appendix G for details.
Revised: See Appendix F for details.

Original: Fitting Real-World Elections.
Revised: Fitting Real-World Elections.

Original: Next, we consider the capabilities of D v,φ Mal , φ -Conitzer, and φWalsh to ﬁt the real-world elections of Boehmer et al.
Revised: Next, we consider the capabilities of D v,φ Mal , φ -Conitzer, and φWalsh to ﬁt the real-world elections of Boehmer et al.

Original: (2021b).
Revised: (2021b).

Original: Overall, we ﬁnd that these threevote distributions have some ability to capture the considered elections, but it certainly is not perfect.
Revised: Overall, we ﬁnd that these three vote distributions have some ability to capture the considered elections, but it certainly is not perfect.

Original: Indeed, the average normalized distance of these elections to the  closest distribution’s frequencymatrix is 0 .
Revised: Indeed, the average normalized distance of these elections to the frequency matrix of the closest distribution is 0 .

Original: To illustrate that some distance is to be expected here, we mention that the averagedistance of an election sampled from impartial culture ( D IC , with 10 candidates and 100 voters) tothe distribution’s expected frequency matrix is 0 . 

Revised: To illustrate that some distance is to be expected here, we mention that the average distance of an election sampled from impartial culture ( D IC , with 10 candidates andvoters) to the distribution’s expected frequency matrix is 0 .

Original: (see Appendix F.4 for a discussion of this andhow it may serve as an estimator for the “variance of a distribution”).
Revised: (see Appendix E.4 for a discussion of this and how it may serve as an estimator for the “variance of a distribution”).

Original: There are also some electionsthat are not captured by any of the considered distributions to an acceptable degree; examples of thisare elections nr. 1 and nr. 2, which are at distance at least 0 .
Revised: There are also some elections that are not captured by any of the considered distributions to an acceptable degree; examples of this are elections nr. 1 and nr. 2, which are at distance at least 0 .

Original: 32 and 0 .
Revised: 32 and 0 .

Original: 25 from all our distributions,respectively.
Revised: 25 from all our distributions, respectively.

Original: Remarkably, while coming from the same dataset, elections nr. 1 and nr. 2 are still quitedifferent from each other and, accordingly, the computed dispersion parameter is also quite different.
Revised: Remarkably, while coming from the same dataset, elections nr. 1 and nr.are still quite different from each other and, accordingly, the computed dispersion parameter is also quite different.

Original: It remains a challenge to ﬁnd distributions capturing such elections.
Revised: It remains a challenge to ﬁnd distributions capturing such elections.

Original: Comparing the power of the three considered models, nearly all of our elections arecaptured best bythe Mallows model instead of φ -Conitzer or φ -Walsh.
Revised: Comparing the power of the three considered models, nearly all of our elections are best captured by the Mallows model rather than φ -Conitzer or φ -Walsh.

Original: There are only twenty elections that are closerto φ -Walsh or φ -Conitzer than to a Mallows model (election nr.
Revised: There are only twenty elections that are closer to φ -Walsh or φ -Conitzer than to a Mallows model (election nr.

Original: 3 is the most extreme example),and, unsurprisingly, both φ -Walsh and φ -Conitzer perform particularly badly at capturing electionsclose to ID (see election nr. 4).
Revised: 3 is the most extreme example), and, unsurprisingly, both φ -Walsh and φ -Conitzer perform particularly badly at capturing elections close to ID (see election nr. 4).

Original: To sum up, φ -Conitzer and φ -Walsh are not needed to ensure a goodcoverage of the space of elections; the average normalized distance of our elections to the closest
Revised: That is, φ -Conitzer and φ -Walsh are not needed to ensure good coverage of the space of elections; the average normalized distance of our elections to the closest

Original: Mallows model is only 0 .
Revised: Mallows model is only 0 .

Original: 0007 higher than their distance to the closest distribution (elections nr.
Revised: 0007 higher than their distance to the closest distribution (elections nr.

Original: 3-6are three examples for elections which are well captured by the Mallows model and distributed overthe whole map).
Revised: 3are three examples of elections which are well captured by the Mallows model and distributed over the entire map).

Original: Nevertheless, φ -Walsh is also surprisingly powerful, as the average normalizeddistance of our elections to the closest φ -Walsh distribution is only 0 .
Revised: Nevertheless, φ -Walsh is also surprisingly powerful, as the average normalized distance of our elections to the closest φ -Walsh distribution is only 0 .

Original: 03 higher than their distanceto the closest distribution (however, this might be also due to the fact that, as observed by Boehmeret al. (2021b), most of the considered real-world elections fall into the same area of the map, whichφ -Walsh happens to capture particularly well).
Revised: 03 higher than their distance to the closest distribution (however, this might be also due to the fact that most of the considered real-world elections fall into the same area of the map, which φ -Walsh happens to capture particularly well (Boehmer et al., 2021b)).

Original: φ -Conitzer performs considerably worse and only forthree elections  produces a (slightly) better result than φ -Walsh.
Revised: φ -Conitzer performs considerably worse: there are only three elections for which it produces a (slightly) better result than φ -Walsh.

Original: Moreover, our results also underline the complex nature of the space of elections: Election nr.
Revised: Moreover, our results also emphasize the complex nature of the space of elections: Election nr.

Original: 7 isvery close to D v, 0 .
Revised: 7 is very close to D v, 0 .

Original: Mal , hinting that its votes are quite chaotic.
Revised: Mal , hinting that its votes are quite chaotic.

Original: At the same time, this election is veryclose to 0 .
Revised: At the same time, this election is very close to 0 .

Original: 63 -Conitzer and 0 .
Revised: 63 -Conitzer and 0 .

Original: 69 -Walsh distributions, which suggests at least a certain level of structureamong its votes (because votes from Conitzer and Walsh distributions are very structured, and
Revised: 69 -Walsh distributions, which suggests at least a certain level of structure among its votes (because votes from Conitzer and Walsh distributions are very structured, and

Original: the
Revised: the

Original: Mallows ﬁlter with dispersion between 0 . 63 and 0 .
Revised: Mallows ﬁlter with dispersion between 0 . 63 and 0 .

Original: 69 does not destroy this structure fully).
Revised: 69 does not destroy this structure fully).

Original: However,as witnessed by the fact that the frequency matrix of GS/balanced (which is highly structured) is UN ,such phenomena can happen.
Revised: However, as witnessed by the fact that the frequency matrix of GS/balanced (which is highly structured) is UN , such phenomena can happen.

Original: Lastly, note that most of our datasets are quite "homogenous" in that theclosest distributions for elections from the dataset are similar and also at a similar distance.
Revised: Lastly, note that most of our datasets are quite “homogenous”, in that the closest distributions for elections from the dataset are similar and also at a similar distance.

Original: However,there are also clear exceptions, for instance, elections nr. 1 and nr.
Revised: However, there are also clear exceptions, for instance, elections nr. 1 and nr.

Original: 4 from the ﬁgure skating dataset.
Revised: 4 from the ﬁgure skating dataset.

Original: Moreover, there are two elections from the speed skating dataset where one election is captured bestby D v, 0 .
Revised: Moreover, there are two elections from the speed skating dataset where one election is captured best by D v, 0 .

Original: 76 Mal and the other by D v, 0 .
Revised: 76 Mal and the other by D v, 0 .

Original: 32 Mal .
Revised: 32 Mal .

Original:  Summary
Revised:  Summary

Original: We have computed the frequency matrices (Szufa et al., 2020; Boehmer et al., 2021b) of severalwell-known distributions of votes.
Revised: We have computed the frequency matrices (Szufa et al., 2020; Boehmer et al., 2021b) of several well-known distributions of votes.

Original: Using them, we have drawn a “skeleton map” which shows howthese distributions relate to each other, and we have analyzed its properties.
Revised: Using them, we have drawn a “skeleton map”, which shows how these distributions relate to each other, and we have analyzed its properties.

Original: Moreover, we havedemonstrated how our results can be used to ﬁt vote distributions to capture real-world elections.
Revised: Moreover, we have demonstrated how our results can be used to ﬁt vote distributions to capture real-world elections.

Original: For future work, it would be interesting to compute the frequency matrices of further popular votedistributions, such as the Placket-Luce model (we conjecture that its frequency matrix is computablein polynomial time).
Revised: For future work, it would be interesting to compute the frequency matrices of further popular vote distributions, such as the Plackett–Luce model (we conjecture that its frequency matrix is computable in polynomial time).

Original: Moreover, it would  be interesting to use our approach to ﬁt more complexmodels, such as mixtures of Mallows models, to real-world elections.Checklist For all authors...
Revised: It would also be interesting to use our approach to ﬁt more complex models, such as mixtures of Mallows models, to real-world elections.

Original: (a) Do the main claims made in the abstract and introduction accurately reﬂect the paper’s

Original: contributions and scope?

