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 analyze several 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 captures the probability that a given candidate appears in a given position in a sampled vote.
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, evaluate its 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: Finally, we develop a general and uniﬁed framework for learning the distribution of real-world preferences using the frequency 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 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).
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 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
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 work often 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 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.
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 the available 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 introduced by Szufa et al.
Revised: One way to achieve this is to form a so-called “map of elections,” recently introduced by Szufa et al.

Original: and extended by Boehmer et al.
Revised: 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 measure between elections.
Revised: First, we ﬁx a distance measure between elections.

Original: 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.
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 so that the Euclidean distances between points are approximately equal to the distances between the respective 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 indicate their origin (e.g., the distribution from which a given election was sampled); see
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

Original: Figure 2 later in the paper for an example of such a map.
Revised: Figure 2 later in the paper for an example of such a map.

Original: A location of an election on a map provides useful information 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 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.
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, 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: 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).

Original: 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.
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 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.
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 a way 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, which we 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 placing them 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 aroundpoints 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 the fact that prior work on the “map of elections” framework represented elections by their frequency matrices , 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 that this 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 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: 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.

Original: Our Contribution.
Revised: Our Contribution.

Original: We provide three sets of results.
Revised: We provide three sets of results.

Original: 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.
Revised: 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.

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

Original: 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: 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.

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

Original: In more detail, we work in the setting of preference learning, where we are given an election
Revised: In more detail, we work in the setting of preference learning, where we are given an election

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

Original: Mallows (1957).
Revised: Mallows (1957).

Original: 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).
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 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.
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: Murphy & Martin (2003); Awasthi et al.
Revised: Murphy & Martin (2003); Awasthi et al.

Original: (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).
Revised: (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: Using frequency matrices, we offer a more general approach.
Revised: Using frequency matrices, we offer a more general approach.

Original: 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.
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 problem may be quite challenging, our approach offers a uniform framework for dealing with multiple kinds of distributions at the same time.
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 ﬁnd that for the case of the Mallows distribution, our approach learns parameters very similar to those established using 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.

Original: The source code used for the experiments is available in a GitHub repository 1 .
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 denote 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 × 1 matrix (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 to total orders over C as preference orders (or, equivalently, votes ), and denote the set of all preference orders 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 a is 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 of vote 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 ) to be the fraction of the votes from V that 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 columns sums up to one.
Revised: Frequency matrices are bistochastic, i.e., their entries are nonnegative and each of their rows and columns sums up to one.

Original: Example 2.1.
Revised: Example 2.1.

Original: Let E =
Revised: Let E =

Original: ( C, V ) be an election with candidate set C
Revised: ( 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 .
Revised: = { a, b, c, d, e } and four voters, v 1 , v 2 , v 3 , and v 4 .

Original: Below, we show the voters’ preference orders (on the left) and the election’s frequency matrix (on the right).
Revised: Below, we show the voters’ preference orders (on the left) and the election’s frequency matrix (on the right).

Original: v 1 : a (cid:31) b (cid:31)
Revised: v 1 : a (cid:31) b (cid:31)

Original: c
Revised: c

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

Original: (cid:31) a (cid:31) e, v 3 : d (cid:31) e (cid:31) c
Revised: (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
Revised: (cid:31) b (cid:31) a, v 4 : b (cid:31) c

Original: (cid:31) a (cid:31) d (cid:31) e.
Revised: (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 / 
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 / 

Original: 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.
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 corresponds 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 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).
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 (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).
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: / 4 1 / 4 1 / 4 1 / 4 1 /1 / 4 1 / 4 1 / 4 1 / 4 1 / 4 1 / 4 1 / 4 1 /
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:  , 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: 0
Revised: 0

Original:  , 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 /  .
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 entries that 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 x into 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 δ amount 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 by a 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 will normalize 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 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)
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)

Original: Txa xb xc
Revised: Txa xb xc

Original: d (c) T
Revised: 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 their frequency 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.
Revised: Boehmer et al.

Original: (2021b) showed that if we take their afﬁne combination Z = αX + (1 − α ) Y ( 0 ≤ α ≤ 1 ) then nPOS
Revised: (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 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).
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 elections (Black, 1958), and group-separable elections (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 motivation behind them, see the original papers and the overviews 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 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.
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: The axis 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 2.2.
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) if it 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.

