$$POS(ID_m, ST_m) = 2 \cdot POS(ID_{\frac{m}{2}}, UN_{\frac{m}{2}}) = \frac{2}{3}(\frac{m^2}{4} - 1)$$

**AN<sub>m</sub> and ST<sub>m</sub>:** We now turn to computing the distance between AN<sub>m</sub> = (an<sub>1</sub>, ..., an<sub>m</sub>) and ST<sub>m</sub> = (st<sub>1</sub>, ..., st<sub>m</sub>). As all column vectors of AN<sub>m</sub> are palindromes, each column vector of AN<sub>m</sub> has the same emd distance to all column vectors of ST<sub>m</sub>, i.e., for i ∈ [m] it holds that emd(an<sub>i</sub>, st<sub>i</sub>) = emd(an<sub>i</sub>, st<sub>i'</sub>) for all j, j' ∈ [m]. Thus, the distance between AN<sub>m</sub> and ST<sub>m</sub> is the same for all permutation between the columns of the two matrices. Thus, we again use the identity permutation. We start by computing emd(an<sub>i</sub>, st<sub>i</sub>) for different i ∈ [m] separately distinguishing two cases. Let i ∈ [m/4]. Recall that an<sub>i</sub> has a 0.5 at position i and position m - i + 1 and that st<sub>i</sub> has a 2/m at entries j ∈ [m/2]. We now analyze how to transform an<sub>i</sub> to st<sub>i</sub>. For all j ∈ [i - 1], it is clear that it is optimal that the value 2/m moved to position j comes from position i. The overall cost of this is Σ<sub>j=1</sub><sup>i-1</sup> 2j/m. Moreover, the remaining surplus value at position i (that is, 1/2 - 2j/m) needs to be moved toward the end. Thus, for j ∈ [i + 1, m/4], we move value 2/m from position i to position j. The overall cost of this is Σ<sub>j=1</sub><sup>m/4-i</sup> 2j/m. Lastly, we need to move value 2/m to positions j ∈ [m/4 + 1, m/2]. This needs to come from position m - i + 1. Thus, for each j ∈ [m/4 + 1, m/2], we move value 2/m from position m - i + 1 to position j. The overall cost of this is 1/2 · (m/2 - i) + Σ<sub>j=1</sub><sup>m/4</sup> 2j/m = 1/2(m/2 - i) + m/16 + 1/4. Now, let i ∈ [m/4 + 1, m/2]. For j ∈ [m/4], we need to move value 2/m from position i to position j. The overall cost of this is 1/2 · (i - m/4 - 1) + Σ<sub>j=1</sub><sup>m/4</sup> 2j/m = 1/2 · (i - m/4 - 1) + m/16 + 1/4. For j ∈ [m/4 + 1, m/2], we need to move value 2/m from position m - i + 1 to position j. The overall cost of this is 1/2 · (m/2 - i) + Σ<sub>j=1</sub><sup>m/4</sup> 2j/m = 1/2 · (m/2 - i) + m/16 + 1/4.

Observing that the case i ∈ [m/4 + 1, m] is symmetric to i ∈ [m/4] and the case i ∈ [m/2 + 1, 3m/4] is symmetric to i ∈ [m/4 + 1, m/2] the emd distance between AN<sub>m</sub> and ST<sub>m</sub> can be computed as follows:

$$\begin{aligned}
POS(AN_m, ST_m) &= 2 \cdot \left(A + \frac{1}{2} \cdot \left(\sum_{i=1}^{m/4} \frac{m}{2} - i\right) + \frac{m}{4} \cdot \left(\frac{m}{16} + \frac{1}{4}\right) + \frac{1}{2} \cdot \left(\sum_{i=m/4+1}^{m/2} (i - \frac{m}{4} - 1)\right) + \frac{m}{4} \cdot \left(\frac{m}{16} + \frac{1}{4}\right)\right. \\
&\qquad \left. + \frac{1}{2} \cdot \left(\sum_{i=m/4+1}^{m/2} \frac{m}{2} - i\right) + \frac{m}{4} \cdot \left(\frac{m}{16} + \frac{1}{4}\right)\right) \\
&= \frac{m^2}{48} - \frac{1}{3} + \frac{3m^2-4m}{32} + \frac{m}{2} \cdot \left(\frac{m}{16} + \frac{1}{4}\right) + \frac{m^2-4m}{32} + \frac{m}{2} \cdot \left(\frac{m}{16} + \frac{1}{4}\right) + \frac{m^2-4m}{32} + \frac{m}{2} \cdot \left(\frac{m}{16} + \frac{1}{4}\right) \\
&= \frac{m^2}{48} - \frac{1}{3} + \frac{3m^2-4m}{32} + \frac{3m}{2} \cdot \left(\frac{m}{16} + \frac{1}{4}\right) + \frac{m^2-4m}{16} = \left(\frac{1}{48} + \frac{3}{32} + \frac{3}{32} + \frac{1}{16}\right)m^2 + \left(-\frac{4}{32} + \frac{3}{8} - \frac{4}{16}\right)m \frac{1}{3} = \frac{13}{48}m^2 - \frac{1}{3}
\end{aligned}$$

with

$$A = \sum_{i=1}^{m/4} \left( \sum_{j=1}^{i-1} \frac{2j}{m} + \sum_{j=1}^{i-4} \frac{2j}{m} \right) = (*) = \frac{1}{6} \left( \frac{m^2}{16} - 1 \right) = \frac{1}{2} \left( \frac{m^2}{48} - \frac{1}{3} \right)$$