Original: [Yes] (b) Did you describe the limitations of your work? [Yes] (c) Did you discuss any potential negative societal impacts of your work?
Revised: Further, it may be interesting to use expected frequency matrices to reason about the asymptotic behavior of our models.

Original: [No](d) Have you read the ethics review guidelines and ensured that your paper conforms tothem?

Original: [Yes]

Original: If you are including theoretical results...

Original: (a) Did you state the full set of assumptions of all theoretical results?

Original: [Yes]

Original: (b) Did you include complete proofs of all theoretical results?

Original: [Yes]

Original: If you ran experiments...

Original: (a) Did you include the code, data, and instructions needed to reproduce the main experi-

Original: mental results (either in the supplemental material or as a URL)? [Yes] (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they
Revised: For example, it might be possible to formally show where, in the limit, do the matrices of our models end up on the map as we increase the number of candidates.

Original: were chosen)?

Original: [N/A]

Original: (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [N/A] (d) Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)? [No]
Revised: Acknowledgments

Original: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...

Original: (a) If your work uses existing assets, did you cite the creators?

Original: [Yes] (b) Did you mention the license of the assets? [N/A] (c) Did you include any new assets either in the supplemental material or as a URL?
Revised: NB was supported by the DFG project MaMu (NI 369/19) and by the DFG project ComSoc-MPMS (NI 369/22).

Original: [Yes]

Original: (d) Did you discuss whether and how consent was obtained from people whose data you’re

Original: using/curating?

Original: [N/A]

Original: (e) Did you discuss whether the data you are using/curating contains personally identiﬁable

Original: information or offensive content?

Original: [N/A]

Original: If you used

Original: crowdsourcing

Original: or conducted research with human subjects

Original: (a) Did you include the full text of instructions given to participants and screenshots, if

Original: applicable?

Original: [N/A]

Original: (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount
Revised: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 101002854).

Original: spent on participant compensation?

Original: [N/A]

Original: A Missing Material From Section 2 
Revised: A Missing Material From Section 3.

Original: Example A.1.

Original: Let E =

Original: ( C, V ) be an election with candidate set C

Original: = { a, b, c, d, e } and four voters,v 1 , v 2 , v 3 , and v 4 .

Original: Below, we show their preference orders (on the left) and the election’s frequencymatrix (on the right).v 1 : a (cid:31) b (cid:31) c (cid:31) d (cid:31) e, v 2 : c (cid:31) b (cid:31) d (cid:31) a (cid:31) e, v 3 : d (cid:31) e (cid:31) c

Original: (cid:31) b (cid:31) a, v 4 : b (cid:31) c

Original: (cid:31) a (cid:31) d (cid:31) e.

Original:   a b c d e 1 / 4 1 / 4 1 / 4 1 /   1 / 2 1 / 1 / 1 / 1 / 2 1 /  1 / 4 1 / 1 /  1 /   3 / 

Original: [-Note that the election from Example A.1 is single-peaked with respect to the axis a (cid:67) b (cid:67) c (cid:
-]
Revised: [-Note that the election from Example A.1 is single-peaked with respect to the axis a (cid:67) b (cid:67) c (cid:
-]

Original: ) d (cid:67) e .

Original: B Missing Material From Section 3.2

Original: Theorem 3.2.
Revised: Theorem 3.2.

Original: Given a tree T , one can compute #freq( D T GS ) using polynomially many arithmeticoperations with respect to the number of nodes in T .
Revised: There is an algorithm that given a tree T computes #freq( D T GS ) using polynomially many arithmetic operations with respect to the number of nodes in T .

Original: Proof.
Revised: Proof.

Original: Let T be the input tree, and let C
Revised: Let T be the input tree, and let C

Original: = { c 1 , . .
Revised: = { c 1 , . .

Original: , c m } be its candidate set.
Revised: , c m } be its candidate set.

Original: We will give analgorithm for computing the probability that a given candidate c j appears in position i ∈
Revised: We will give an algorithm for computing the probability that a given candidate c j appears in position i ∈

Original: [ m ] in avote sampled from D T GS .
Revised: [ m ] in a vote sampled from D T GS .

Original: Let x be some node of T (either internal or a leaf).
Revised: Let x be some node of T (either internal or a leaf).

Original: Let T x be the tree obtained from T by deletingall descendants of x , so that x becomes a leaf, and for each subset S of internal nodes of T x , let
Revised: Let T x be the tree obtained from T by deleting all descendants of x , so that x becomes a leaf, and for each subset S of internal nodes of T x , let

Original: T Sx be the ordered tree obtained by starting with T x and reversing the nodes in the set S .
Revised: T Sx be the ordered tree obtained by starting with T x and reversing the nodes in the set S .

Original: For eacht ∈ { 0 } ∪ [ m − 1] we deﬁne f ( x, t ) to be the probability that if we reversed each internal node of T xwith probability 1 / 2 then x would be preceded by exactly t candidates in the frontier of the resultingordered tree.
Revised: For each t ∈ { 0 } ∪ [ m − 1] we deﬁne f ( x, t ) to be the probability that if we reversed each internal node of T x with probability 1 / 2 then x would be preceded by exactly t candidates in the frontier of the resulting ordered tree.

Original: We compute f
Revised: We compute f

Original: ( x, t ) using dynamic programming.
Revised: ( x, t ) using dynamic programming.

Original: Let root be the root of T .
Revised: Let root be the root of T .

Original: Then f ( root , 0) = 1 , and f ( root , t ) = 

Revised: Then f ( root , 0) = 1 , and f ( root , t ) =

Original: for t ∈ [1 , m − 1] .
Revised: for t ∈ [1 , m − 1] .

Original: Next, let x besome node of T other than the root, let p be the parent of x , and let (cid:96) and r be the number of leavesthat are descendants of x ’s left siblings and x ’s right siblings in T , respectively.
Revised: Next, let x be some node of T other than the root, let p be the parent of x , and let (cid:96) and r be the number of leaves that are descendants of x ’s left siblings and x ’s right siblings in T , respectively.

Original: We claim that foreach t ∈ { 0 } ∪ [ m − 1] we have:
Revised: We claim that for each t ∈ { 0 } ∪ [ m − 1] we have:

Original: [Equation]
Revised: [Equation]

Original: To see why this formula is correct, observe that if p (cid:54)∈ S then x appears in position t in the frontierof T Sx if and only if p appears in position t − (cid:96) in the frontier of T Sp : indeed, in the frontier of T Sxthe node x appears after all predecessors of p in the frontier of T Sp as well as after the (cid:96) leaves thatare the descendants of x ’s left siblings in T .
Revised: To see why this formula is correct, observe that if p (cid:54)∈ S then x appears in position t in the frontier of T Sx if and only if p appears in position t − (cid:96) in the frontier of T Sp : indeed, in the frontier of T Sx the node x appears after all predecessors of p in the frontier of T Sp as well as after the (cid:96) leaves that are the descendants of x ’s left siblings in T .

Original: Similarly, if p ∈ S then x appears in position t in thefrontier of T Sx if and only if p appears in position t − r in the frontier of T S \{ p } p : indeed, in thefrontier of T S x the node x appears after all predecessors of p in the frontier of T S \{ p } p as well as afterthe r leaves that are the descendants of x ’s right siblings in T .
Revised: Similarly, if p ∈ S then x appears in position t in the frontier of T Sx if and only if p appears in position t − r in the frontier of T S \{ p } p : indeed, in the frontier of T Sx the node x appears after all predecessors of p in the frontier of T S \{ p } p as well as after the r leaves that are the descendants of x ’s right siblings in T .

Original: Since p is reversed with probability 12 ,the recurrence follows.
Revised: Since p is reversed with probability 12 , the recurrence follows.

Original: The above formula and standard dynamic programming allow us to compute all the values of f using
Revised: The above formula and standard dynamic programming allow us to compute all the values of f using

Original: O ( m 2 ) arithmetic operations (note that there are at most O ( m ) internal nodes).
Revised: O ( m 2 ) arithmetic operations (note that there are at most O ( m ) internal nodes).

Original: To complete theproof, observe that the probability that candidate c j ends up in position i is f ( c j , i − 1) .
Revised: To complete the proof, observe that the probability that candidate c j ends up in position i is f ( c j , i − 1) .

Original: C Missing Material From Section 3.2 
Revised: B Missing Material From Section 3.

Original: We use the following lemma, which is implicit in the work of Faliszewski et al.
Revised: We use the following lemma, which is implicit in the work of Faliszewski et al.

Original: Lemma C.1.
Revised: Lemma B.1.

Original: Let T = CP( c 1 , . . .
Revised: Let T = CP( c 1 , . . .

Original: , c m ) .
Revised: , c m ) .

Original: A ranking v over { c 1 , . . .
Revised: A ranking v over { c 1 , . . .