Original: 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 .
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 will be 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 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.
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 its frontier 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 2.3.
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 whose leaves 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 2.3 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 2.4.
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 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 .
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 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 ).
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 and
Revised: Formally, we require that for each v ∈ L ( C ) it holds that D ( v ) ≥ 0 and

Original: (cid:80) v ∈L ( C ) D ( v )
Revised: (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 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).
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 candidate set 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: 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 )
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: 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.
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 variable equal to the frequency matrix of an n -voter election generated according to D , then it holds that lim 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 by elements 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 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 .
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 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.
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. 2.
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 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.
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 similar terminology 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 , and if 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 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:
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 position i ∈ [ 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 r precedes 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 ranks c
Revised: Then v ranks c

Original: j above each of c j +1 , . . .
Revised: 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 be that 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 that x 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 is reversed, 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 be ranked exactly in position i it must hold that i > m − j and exactly ( i − 1) − ( m − j ) nodes among x 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) .
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: 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 .
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: 3.3 From Caterpillars to Single-Peaked Preferences.
Revised: 3.3 From Caterpillars to Single-Peaked Preferences.

Original: 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.
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 another ranking over C such that, 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 with respect 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: 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 .
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) 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 .
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: 3.4 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 due to 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) has equal 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 transposed matrix 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 probability that 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: 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.
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 −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.
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 a random 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: P [pos v ( c m − j +1 )
Revised: P [pos v ( c m − j +1 )

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

Original: 3.5 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 ∗ over m 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 this distribution (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 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 .
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 that lie 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 ranked in position j in the central vote, we want to compute the probability that he or she appears in a given 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 a position 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 votes in L ( C ( m ))
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 ))

Original: 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 .
Revised: 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 in the 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 a parameter 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 also compute 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] , we deﬁne 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
Revised: Mal as p · D v,φ Mal + (1 − p ) · D rev( v ) ,ψ Mal , i.e., with probability p

Original: we sample a vote from D v,φ Mal and with probability 1 − p
Revised: 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 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 ) .
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 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).
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 in position 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 a vote 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 #freq( D (cid:48) ) =
Revised: In terms of matrix multiplication, this means that #freq( D (cid:48) ) =

Original: #freq( D v ∗ ,φ Mal ) · #freq( D ) , where v ∗ is c 1 (cid:31) · · · (cid:31)
Revised: #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 and normalized 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 mean the positionwise distance (occasionally we will also refer to the Euclidean distances on our maps, but we 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 dispersion parameters 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 Boehmer et 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 (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
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 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).
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 matrices we 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 to the positionwise distances as possible (we use the MDS algorithm, as implemented in the Python sklearn.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).

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

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  lines between some points/matrices show their positionwise distances; to maintain clarity, we only provide some of them).
Revised: 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 expect its 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, seems to 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 is the 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 (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
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 1 . 05 .
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 relations between 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 by almost 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, while one 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: This conclusion is particularly valuable when we compare the skeleton map and the map of Boehmer et 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) appear in 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 the skeleton map and Conitzer elections in the map of Boehmer et al., but even this difference is not huge.
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 for the 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 results allow us to make this claim formal.
Revised: (2021b) make a similar observation; our results allow us to make this claim formal.

Original: In Appendix E, we analyze the robustness of the skeleton map with 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 distance between 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 vote distributions 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 ) and the 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 on showcasing 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, we verify that for Mallows model our approach is strongly correlated with existing maximumlikelihood approaches.
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 distributions to 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 frequency matrix of a given vote distribution, we can obtain an approximate matrix by sampling sufﬁciently many votes 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
Revised: In principle, it is also possible to deal with distributions

Original: [Table]
Revised: [Table]

Original: 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.
Revised: 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 want to compute the distance of such a distribution, we sample sufﬁciently many elections and 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 this approach 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 to the 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 the distribution corresponding to the closest matrix.
Revised: We select the distribution corresponding to the closest matrix.

Original: Data.
Revised: Data.

Original: We consider elections from the real-world datasets 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 eleven different real-world election datasets (so, altogether, they generated 165 elections, 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 datasets of political elections (from North
Revised: They used four datasets of political elections (from North

Original: 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).
Revised: 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 our analysis for seven illustrative and particularly interesting elections in Table 1 and also include them in 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 & Meil ˇ a, 2009)) that given an election provides the most likely dispersion parameter of 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 average absolute difference between the dispersion parameter calculated by our approach and the MLE is only 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 F 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 three vote 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 frequency matrix of the closest distribution 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 average distance of an election sampled from impartial culture ( D IC , with 10 candidates andvoters) to the 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 E.4 for a discussion of this and how 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 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 .
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.are still quite different 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 are best captured by the Mallows model rather than 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 closer to 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 elections close 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: 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
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: 3are three examples of elections which are well captured by the Mallows model and distributed over the entire 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 normalized distance 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 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)).
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: there are only three elections for which it 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 emphasize 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 is very 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 very close 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 structure among 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
Revised: However, as witnessed by the fact that the frequency matrix of GS/balanced (which is highly structured) is

Original: UN , such phenomena can happen.
Revised: UN , such phenomena can happen.