Original: , c m } belongs to the support of
Revised: , c m } belongs to the support of

Original: D T GS if and only if there exists a subset C (cid:48) ⊆
Revised: D T GS if and only if there exists a subset C (cid:48) ⊆

Original: C \ { c m } such that in v :
Revised: C \ { c m } such that in v :

Original: (1) all alternatives in C (cid:48) are ranked above c m and all alternatives in ( C \ { c m } ) \ C (cid:48) are ranked below c m ; 
Revised: (1) all alternatives in C (cid:48) are ranked above c m and all alternatives in ( C \ { c m } ) \ C (cid:48) are ranked below c m ; 2.

Original: (2) for all c
Revised: (2) for all c

Original: i , c j ∈ C (cid:48) with i <
Revised: i , c j ∈ C (cid:48) with i <

Original: j the alternative c i is ranked above c j ; 
Revised: j the alternative c i is ranked above c j ; 3.

Original: (3) for all c
Revised: (3) for all c

Original: i , c j (cid:54)∈ C (cid:48) ∪ { c m } with i <
Revised: i , c j (cid:54)∈ C (cid:48) ∪ { c m } with i <

Original: j the alternative c i is ranked below c j .
Revised: j the alternative c i is ranked below c j .

Original: That is, in v the alternatives in C (cid:48) appear in the increasing order of indices, followed by c m , followedby the remaining alternatives in the decreasing order of indices, i.e., the sequence of indices in v is“single-peaked”.
Revised: That is, in v the alternatives in C (cid:48) appear in the increasing order of indices, followed by c m , followed by the remaining alternatives in the decreasing order of indices, i.e., the sequence of indices in v is “single-peaked”.

Original: Using this observation, we establish a bijection between the votes in the support of
Revised: Using this observation, we establish a bijection between the votes in the support of

Original: D T GS and single-peaked votes.
Revised: D T GS and single-peaked votes.

Original: Theorem 3.3.
Revised: Theorem 3.3.

Original: Given a ranking v over C = { c 1 , . . .
Revised: Given a ranking v over C = { c 1 , . . .

Original: , c m } , let (cid:98) v be another ranking over C suchthat, for each j ∈ [ m ] , if c j is ranked in position i in v then
Revised: , c m } , let (cid:98) v be another ranking over C such that, for each j ∈ [ m ] , if c j is ranked in position i in v then

Original: c
Revised: c

Original: i is ranked in position m − j + 1 in (cid:98) v .
Revised: i is ranked in position m − j + 1 in (cid:98) v .

Original: Suppose that v is in the support of D T GS , where T = CP( c 1 , . .
Revised: Suppose that v is in the support of D T GS , where T = CP( c 1 , . .

Original: c m ) .
Revised: c m ) .

Original: Then (cid:98) v is single-peaked withrespect to c 1 (cid:67) · · · (cid:67) c m .
Revised: Then (cid:98) v is single-peaked with respect to c 1 (cid:67) · · · (cid:67) c m .

Original: Proof.
Revised: Proof.

Original: Suppose that c m is ranked in position z in v .
Revised: Suppose that c m is ranked in position z in v .

Original: Then c z is ranked ﬁrst in (cid:98) v .
Revised: Then c z is ranked ﬁrst in (cid:98) v .

Original: Consider two candidates c x , c y with x < y <
Revised: Consider two candidates c x , c y with x < y <

Original: z .
Revised: z .

Original: We will prove that in (cid:98) v candidate c y is ranked abovec x , i.e., pos (cid:98) v ( c x ) > pos (cid:98) v ( c y ) .
Revised: We will prove that in (cid:98) v candidate c y is ranked above c x , i.e., pos (cid:98) v ( c x ) > pos (cid:98) v ( c y ) .

Original: Let k = pos (cid:98) v ( c x ) , (cid:96) = pos (cid:98) v ( c y ) .
Revised: Let k = pos (cid:98) v ( c x ) , (cid:96) = pos (cid:98) v ( c y ) .

Original: Then, in v , alternative c m − k +1 is570 ranked in position x and alternative c m − (cid:96) +1 is ranked in position y .
Revised: Then, in v , alternative c m − k +1 is ranked in position x and alternative c m − (cid:96) +1 is ranked in position y .

Original: As we have x < y <
Revised: As we have x < y <

Original: z and v issampled from D T GS , by Lemma [-C.
-] we have m − k + 1 < m − (cid:96) + 1 , and hence k > (cid:96) .
Revised: z and v is sampled from D T GS , by Lemma [-C.
-]B.1 we have m − k + 1 < m − (cid:96) + 1 , and hence k > (cid:96) .

Original: Similarly, if we have two alternatives c x , c y with z < y < x , we can show that pos (cid:98) v ( c x ) > pos (cid:98) v ( c y ) .
Revised: Similarly, if we have two alternatives c x , c y with z < y < x , we can show that pos (cid:98) v ( c x ) > pos (cid:98) v ( c y ) .

Original: Thus, (cid:98) v is single-peaked with respect to c 1 (cid:67) · · · (cid:67) c m , as claimed.
Revised: Thus, (cid:98) v is single-peaked with respect to c 1 (cid:67) · · · (cid:67) c m , as claimed.

Original: D Missing Material From Section 3.3 
Revised: C Missing Material From Section 3.

Original: Recall that our candidate set is C
Revised: Recall that our candidate set is C

Original: = { c 1 , . .
Revised: = { c 1 , . .

Original: , c m } and the societal axis is c 1 (cid:67) · · · (cid:67) c m .
Revised: , c m } and the societal axis is c 1 (cid:67) · · · (cid:67) c m .

Original: We considerthe Conitzer distribution.
Revised: We consider the Conitzer distribution.

Original: Let f ( i, j ) be the probability that in a sampled vote candidates c i , . .
Revised: Let f ( i, j ) be the probability that in a sampled vote candidates c i , . .

Original: , c jappear in the top j − i + 1 positions.
Revised: c j appear in the top j − i + 1 positions.

Original: Next we ﬁnd the values of f ( i, j ) for all i, j ∈ [ m ] , and usingthem we establish the frequency matrix for the Conitzer distribution.
Revised: Next we ﬁnd the values of f ( i, j ) for all i, j ∈ [ m ] , and using them we establish the frequency matrix for the Conitzer distribution.

Original: Proposition D.1.
Revised: Proposition C.1.

Original: Let i, j be two integers with 1 < i ≤ j < m .
Revised: Let i, j be two integers with 1 < i ≤ j < m .

Original: Then f ( (cid:96), (cid:96) )
Revised: Then f ( (cid:96), (cid:96) )

Original: = 1 / m for all (cid:96) ∈ [ m ] ,f (1 , m )
Revised: = 1 / m for all (cid:96) ∈ [ m ] , f (1 , m )

Original: = 1 , f
Revised: = 1 , f

Original: ( i, j )
Revised: ( i, j )

Original: = 1 / m , f (1 , i ) =
Revised: = 1 / m , f (1 , i ) =

Original: ( i +1) / 2 m , and f ( j, m ) =
Revised: ( i +1) / 2 m , and f ( j, m ) =

Original: ( m − j +2) / 2 m .
Revised: ( m − j +2) / 2 m .

Original: Proof.
Revised: Proof.

Original: The quantity f ( (cid:96), (cid:96) ) is simply the probability that c (cid:96) is ranked ﬁrst, so we have f ( (cid:96), (cid:96) )
Revised: The quantity f ( (cid:96), (cid:96) ) is simply the probability that c (cid:96) is ranked ﬁrst, so we have f ( (cid:96), (cid:96) )

Original: = 1 / mby the deﬁnition of the Conitzer distribution.
Revised: = 1 / m by the deﬁnition of the Conitzer distribution.

Original: The equality f (1 , m ) = 1 is immediate from the deﬁnition of f ( i, j ) .
Revised: The equality f (1 , m ) = 1 is immediate from the deﬁnition of f ( i, j ) .

Original: To show that f ( i, j )
Revised: To show that f ( i, j )

Original: = 1 / m , we give a proof by induction on j − i .
Revised: = 1 / m , we give a proof by induction on j − i .

Original: The base step holds because foreach (cid:96) ∈ [ m ] we have f ( (cid:96), (cid:96) )
Revised: The base step holds because for each (cid:96) ∈ [ m ] we have f ( (cid:96), (cid:96) )

Original: = 1 / m .
Revised: = 1 / m .

Original: Assume that for all integers x , y such that 1 < x ≤ y < m andy − x < j − i we have f ( x, y )
Revised: Assume that for all integers x , y such that 1 < x ≤ y < m and y − x < j − i we have f ( x, y )

Original: = 1 / m .
Revised: = 1 / m .

Original: The only way for candidates c i , . .
Revised: The only way for candidates c i , . .

Original: c j to be ranked in topj − i + 1 positions under the Conitzer model is that, while generating the vote, we placed candidatesc i , . . .
Revised: c j to be ranked in top j − i + 1 positions under the Conitzer model is that, while generating the vote, we placed candidates c i , . . .

Original: , c j − 1 in top j − i positions and then extended the vote with c j (the probability of this latterstep is 1 / 2 ), or we placed candidates c i +1 , . ..
Revised: , c j − 1 in top j − i positions and then extended the vote with c j (the probability of this latter step is 1 / 2 ), or we placed candidates c i +1 , . .

Original: , c j in top j − i positions and then extended the votewith c
Revised: c j in top j − i positions and then extended the vote with c

Original: i (again, the probability of the latter step is 1 / 2 ).
Revised: i (again, the probability of the latter step is 1 / 2 ).

Original: Thus:
Revised: Thus:

Original: [Equation]
Revised: [Equation]

Original: This proves the claim for f ( i, j ) .
Revised: This proves the claim for f ( i, j ) .

Original: Next we show that f (1 , i ) =
Revised: Next we show that f (1 , i ) =

Original: ( i +1) / 2 m , using induction over i < m .
Revised: ( i +1) / 2 m , using induction over i < m .

Original: For i = 1 we have f (1 , 1) = 593/ m
Revised: For i = 1 we have f (1 , 1) =/ m

Original: = 2 / 2 m , so our claim holds.
Revised: = 2 / 2 m , so our claim holds.

Original: Assume that it holds up to i − 1 .
Revised: Assume that it holds up to i − 1 .

Original: There are only two ways in whichcandidates c 1 , . . .
Revised: There are only two ways in which candidates c 1 , . . .

Original: , c i can be ranked in the top i positions: Either we ﬁrst place c 1 , . . .
Revised: , c i can be ranked in the top i positions: Either we ﬁrst place c 1 , . . .

Original: c i − 1 in topi − 1 positions and then extend the vote with c
Revised: c i − 1 in top i − 1 positions and then extend the vote with c

Original: i (the latter step has probability 1 , since there is nocandidate to the left of c 1 ), or we ﬁrst place candidates c 2 , . . .
Revised: i (the latter step has probability 1 , since there is no candidate to the left of c 1 ), or we ﬁrst place candidates c 2 , . . .

Original: , c i in the top i − 1 positions and thenextend the vote with c 1 (the latter step has probability 1 / 2 , since the vote could also be extended withc i +1 ).
Revised: , c i in the top i − 1 positions and then extend the vote with c 1 (the latter step has probability 1 / 2 , since the vote could also be extended with c i +1 ).

Original: Thus, we have:
Revised: Thus, we have:

Original: [Equation]
Revised: [Equation]

Original: This completes the proof for 
Revised: This completes the proof for f (1 , i ) .

Revised: The expression for f ( j, m ) can be derived by symmetry: we have f ( j, m ) =

Revised: f (1 , m − j + 1) .

Original: Theorem 3.5.
Revised: Theorem 3.5.

Original: Let c 1 (cid:67) · · · (cid:67)
Revised: Let c 1 (cid:67) · · · (cid:67)

Original: c m be the societal axis, where m is an even number, and let v be arandom vote sampled from D ConSP for this axis.
Revised: c m be the societal axis, where m is an even number, and let v be a random vote sampled from D ConSP for this axis.

Original: For j ∈ [ m 2 ] and i ∈ [ m ] we have:P [pos v [Equation] c j
Revised: For j ∈ [ m 2 ] and i ∈ [ m ] we have:

Original: [Equation]

Original: = i ] =

Original:   2 /

Original: [Equation]

Original: m if i < j , ( j

Original: [Equation]

Original: ) / 2 m if i

Original: [Equation]

Original: j , 1 /

Original: [Equation]

Original: m if j

Original: [Equation]

Original: i

Original: [Equation]

Original: [Equation]

Original: if

Original: [Equation]
Revised: [Equation]

Original: 1 , if i

Original: [Equation]

Original: j [Equation] m . Further, for each candidate c j ∈ C and each position i ∈ [ m ] we have P [pos v ( c j ) =
Revised: Further, for each candidate c j ∈ C and each position i ∈ [ m ] we have P [pos v ( c j ) =

Original: i ] 
Revised: i ] =

Original: = 604 P [pos v ( c m − j +1 )
Revised: P [pos v ( c m − j +1 )

Original: = i ] .
Revised: = i ] .

Original: Proof.
Revised: Proof.

Original: Consider a candidate c j , j ∈ [ m 2 ] , and a position i ∈ [ m ] .
Revised: Consider a candidate c j , j ∈ [ m 2 ] , and a position i ∈ [ m ] .

Original: We proceed by case analysis:1.
Revised: We proceed by case analysis:

Original: If i < j, then there are two ways to generate a vote with c j in position i : Either candidatesc j +1 , . .
Revised: If i < j then there are two ways to generate a vote with c j in position i : Either candidates

Original:  , c i + j − 1 are ranked in the top i − 1 positions or candidates c j − i +1 , . . .
Revised: c j +1 , . . , c i + j − 1 are ranked in the top i − 1 positions or candidates c j − i +1 , . . .

Original: , c j − 1 areranked in the top i − 1 positions. 
Revised: , c j − 1 are ranked in the top i − 1 positions. In both cases, c j is ranked in the i -th position with probability indeed , we have

Revised: i

Revised: +

Original: In both cases, c j is ranked in the i -th position withprobability 1 / 2 (indeed, we have i + j < m/ 2 + m/ 2 = m , so in the former case both c jandc i + j could have been placed in position i , andalso j − i ≥ 1 , so in the latter case both cjand c j − i could have been placed in position i ).
Revised: j < m 2 + m 2 = m , so in the former case both c j and c i + j

Revised: could have been placed

Revised: in position i , and also

Revised: j

Original: Thus we have:
Revised: − i ≥ 1 , so in the latter case both c j and c j − i could have been

Revised: placed

Revised: in

Revised: position

Revised: i )

Revised: Thus we have:

Original: [Equation]
Revised: [Equation]

Original: If i = j , then either candidates c 1 , . . .
Revised: If i = j , then either candidates c 1 , . . .

Original: , c j − 1 are ranked in the top j − 1 positions and thevote is extended with c j ( with probability 1 ), or candidates c j +1 , . . .
Revised: , c j − 1 are ranked in the top j − 1 positions and the vote is extended with c j (with probability 1 ), or candidates c j +1 , . . .

Original: , c 2 j − 1 are ranked inthe top j − 1 positions and the vote is extended with c j (with probability 1 / 2 ).
Revised: , c 2 j − 1 are ranked in the top j − 1 positions and the vote is extended with c j (with probability 1 / 2 ).

Original: Thus, wehave:
Revised: Thus, we have:

Original: [Equation]
Revised: [Equation]

Original: If j < i < m − j + 1 then there is only one possibility for c j to be ranked i
Revised: If j < i < m − j + 1 then there is only one possibility for c j to be ranked i

Original: -th:
Revised: -th:

Original: It must be that candidates c j +1 , . . .
Revised: It must be that candidates c j +1 , . . .

Original: , c j + i − 1 are ranked in the top i − 1 positions and the vote is extended with c j (which happens with probability 12 because j + i ≤ m ).
Revised: , c j + i − 1 are ranked in the top i − 1 positions and the vote is extended with c j (which happens with probability 12 because j + i ≤ m ).

Original: Thus, we have:
Revised: Thus, we have:

Original: [Equation]
Revised: [Equation]

Original: If i = m − j + 1 , the analysis is similar to the case i
Revised: If i = m − j + 1 , the analysis is similar to the case i

Original: = j : candidates c j +1 , . .
Revised: = j : candidates c j +1 , . .

Original:  c m mustbe ranked in the top m − j positions, in which case c j gets ranked in the ( m − j + 1) -stposition
Revised: , c m must be ranked in the top m − j positions, in which case c j gets ranked in the ( m − j + 1) -st

Original:  (with probability 1 ).
Revised: position (with probability 1 ).

Original: Thus, we have:
Revised: Thus, we have:

Original: [Equation]
Revised: [Equation]

Original: If i > m − j + 1 then P [pos v ( c j ) = i ] = 0 , because both to the left of c j and to the right ofc j there are fewer than i − 1 candidates.
Revised: If i > m − j + 1 then P [pos v ( c j ) = i ] = 0 , because both to the left of c j and to the right of c j there are fewer than i − 1 candidates.

Original: The fact that for each candidate c j ∈ C and each position i ∈ [ m ] we have P [pos v ( c j ) = i ] 
Revised: The fact that for each candidate c j ∈ C and each position i ∈ [ m ] we have P [pos v ( c j ) = i ] =

Original: = 625 P [pos v ( c m − j +1 )
Revised: P [pos v ( c m − j +1 )

Original: =
Revised: =

Original: i ] follows directly from the symmetry of the Conitzer distribution and the factthat m is even.
Revised: i ] follows directly from the symmetry of the Conitzer distribution and the fact that m is even.