Original: 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.
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 best by 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 several well-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 how these 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 have demonstrated 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 vote distributions, such as the Plackett–Luce model (we conjecture that its frequency matrix is computable in 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: It would also be interesting to use our approach to ﬁt more complex models, such as mixtures of Mallows models, to real-world elections.
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: Further, it may be interesting to use expected frequency matrices to reason about the asymptotic behavior of our models.
Revised: Further, it may be interesting to use expected frequency matrices to reason about the asymptotic behavior of our models.

Original: 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.
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: Acknowledgments
Revised: Acknowledgments

Original: NB was supported by the DFG project MaMu (NI 369/19) and by the DFG project ComSoc-MPMS (NI 369/22).
Revised: NB was supported by the DFG project MaMu (NI 369/19) and by the DFG project ComSoc-MPMS (NI 369/22).

Original: 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).A Missing Material From Section 3.  Theorem 3.2. 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 .
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: Proof.

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

Original: = { c 1 , . .

Original: , c m } be its candidate set.

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

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

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

Original: 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 .

Original: 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

Original: ( x, t ) using dynamic programming.

Original: 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] .

Original: 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 for each t ∈ { 0 } ∪ [ m − 1] we have:

Original: [Equation]

Original: 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 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.

Original: 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).

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

Original: B Missing Material From Section 3.

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

Original: Lemma B.1.

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

Original: , c m ) .

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

Original: , c m } belongs to the support of

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

Original: 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 ; 2.

Original: (2) for all c

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

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

Original: (3) for all c

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

Original: 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 , 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

Original: D T GS and single-peaked votes.

Original: Theorem 3.3.

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

Original: , 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

Original: 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 , . .

Original: c m ) .

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

Original: Proof.

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

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

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

Original: z .

Original: 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 ) .

Original: 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 <

Original: z and v is sampled from D T GS , by Lemma 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 ) .

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

Original: C Missing Material From Section 3.

Original: Recall that our candidate set is C

Original: = { c 1 , . .

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

Original: We consider the Conitzer distribution.

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

Original: 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 using them we establish the frequency matrix for the Conitzer distribution.

Original: Proposition C.1.

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

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

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

Original: = 1 , f

Original: ( i, j )

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

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

Original: ( m − j +2) / 2 m .

Original: 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) )

Original: = 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 ) .

Original: To show that f ( i, j )

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

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

Original: = 1 / m .

Original: 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 .

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

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

Original: c i , . . .

Original: , 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

Original: c i +1 , . .

Original: 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 ).

Original: Thus:

Original: [Equation]

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

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

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

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

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

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

Original: 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 , . . .

Original: 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 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 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:

Original: [Equation]

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

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

Original: f (1 , m − j + 1) .

Original: Theorem 3.5.

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

Original: 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:

Original: [Equation]

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

Original: i ] =

Original: P [pos v ( c m − j +1 )

Original: = i ] .

Original: Proof.

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

Original: We proceed by case analysis:

Original: If i < j then there are two ways to generate a

Original: vote

Original: with c j in position i : Either

Original: candidates

Original: c

Original: j +1 , . .

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

Original: , c j − 1 are ranked in the top i − 1 positions.

Original: In both cases, c j is ranked in the i -th position

Original: with probability indeed

Original: , we have

Original: i

Original: +

Original: j

Original: < m

Original: 2 + m

Original: 2 = m , so in the former case both c j and c i + j

Original: could have been placed

Original: in position i , and also

Original: j

Original: − i ≥ 1 , so in the latter case both c j

Original: and

Original: c j − i

Original: could have been

Original: placed

Original: in

Original: position

Original: i )

Original: Thus we have:

Original: [Equation]

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

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

Original: Thus, we have:

Original: [Equation]

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

Original: -th:

Original: 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 ).

Original: Thus, we have:

Original: [Equation]

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

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

Original: , 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: position (with probability 1 ).

Original: Thus, we have:

Original: [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 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 ] =

Original: P [pos v ( c m − j +1 )

Original: =

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

Original: D Missing Material From Section 3.

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

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

Original: Proof.

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

Original: = 1 .

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

Original: [ k ] 0 the following recursion holds:

Original: [Equation]

Original: 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 ) .

Original: Lemma D.2.

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

Original: Proof.

Original: Our algorithm is based on dynamic programming.

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

Original: [ m ] .

Original: We claim that:

Original: [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 , . .

Original: c m − 1 } .

Original: 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 distance k (cid:48)

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

Original: i additional swaps.

Original: 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:

Original: [Equation]

Original: 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 } .

Original: 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 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 .

Original: 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)

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 with respect to m .

Original: E Missing Material From SectionE.1 Distance to Compass Matrices

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

Original: 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 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 compass matrices, for varying number of candidates.