Original: E Missing Material From Section 3. 
Revised: D Missing Material From Section 3.

Original: Proposition E.1 (OEIS Foundation Inc. (2020)) .
Revised: Proposition D.1 (OEIS Foundation Inc. (2020)) .

Original: There is an algorithm that computes S ( m, k ) usingat most polynomially many operations.
Revised: There is an algorithm that computes S ( m, k ) using at most polynomially many operations.

Original: Proof.
Revised: Proof.

Original: First, we note that for each m (cid:48) ∈ [ m ] we have S ( m (cid:48) , 0)
Revised: First, we note that for each m (cid:48) ∈ [ m ] we have S ( m (cid:48) , 0)

Original: = 1 .
Revised: = 1 .

Original: Further, for each m (cid:48) ∈ [ m ] andk (cid:48) ∈
Revised: Further, for each m (cid:48) ∈ [ m ] and k (cid:48) ∈

Original: [ k ] 0 the following recursion holds:
Revised: [ k ] 0 the following recursion holds:

Original: [Equation]
Revised: [Equation]

Original: Using these two facts and standard dynamic programming, we can compute S ( m, k ) using O ( mk )arithmetic operations.
Revised: Using these two facts and standard dynamic programming, we can compute S ( m, k ) using O ( mk ) arithmetic operations.

Original: Since k is at most O ( m 2 ) , the running time is at most O ( m 3 ) .
Revised: Since k is at most O ( m 2 ) , the running time is at most O ( m 3 ) .

Original: Lemma E.2.
Revised: Lemma D.2.

Original: There is an algorithm that computes T ( m, k, j, i ) in polynomial time with respect tom .
Revised: There is an algorithm that computes T ( m, k, j, i ) in polynomial time with respect to m .

Original: Proof.
Revised: Proof.

Original: Our algorithm is based on dynamic programming.
Revised: Our algorithm is based on dynamic programming.

Original: Fix some m > 0 , k ∈ [ m ( m − 1) / 2 ] , andj, i ∈ [ m ] .
Revised: Fix some m > 0 , k ∈ [ m ( m − 1) / 2 ] , and j, i ∈ [ m ] .

Original: We claim that:
Revised: We claim that:

Original: [Equation]
Revised: [Equation]

Original: Indeed, let v be a vote over C ( m ) that ranks c m in position i , and let v (cid:48) be its restriction to{ c 1 , . .
Revised: Indeed, let v be a vote over C ( m ) that ranks c m in position i , and let v (cid:48) be its restriction to { c 1 , . .

Original: c m − 1 } .
Revised: c m − 1 } .

Original: Then v can be obtained from v (cid:48) by inserting c m right behind the candidate inposition i − 1 .
Revised: Then v can be obtained from v (cid:48) by inserting c m right behind the candidate in position i − 1 .

Original: If v (cid:48) is at swap distance k (cid:48) from v ∗ m − 1 , then the resulting vote is at swap distancek (cid:48)
Revised: If v (cid:48) is at swap distance k (cid:48) from v ∗ m − 1 , then the resulting vote is at swap distance k (cid:48)

Original: + ( m − i ) from v ∗ m , since c m contributes m −
Revised: + ( m − i ) from v ∗ m , since c m contributes m −

Original: i additional swaps.
Revised: i additional swaps.

Original: Thus, T ( m, k, m, i ) is equal tothe number of votes in L ( C ( m − 1)) at swap distance k − ( m − i ) from v ∗ m − 1 .
Revised: Thus, T ( m, k, m, i ) is equal to the number of votes in L ( C ( m − 1)) at swap distance k − ( m − i ) from v ∗ m − 1 .

Original: Next, we claim that for each j < m , we have:
Revised: Next, we claim that for each j < m , we have:

Original: [Equation]
Revised: [Equation]

Original: To see why this holds, again consider inserting c m at some position in a vote v (cid:48) over { c 1 , . . .
Revised: To see why this holds, again consider inserting c m at some position in a vote v (cid:48) over { c 1 , . . .

Original: c m − 1 } .
Revised: c m − 1 } .

Original: Candidate c j will end up in position i in the resulting vote if (1)
Revised: Candidate c j will end up in position i in the resulting vote if (1)

Original: c j was in position i in v (cid:48) and c m wasinserted after c j , or if (2)
Revised: c j was in position i in v (cid:48) and c m was inserted after c j , or if (2)

Original: c j was in position i − 1 in v (cid:48) and c m was inserted ahead of c j .
Revised: c j was in position i − 1 in v (cid:48) and c m was inserted ahead of c j .

Original: Consideringall positions in which c m can be inserted, we obtain the above equality.
Revised: Considering all positions in which c m can be inserted, we obtain the above equality.

Original: Using the above equalities and the fact that T (1 , 0 , 1 , 1)
Revised: Using the above equalities and the fact that T (1 , 0 , 1 , 1)

Original: = 1 (as there is just a single vote over C (1) ),we can compute T ( m, k, j, i ) by dynamic programming; our algorithm runs in polynomial time withrespect to m .
Revised: = 1 (as there is just a single vote over C (1) ), we can compute T ( m, k, j, i ) by dynamic programming; our algorithm runs in polynomial time with respect to m .

Original: F Missing Material From Section 4F.1 Distance to Compass Matrices
Revised: E Missing Material From SectionE.1 Distance to Compass Matrices

Original: We analyze the distances between our matrices for different numbers of candidates.
Revised: We analyze the distances between our matrices for different numbers of candidates.

Original: In Figure 5 weshow these distances for the Conitzer and Walsh matrices and the compass matrices: For Conitzer,they are nearly constant, and for Walsh they vary signiﬁcantly.
Revised: In Figure 5 we show these distances for the Conitzer and Walsh matrices and the compass matrices: For Conitzer, they are nearly constant, and for Walsh they vary signiﬁcantly.

Original: Indeed, the more candidates wehave, the closer the Walsh matrix is to ID (e.g., for 10 candidates their distance is 0 . 44 , and for 300candidates it is 0 . 09 ).
Revised: Indeed, the more candidates we have, the closer the Walsh matrix is to ID (e.g., for 10 candidates their distance is 0 . 44 , and forcandidates it is 0 . 09 ).

Original: Figure 6 depicts the distance between the frequency matrix for GS/caterpillar and the four compassmatrices, for varying number of candidates.
Revised: Figure 6 depicts the distance between the frequency matrix for GS/caterpillar and the four compass matrices, for varying number of candidates.

Original: As for the matrix for the Walsh model, its distance tothe compass matrices changes as the number of candidates increases: The matrix moves closer andcloser to AN .
Revised: As for the matrix for the Walsh model, its distance to the compass matrices changes as the number of candidates increases: The matrix moves closer and closer to AN .

Original: In Figure 7, we display the distance between the frequency matrix for the Mallows model for differentvalues of the dispersion parameter φ and the compass matrices (in contrast to the previous ﬁgures,we only consider up to 100 candidates, as for more than 100 candidates computing the matrix forthe Mallows model becomes very memory-consuming).
Revised: In Figure 7, we display the distance between the frequency matrix for the Mallows model for different values of the dispersion parameter φ and the compass matrices (in contrast to the previous ﬁgures, we only consider up to 100 candidates, as for more than 100 candidates computing the matrix for the Mallows model becomes very memory-consuming).

Original: Independent of the chosen value of thedispersion parameter, the distance of the respective matrix to the four compass matrices changes [Figure] i s [Figure]
Revised: Independent of the chosen value of the dispersion parameter, the distance of the respective matrix to the four compass matrices changes signiﬁcantly when we increase the number of candidates.

Original: d i s t a

Original: [Figure]

Original: ID

Original: UN

Original: Conitzer AN

Original: ST

Original: Walsh

Original: [Figure]

Original: [Figure]

Original: [Figure] varying number of candidates. signiﬁcantly when we increase the number of candidates. In fact, for any ﬁxed dispersion parameter φ ,the resulting matrix will always move closer and closer to ID as the number of candidates increases.
Revised: In fact, for any ﬁxed dispersion parameter φ , the resulting matrix will always move closer and closer to ID as the number of candidates increases.

Original: In contrast, if we use the normalized version of the Mallows model, the matrices remain more orless at a constant distance from the compass matrices.
Revised: In contrast, if we use the normalized version of the Mallows model, the matrices remain more or less at a constant distance from the compass matrices.

Original: Figure 8 shows the distance of the frequencymatrix of the normalized version of the Mallows model for different values of norm - φ , as the numberof candidates increases.
Revised: Figure 8 shows the distance of the frequency matrix of the normalized version of the Mallows model for different values of norm - φ , as the number of candidates increases.

Original: F.2 Distances of Pairs of Vote Distributions on the Skeleton Map
Revised: E.2 Distances of Pairs of Vote Distributions on the Skeleton Map

Original: As we have observed above, some vote distributions stay more or less at constant normalizedpositionwise distance from the four compass matrices.
Revised: As we have observed above, some vote distributions stay more or less at constant normalized positionwise distance from the four compass matrices.

Original: This raises the question whether thesematrices also stay at a constant normalized positionwise distance from each other.
Revised: This raises the question whether these matrices also stay at a constant normalized positionwise distance from each other.

Original: This would implythat the data on which the skeleton map is based is independent of the number of candidates, andthus that the map is likely to look very similar for different numbers of candidates.
Revised: This would imply that the data on which the skeleton map is based is independent of the number of candidates, and thus that the map is likely to look very similar for different numbers of candidates.

Original: To check this,we conducted the following experiment.
Revised: To check this, we conducted the following experiment.

Original: We put together a set of vote distributions/matrices that do numberofcandidates m a x i m u m p a i r
Revised: We put together a set of vote distributions/matrices that do not structurally change when increasing the number of candidates (like the change happening for the Walsh model). First, we add the four compass matrices and the frequency matrix of the Conitzer model. Second, we add the frequency matrices of different variants of the Mallows model (similar as on the skeleton map as described in Section 4): the normalized Mallows model, the normalized Mallows model where the central vote is reversed with probability 1 / 2 , the normalized Mallows model

Revised: [Figure]

Original: w i s e d i ff e r e n ce absolutedifferencerelativedifference
Revised: i s [Figure] d i s t a [Figure]

Revised: ID

Revised: UN

Revised: Conitzer AN

Revised: ST

Revised: Walsh

Revised: [Figure]

Revised: [Figure]

Revised: [Figure]

Revised: varying number of candidates.

Revised: where the central vote is reversed with probability 1 / 4 , and the distribution where we ﬁrst sample a vote v from the Conitzer distribution and then sample the ﬁnal vote from the normalized Mallows model with v as the central vote.

Revised: For each of these variants, we include their frequency matrix for norm - φ ∈ { 0 . 2 , 0 . 4 , 0 . 6 , 0 . 8 } .

Revised: For each pair of matrices from the created set, we compute their normalized positionwise distance for 100 candidates.

Original: [Figure]
Revised: Subsequently, for m ∈ { 4 , 6 , . . . , 98 , 100 } candidates, we compute the normalized positionwise distance of the frequency matrices of the two considered models for this number of candidates as well as the absolute and relative difference between their normalized distance for m and 100 candidates (where we normalize by their normalized distance for 100 candidates). Finally, for each m ∈ { 4 , 6 , . . . , 98 , 100 } , we take the maximum over the computed absolute/relative differences for all pairs of matrices. In Figure 9, we present these maxima for all considered values of m .

Original: [Figure]
Revised: Examining the maximum absolute difference (the blue line in Figure 9), what stands out is that for 20 or more candidates the normalized positionwise distance of any pair of considered vote distributions/matrices differs only by at most 0 . 0287 from the pair’s normalized positionwise distance for 100 candidates (for 10 or more candidates the error is at most 0 . 0547 ). As the diameter of our numberofcandidates m a x i m u m p a i r

Revised: w i s e d i ff e r e n ce absolutedifferencerelativedifference

Original: ID [Figure]
Revised: ID UN AN

Original: ST
Revised: ST

Revised: WAL

Original: [Figure]
Revised: [Figure]

Original: [Figure]

Original: [Figure] [Figure] [Figure] [Figure]
Revised: MID

Original: [Figure]
Revised: [Figure]

Revised: candidates 0.

Original: ID UN
Revised: ID UN

Original: AN
Revised: AN

Original: ST
Revised: ST

Original: WAL
Revised: WAL

Original: CON
Revised: CON

Original: CAT
Revised: CAT

Original: MID   5  (
Revised: MID   5  (

Original: [Figure]
Revised: [Figure]

Original:  10 candidates 0.
Revised: ) 10 candidates 0.

Original: ID 
Revised: ID UN

Original: [Figure]

Original: AN
Revised: AN

Original: ST
Revised: ST

Original: WAL
Revised: WAL

Original: CON
Revised: CON

Original: CAT
Revised: CAT

Original: MID (c) 25 candidates 0.
Revised: MID (c) 25 candidates 0.

Original: [Figure]

Original: ID
Revised: ID

Original: [Figure]
Revised: [Figure]

Original: not structurally change when increasing the number of candidates (like the change happening forthe Walsh model).

Original: First, we add the four compass matrices and the frequency matrix of the Conitzermodel.

Original: Second, we add the frequency matrices of different variants of the Mallows model (similaras on the skeleton map as described in Section 4): the normalized Mallows model, the normalized

Original: Mallows model where the central vote is reversed with probability 1 / 2 , the normalized Mallows modelwhere the central vote is reversed with probability 1 / 4 , and the distribution where we ﬁrst sample avote v from the Conitzer distribution and then sample the ﬁnal vote from the normalized Mallowsmodel with v as the central vote.

Original: For each of these variants, we include their frequency matrix fornorm - φ ∈ { 0 . 2 , 0 . 4 , 0 . 6 , 0 . 8 } . For each pair of matrices from the created set, we compute their normalized positionwise distance for100 candidates. Subsequently, for m ∈ { 4 , 6 , . . . , 98 , 100 } candidates, we compute the normalizedpositionwise distance of the frequency matrices of the two considered models for this number ofcandidates as well as the absolute and relative difference between their normalized distance for m and100 candidates (where we normalize by their normalized distance for 100 candidates).
Revised: AN

Original: Finally, foreach m ∈ { 4 , 6 , . . . , 98 , 100 } , we take the maximum over the computed absolute/relative differencesfor all pairs of matrices. In Figure 9, we present these maxima for all considered values of m . Examining the maximum absolute difference (the blue line in Figure 9), what stands out is thatfor 20 or more candidates the normalized positionwise distance of any pair of considered votedistributions/matrices differs only by at most 0 .
Revised: [Figure]

Original: 0287 from the pair’s normalized positionwise distancefor 100 candidates (for 10 or more candidates the error is at most 0 . 0547 ).

Original: As the diameter of ourspace has at least length 1 , this change is quite small, and the considered vote distributions indeedremain at nearly the same distance for more than 20 candidates.
Revised: space has at least length 1 , this change is quite small, and the considered vote distributions indeed remain at nearly the same distance for more than 20 candidates.

Original: Considering the relative difference(the red line in Figure 9), the picture appears to be a bit worse: for 20 or more candidates, thenormalized positionwise distance of any pair of considered vote distributions/matrices differs at mostby 11 .
Revised: Considering the relative difference (the red line in Figure 9), the picture appears to be a bit worse: for 20 or more candidates, the normalized positionwise distance of any pair of considered vote distributions/matrices differs at most by 11 .

Original: 44 % from their normalized positionwise distance for 100 candidates.
Revised: 44 % from their normalized positionwise distance for 100 candidates.

Original: Nevertheless, this valueis still relatively low, indicating an overall high robustness of the normalized positionwise distancesof each pair of considered distributions with respect to the number of candidates.
Revised: Nevertheless, this value is still relatively low, indicating an overall high robustness of the normalized positionwise distances of each pair of considered distributions with respect to the number of candidates

Original: F.
Revised: [Figure]

Original: 3 Skeleton Map for Different Number of Candidates
Revised: E.3 Skeleton Map for Different Number of Candidates

Original: After we have provided various arguments for why large parts of the skeleton map are presumablyquite robust with respect to changing the number of candidates in the previous two subsections, in
Revised: After we have provided various arguments for why large parts of the skeleton map are presumably quite robust with respect to changing the number of candidates in the previous two subsections, in

Original: Figure 10, we present the skeleton map for 5 / 10 / 25 / 50 candidates.
Revised: Figure [Figure] we present the skeleton map for 5 / 10 / 25 / 50 candidates.

Original: While the map for 5 candidateslooks a bit different from the other maps, the maps for 10 , 25 , and 50 candidates differ only in thatthe frequency matrix for GS/caterpillar moves closer to AN and that the frequency matrix for the
Revised: While the map for 5 candidates looks a bit different from the other maps, the maps for 10 , [Figure] and 50 candidates differ only in that the frequency matrix for GS/caterpillar moves closer to AN and that the frequency matrix for the

Original: Walsh model moves closer to ID (both phenomena that we have already observed earlier).
Revised: Walsh model moves closer to ID (both phenomena that we have already observed earlier) [Figure] [Figure]

Revised: r

Original: [Figure]
Revised: [Figure]

Revised: ze dpo

Original: [Figure]
Revised: [Figure]

Revised: w i s e

Revised: d

Revised: i s t a n ce m =

Revised: Mean m = 90thquantile m

Revised: = 10thquantile

Original: F.4 Variance of a Vote Distribution
Revised: [Figure] [Figure] numberofvoters no r m a li ze dpo

Revised: [Figure]

Revised: iti on w i s e

Revised: d

Revised: i s t a n ce m =

Revised: Mean m

Original:  As argued in Section 3.1, a frequency matrix of a distribution can be interpreted as a matrix ofan “ideal” election sampled from this distribution.
Revised: [Figure] [Figure] E.4 Variance of a Vote Distribution As argued in Section 3.1, a frequency matrix of a distribution can be interpreted as a matrix of an “ideal” election sampled from this distribution.