Original: 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 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 the dispersion parameter, the distance of the respective matrix to the four compass matrices changes signiﬁcantly when we increase the number of candidates.

Original: 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 or less at a constant distance from the compass matrices.

Original: 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: 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 normalized positionwise distance from the four compass matrices.

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

Original: 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.

Original: 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).

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

Original: 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

Original: Mallows model where the central vote is reversed with probability 1 / 2 , the normalized Mallows model

Original: [Figure]

Original: i s

Original: [Figure]

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]

Original: varying number of candidates.

Original: 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.

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

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

Original: Subsequently, for m ∈ { 4 , 6 , . .

Original: . , 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).

Original: Finally, for each m ∈ { 4 , 6 , . . . , 98 , 100 } , we take the maximum over the computed absolute/relative differences for all pairs of matrices.

Original: In Figure 9, we present these maxima for all considered values of m .

Original: 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 .

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

Original: As the diameter of our numberofcandidates m a x

Original: i m u m p a i r

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

Original: ID UN

Original: AN

Original: ST

Original: WAL

Original: [Figure]

Original: MID

Original: [Figure]

Original: candidates 0.

Original: ID UN

Original: AN

Original: ST

Original: WAL

Original: CON

Original: CAT

Original: MID   5  (

Original: [Figure]

Original: ) 10 candidates 0.

Original: ID UN

Original: AN

Original: ST

Original: WAL

Original: CON

Original: CAT

Original: MID (c) 25 candidates 0.

Original: ID

Original: [Figure]

Original: AN

Original: [Figure]

Original: 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, 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.

Original: 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: [Figure]

Original: 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 presumably quite robust with respect to changing the number of candidates in the previous two subsections, in

Original: Figure

Original: [Figure]

Original: we present the skeleton map for 5 / 10 / 25 / 50 candidates.

Original: While the map for 5 candidates looks a bit different from the other maps, the maps for 10 ,

Original: [Figure]

Original: 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)

Original: [Figure]

Original: [Figure]

Original: r

Original: [Figure]

Original: ze dpo

Original: [Figure]

Original: w i s e

Original: d

Original: i s t a n ce m =

Original: Mean m = 90thquantile m

Original: = 10thquantile

Original: [Figure]

Original: [Figure]

Original: numberofvoters no r m a li

Original: ze dpo

Original: [Figure]

Original: iti on w i s e

Original: d

Original: i s t a n ce m =

Original: Mean m

Original: [Figure]

Original: [Figure]

Original: E.4 Variance of a Vote Distribution

Original: 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, the elections sampled from our distributions land from the “ideal” ones.

Original: 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 the number of voters from 10 to 200 (with a step of 5 ).

Original: 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.

Original: 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 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 of candidates.

Original: 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 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

Original: 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: 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 of voters).

Original: 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, 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).

Original: We also performed the same experiment for the Mallows model with different values of the normalized dispersion parameter (see Figure 14):

Original: 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, 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.

Original: [Figure]

Original: w i s e

Original: d

Original: i s t a n ce

Original: Conitzer

Original: Walsh

Original: IC GS/balanced

Original: GS/caterpillar

Original: [Figure]

Original: no

Original: [Figure]

Original: m a li ze

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

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

Original: (2021b) in Figure 2.

Original: Boehmer et al.

Original: considered 10 candidates and 100 voters.

Original: 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

Original: Figure 2.

Original: Similar observations hold for the other distributions too.

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

Original: F Validation of Location Framework

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

Original: 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 .

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

Original: 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 .

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

Original: 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 most 0 .

Original: Thus, the dispersion parameter returned by our approach is always very close to the maximumlikelihood estimate.

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

Original: [-On average, the absolute difference between the dispersion parameter returned by our approach and the underlying dispersion parameter is 0 . 
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Revised: [-On average, the absolute difference between the dispersion parameter returned by our approach and the underlying dispersion parameter is 0 . 
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Original: with the maximum difference being 0 . 069 ; for Kemeny, the average is 0 .

Original: and the maximum is 0 .

Original: 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 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 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 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 .

Original: 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

Original: [Figure]

Original: sushiirishglasgowﬁgureskatingtshirtcitiesaspenerscyclingtdfcyclinggdispeedskating subsequently ﬂip the sampled vote with probability p ).

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

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

Original: , 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 between the estimated and underlying dispersion parameter is 0 . 
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Revised: [-The average difference between the estimated and underlying dispersion parameter is 0 . 
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Original: and the average difference between the estimated and underlying ﬂipping probability is 0 .

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

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

Original: 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 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 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 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)

Original: ; however, this time instead of considering elections sampled from Mallows model, we examinereal-world elections used by Boehmer et al.

Original: (2021b)

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

Original: 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 .

Original: 017 , median difference of 0 .

Original: 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).

Original: 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.