Original: In this section we ask how far, on average, theelections sampled from our distributions land from the “ideal” ones.
Revised: In this section we ask how far, on average, the elections sampled from our distributions land from the “ideal” ones.

Original: This distance may also serve asa measure of “diversity” for elections sampled from a given distribution.
Revised: This distance may also serve as a measure of “diversity” for elections sampled from a given distribution.

Original: For each of our distributions, we consider elections with 10 , 25 , and 50 candidates, and vary thenumber of voters from 10 to 200 (with a step of 5 ).
Revised: For each of our distributions, we consider elections with 10 , 25 , and 50 candidates, and vary the number of voters from 10 to 200 (with a step of 5 ).

Original: For each combination of these parameters, wesample 600 elections and, for each election, compute the positionwise distance between its frequencymatrix and the matrix of the respective distribution.
Revised: For each combination of these parameters, we sample 600 elections and, for each election, compute the positionwise distance between its frequency matrix and the matrix of the respective distribution.

Original: We show the results in Figures 11 and 12.
Revised: We show the results in Figures 11 and 12.

Original: Asexpected, for all vote distributions, increasing the number of votes decreases the average distance ofan election from the distribution’s matrix (indeed, in the limit this distance is zero).
Revised: As expected, for all vote distributions, increasing the number of votes decreases the average distance of an election from the distribution’s matrix (indeed, in the limit this distance is zero).

Original: What is moresurprising, this distance does not seem to depend on the number of candidates for the Conitzer, IC ,and GS/balanced distribution.
Revised: What is more surprising, this distance does not seem to depend on the number of candidates for the Conitzer, IC , and GS/balanced distribution.

Original: For the Walsh distribution (and, to a lesser extent, for GS/caterpillar),the sampled elections get slightly closer to the respective matrix as we increase the number ofcandidates.
Revised: For the Walsh distribution (and, to a lesser extent, for GS/caterpillar), the sampled elections get slightly closer to the respective matrix as we increase the number of candidates.

Original: Moreover, if we ﬁx the number of candidates and voters, then for all our distributions theelections sampled from them are, roughly, at the same distance from the distribution’s matrix.
Revised: Moreover, if we ﬁx the number of candidates and voters, then for all our distributions the elections sampled from them are, roughly, at the same distance from the distribution’s matrix.

Original: For anillustration of this effect consider Figure 11a for the Conitzer distribution; there, the 10th quantile(dotted) and the 90th quantile (dashed) only differ by a factor of four.
Revised: For an illustration of this effect consider Figure 11a for the Conitzer distribution; there, the 10th quantile (dotted) and the 90th quantile (dashed) only differ by a factor of four.

Original: Lastly, comparing the plots for
Revised: Lastly, comparing the plots for

Original: IC and GS/balanced (Figure 12), who both have UN as their frequency matrix, the average distance [Figure] r m a li ze dpo s iti on w i s e
Revised: IC and GS/balanced (Figure 12), who both have UN as their frequency matrix, the average distance of elections sampled form one of these two models to UN (which is their frequency matrix) is the same for both distributions.

Original: d i s t a n ce

Original: Conitzer

Original: Walsh

Original: IC GS/balanced

Original: GS/caterpillar

Original: [Figure]

Original: n no r m a li ze d m ea

Original: nd i s t a n ce 0 .0 .0 . of elections sampled form one of these two models to UN (which is their frequency matrix) is thesame for both distributions. However, for IC, the 10th and 90th quantile of the distances of electionsto UN are closer to the average than for GS/balanced, which indicates that IC produces in some senseless varied elections than GS/balanced.
Revised: However, for IC, the 10th and 90th quantile of the distances of elections to UN are closer to the average than for GS/balanced, which indicates that IC produces in some sense less varied elections than GS/balanced.

Original: In Figure 13 we compare the average distances between elections sampled from various vote distributions and the distribution’s matrices (we ﬁx the number of candidates to 50 and vary the number ofvoters).
Revised: In Figure 13 we compare the average distances between elections sampled from various vote distributions and the distribution’s matrices (we ﬁx the number of candidates to 50 and vary the number of voters).

Original: While, on average, IC elections and GS/balanced elections end up at nearly the same distancefrom UN (which is their frequency matrix), Conitzer elections and GS/caterpillar elections end upcloser to their distribution’s matrices, and for Walsh this effect is considerably stronger.
Revised: While, on average, IC elections and GS/balanced elections end up at nearly the same distance from UN (which is their frequency matrix), Conitzer elections and GS/caterpillar elections end up closer to their distribution’s matrices, and for Walsh this effect is considerably stronger.

Original: Overall, itis remarkable that even for 200 voters, for the Conitzer, IC , GS/balanced, and GS/caterpillar, theaverage distance of a sampled election from the respective matrix is still above 0 .
Revised: Overall, it is remarkable that even for 200 voters, for the Conitzer, IC , GS/balanced, and GS/caterpillar, the average distance of a sampled election from the respective matrix is still above 0 .

Original: (so at least 5%of the diameter of the whole space).
Revised: (so at least 5% of the diameter of the whole space).

Original: We also performed the same experiment for the Mallows modelwith different values of the normalized dispersion parameter (see Figure 14): For a varying numberof voters, we depict the average distance of 600 elections with 50 candidates sampled from Mallowsmodel for different values of norm - φ to the distribution’s frequency matrix.
Revised: We also performed the same experiment for the Mallows model with different values of the normalized dispersion parameter (see Figure 14): For a varying number of voters, we depict the average distance of 600 elections with 50 candidates sampled from Mallows model for different values of norm - φ to the distribution’s frequency matrix.

Original: Quite intuitively, themore swaps we make to the central vote (i.e., the higher norm - φ is), the higher is the average distanceof a sampled elections from the distribution’s frequency matrix. 
Revised: Quite intuitively, the more swaps we make to the central vote (i.e., the higher norm - φ is), the higher is the average distance of a sampled elections from the distribution’s frequency matrix. [Figure] w i s e d

Original: It is interesting to contrast the data from Figure 11a with the map of Boehmer et al.
Revised: i s t a n ce Conitzer Walsh IC GS/balanced

Revised: GS/caterpillar

Revised: [Figure]

Revised: no

Revised: [Figure]

Revised: m a li ze

Revised: d m ea nd i s t a n ce 0 .0 .0 .

Revised: It is interesting to contrast the data from Figure 11a with the map of Boehmer et al.

Original: (2021b) in Figure 2.
Revised: (2021b) in Figure 2.

Original: Boehmer et al.
Revised: Boehmer et al.

Original: considered 10 candidates and 100 voters.
Revised: considered 10 candidates and 100 voters.

Original: For these parameters, in Figure 11a we seethat 10% of the elections are still farther from the Conitzer matrix than about 12% of the distancefrom UN to ID .
Revised: For these parameters, in Figure 11a we see that 10% of the elections are still farther from the Conitzer matrix than about 12% of the distance from UN to ID .

Original: This is roughly reﬂected by the size of the area taken by Conitzer elections in
Revised: This is roughly reﬂected by the size of the area taken by Conitzer elections in

Original: Figure 2.
Revised: Figure 2.

Original: Similar observations hold for the other distributions too.
Revised: Similar observations hold for the other distributions too.

Original: While this might be a coincidence,it conﬁrms the value of their map.
Revised: While this might be a coincidence, it conﬁrms the value of their map

Original: G Validation of Location Framework
Revised: F Validation of Location Framework

Original: Again, in this section, distances and dispersion parameters are always normalized.
Revised: Again, in this section, distances and dispersion parameters are always normalized.

Original: To validatewhether our approach from Section 5 correctly identiﬁes the “nature” of an election, we test itscapabilities to ﬁnd for a given election the dispersion parameter of the closest Mallows distributionwith a single central vote.
Revised: To validate whether our approach from Section 5 correctly identiﬁes the “nature” of an election, we test its capabilities to ﬁnd for a given election the dispersion parameter of the closest Mallows distribution with a single central vote.

Original: That is, given an election, we computed for all φ ∈ { 0 , 0 .
Revised: That is, given an election, we computed for all φ ∈ { 0 , 0 .

Original: , 1 }its distance to D v,φ Mal and returned the minimizing φ value.
Revised: , 1 } its distance to D v,φ Mal and returned the minimizing φ value.

Original: We compare the computed value to themaximum-likelihood estimator for the dispersion parameter of the underlying Mallows distributioncomputed from the Kemeny consensus ranking (Mandhani & Meil ˇ a, 2009) in two experiments.
Revised: We compare the computed value to the maximum-likelihood estimator for the dispersion parameter of the underlying Mallows distribution computed from the Kemeny consensus ranking (Mandhani & Meil ˇ a, 2009) in two experiments.

Original: We start by sampling for φ ∈ { 0 , 0 .
Revised: We start by sampling for φ ∈ { 0 , 0 .

Original: , 1 } an election with 10 candidates and 100 voters from
Revised: , 1 } an election with 10 candidates and 100 voters from

Original: Mallows model with dispersion parameter φ and computed estimates for the dispersion parameterbased on our and the Kemeny approach.
Revised: Mallows model with dispersion parameter φ and computed estimates for the dispersion parameter based on our and the Kemeny approach.

Original: For all elections, the returned estimates differ by at most0 .
Revised: For all elections, the returned estimates differ by at most 0 .

Original: Thus, the dispersion parameter returned by our approach is always very close to the maximum- [Figure]
Revised: Thus, the dispersion parameter returned by our approach is always very close to the maximumlikelihood estimate.

Original: sushiirishglasgowﬁgureskatingtshirtcitiesaspenerscyclingtdfcyclinggdispeedskating likelihood estimate.

Original: However, the estimated dispersion parameter might deviate a bit from theoriginally used dispersion parameter:
Revised: However, the estimated dispersion parameter might deviate a bit from the originally used dispersion parameter:

Original: On average, the absolute difference between the dispersionparameter returned by our approach and the underlying dispersion parameter is 0 . 0179{+
+} with themaximum difference being 0 . 069 ; for Kemeny, the average is 0 .
Revised: On average, the absolute difference between the dispersion parameter returned by our approach and the underlying dispersion parameter is 0 . {+
+} with the maximum difference being 0 . 069 ; for Kemeny, the average is 0 .

Original: and the maximum is 0 .
Revised: and the maximum is 0 .

Original: Whileit might seem surprising that the estimated dispersion parameter is different from the underlying one,recall from Appendix F.4 that elections sampled from a vote distribution typically have non-zerodistance from the distribution’s frequency matrix.
Revised: While it might seem surprising that the estimated dispersion parameter is different from the underlying one, recall from Appendix E.4 that elections sampled from a vote distribution typically have non-zero distance from the distribution’s frequency matrix.

Original: To illustrate this idea, we can think of a Mallowsdistribution as a normal distribution placed in the space of elections with the dispersion parameterbeing its mean (in particular as soon as the dispersion parameter is greater zero, all elections have anon-zero probability of being sampled).
Revised: To illustrate this idea, we can think of a Mallows distribution as a normal distribution placed in the space of elections with the dispersion parameter being its mean (in particular as soon as the dispersion parameter is greater zero, all elections have a non-zero probability of being sampled).

Original: So multiple Mallows distributions for different dispersionparameters translate to multiple partly overlapping normal distributions and it might as well happenthat an election sampled from a Mallows distribution with one dispersion parameter is in fact closerto the mean of the Mallows distribution with a different dispersion parameter.
Revised: So multiple Mallows distributions for different dispersion parameters translate to multiple partly overlapping normal distributions and it might as well happen that an election sampled from a Mallows distribution with one dispersion parameter is in fact closer to the mean of the Mallows distribution with a different dispersion parameter.

Original: We also repeated the above experiment to measure the capabilities of our approach to estimatethe parameters of a mixed Mallows distribution.
Revised: We also repeated the above experiment to measure the capabilities of our approach to estimate the parameters of a mixed Mallows distribution.

Original: For each pair of p ∈ { 0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 , 0 . 5 }and φ ∈ { 0 . 05 , 0 . 1 , . . . , 0 .
Revised: For each pair of p ∈ { 0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 , 0 . 5 } and φ ∈ { 0 . 05 , 0 . 1 , . . . , 0 .

Original: 95 } we sampled an election with 10 candidates and 100 voters fromp - D v,φ,φ Mal (i.e., we sample a vote from the Mallows distribution with dispersion parameter φ andsubsequently ﬂip the sampled vote with probability p ).
Revised: 95 } we sampled an election with 10 candidates and 100 voters from p - D v,φ,φ Mal (i.e., we sample a vote from the Mallows distribution with dispersion parameter φ and [Figure] sushiirishglasgowﬁgureskatingtshirtcitiesaspenerscyclingtdfcyclinggdispeedskating subsequently ﬂip the sampled vote with probability p ).

Original: Subsequently we computed for which valueof p ∈ { 0 , 0 .
Revised: Subsequently we computed for which value of p ∈ { 0 , 0 .

Original: . , 0 . 5 } and φ ∈ { 0 , 0 .
Revised: . , 0 . 5 } and φ ∈ { 0 , 0 .

Original: , 1 } the distance of the sampled election is closest tothe frequency matrix of the induced distribution p - D v,φ,φ Mal (the Kemeny consensus ranking can nolonger be used here to provide a maximum-likelihood estimate).
Revised: , 1 } the distance of the sampled election is closest to the frequency matrix of the induced distribution p - D v,φ,φ Mal (the Kemeny consensus ranking can no longer be used here to provide a maximum-likelihood estimate).

Original: The average difference betweenthe estimated and underlying dispersion parameter is 0 . 

Revised: The average difference between the estimated and underlying dispersion parameter is 0 .

Original: and the average difference between theestimated and underlying ﬂipping probability is 0 .
Revised: and the average difference between the estimated and underlying ﬂipping probability is 0 .

Original: For 63 out of the 95 elections is the differencebetween the estimated and underlying dispersion parameter and weight smaller equal 0 .
Revised: For 63 out of the 95 elections is the difference between the estimated and underlying dispersion parameter and weight smaller equal 0 .

Original: (whichis the smallest non-zero difference).
Revised: 05 (which is the smallest non-zero difference).

Original: The error of the estimated dispersion parameter here is aroundthree times larger than for Mallows elections with a single central vote.
Revised: The error of the estimated dispersion parameter here is around three times larger than for Mallows elections with a single central vote.

Original: This can be explained by thefact that for mixtures of Mallows distributions, the “overlap” between different distributions is evenlarger; in fact, some paramterizations even result in the same distributions (e.g., for φ = 1 all ﬂippingprobabilities result in the same distribution).
Revised: This can be explained by the fact that for mixtures of Mallows distributions, the “overlap” between different distributions is even larger; in fact, some paramterizations even result in the same distributions (e.g., for φ = 1 all ﬂipping probabilities result in the same distribution).

Original: Second, while producing good estimates for elections that have been sampled from a Mallowsdistribution is a good sanity check, we are ultimately interested in computing to which distribution(unknown) real-world elections are closest.
Revised: Second, while producing good estimates for elections that have been sampled from a Mallows distribution is a good sanity check, we are ultimately interested in computing to which distribution (unknown) real-world elections are closest.

Original: To do so, we again compare the estimated dispersionparameter for a Mallows distribution with a single central vote computed by our approach with theone estimated via the Kemeny consensus ranking (as described in the beginning of this section);however, this time instead of considering elections sampled from Mallows model, we examine 165real-world elections used by Boehmer et al.
Revised: To do so, we again compare the estimated dispersion parameter for a Mallows distribution with a single central vote computed by our approach with the one estimated via the Kemeny consensus ranking (as described in the beginning of this section) ; however, this time instead of considering elections sampled from Mallows model, we examinereal-world elections used by Boehmer et al.

Original: (2021b)
Revised: (2021b)

Original: (see the data part of Section 5 for details onthe dataset).
Revised: (see the data part of Section 5 for details on the dataset).

Original: The estimated dispersion parameters returned by both methods are highly correlatedwith a Pearson correlation coefﬁcient of 0 .
Revised: The estimated dispersion parameters returned by both methods are highly correlated with a Pearson correlation coefﬁcient of 0 .

Original: 976 and an average difference of 0 .
Revised: 976 and an average difference of 0 .

Original: 017 , median differenceof 0 .
Revised: 017 , median difference of 0 .

Original: 0105 , and maximum difference of 0 . 197 , indicating the power of our approach.
Revised: 0105 , and maximum difference of 0 . 197 , indicating the power of our approach.

Original: Interestingly,the correlation is particularly strong for larger dispersion parameters (see Figure Figure 15 for a plot showing the correlation between the two approaches).
Revised: Interestingly, the correlation is particularly strong for larger dispersion parameters (see Figure Figure 15 for a plot showing the correlation between the two approaches).

Original: Together with the estimated normalizeddispersion parameter, both approaches also return the central order v of the closest Mallows model,which are typically again quite similar: the average swap distance between the two estimators is 2 . 81out of 45 possible swaps.
Revised: Together with the estimated normalized dispersion parameter, both approaches also return the central order v of the closest Mallows model, which are typically again quite similar: the average swap distance between the two estimators is 2 .out of 45 possible swaps.