**ID<sub>m</sub> and AN<sub>m</sub>:** Lastly, we consider ID<sub>m</sub> = (id<sub>1</sub>, ..., id<sub>m</sub>) and AN<sub>m</sub> = (an<sub>1</sub>, ..., an<sub>m</sub>). Note that, for i ∈ [m], id<sub>j</sub> contains a 1 at position i and an<sub>i</sub> contains a 0.5 at position i and position m - i. Note further that for i ∈ [m/2] it holds that an<sub>i</sub> = an<sub>m-i+1</sub>. Fix some i ∈ [m/2]. For all j ∈ [i, m - i + 1] it holds that emd(an<sub>i</sub>, id<sub>j</sub>) = m - 2i + 1/2 and for all j ∈ [i - 1] ∪ [m - i + 2, m] it holds that emd(an<sub>i</sub>, id<sub>j</sub>) > m - 2i + 1/2. That is, for every i ∈ [m], an<sub>i</sub> has the same distance to all column vectors of ID<sub>m</sub> where the one entry lies in between the two 0.5 entries of an<sub>i</sub> but a larger distance to all column vectors of ID<sub>m</sub> where the one entry is above the top 0.5 entry of an<sub>i</sub> or below the bottom 0.5 entry of an<sub>i</sub>. Thus, it is optimal to choose a mapping of the column vectors such that for all i ∈ [m] it holds that an<sub>i</sub> is mapped to a vector id<sub>j</sub> where the one entry of id<sub>j</sub> lies between the two 0.5 in an<sub>i</sub>. This is, among others, achieved by the identity permutation, which we use to compute:

$$POS(ID_m, AN_m) = 2 \sum_{i=1}^{m/2} \left( \frac{1}{2}(m - 2i + 1) \right) = \frac{m}{2}m - \frac{m}{2}(\frac{m}{2} + 1) + \frac{m}{2} = \frac{m^2}{4} \quad \square$$

## Appendix B. Different embeddings and their robustness

In this section, we provide the results of the accuracy experiments from Section 4.3 for the embedding algorithms of Fruchterman-Reingold (FR) [71], Kamada-Kawai (KK) [78], and for Multidimensional Scaling algorithm (MDS) [44,83]. The results for KK from the main body of the paper are repeated in this section, for comparison. Overall, we find that KK is either superior or nearly as good as each of the other ones.

In Fig. 14 we show maps of elections of the 100x100 dataset from Section 4 obtained using the three embedding algorithms. While the three maps in Fig. 14 are different in some ways, for example, the MDS one has the most compact clusters of points and the FR one has the most scattered ones, on the high level they are similar. For example, they all have IC, SPOC, and 4-Sphere elections very close to the UN matrix, they all have a path of Mallows elections forming an arc between the UN and ID matrices, and they all have hypercube elections in the same area and in the same relation with respect to each other (and with respect to the other elections).

### B.1. Correlation between the positionwise and Euclidean distances

For each of the maps from Fig. 14, we have computed the PCC between the vector of the original distances and the vector of the embedded ones. For the KK method the PCC is the highest and is equal to 0.9805, for the MDS method it is equal to 0.9748 and for the FR method it is equal 0.9364. In Table 9 we report average PCC values and standard deviations for the three embedding algorithms
<figure><img src="image_29.png" /><figcaption>Fig. 14. Maps of elections obtained using the (a) Fruchterman-Reingold algorithm (FR), (b) the Multidimensional Scaling algorithm (MDS), and the Kamada-Kawai algorithm (KK). The colors of the dots correspond to the statistical culture from which the elections are generated. Elections generated from the Mallows model use the blue color, and the more pale they are, the closer is the norm-$\phi$ parameter to 0. The urn elections use yellow-red colors, and the more red they are, the larger is the contagion parameter $\alpha$.</figcaption></figure>

<caption>**Table 9**</caption>

Average values of the PCC between the original and embedded distances for collections of datasets of various sizes. After the ± signs we report the standard deviations.

<table><thead><tr><th rowspan="2">dataset</th><th colspan="3">average PCC values</th></tr><tr><th>FR</th><th>MDS</th><th>KK</th></tr></thead><tbody><tr><td>4 × 100</td><td>0.9226 ± 0.0083</td><td>0.9664 ± 0.0031</td><td>0.9661 ± 0.0044</td></tr><tr><td>10 × 100</td><td>0.9095 ± 0.0098</td><td>0.9649 ± 0.0032</td><td>0.9686 ± 0.0061</td></tr><tr><td>20 × 100</td><td>0.9135 ± 0.0105</td><td>0.9657 ± 0.0036</td><td>0.9745 ± 0.0015</td></tr><tr><td>100 × 100</td><td>0.9174 ± 0.0088</td><td>0.9731 ± 0.0051</td><td>0.9735 ± 0.0115</td></tr></tbody></table>

and collections of datasets of various sizes (for our three maps of the 100 × 100 dataset we chose very good embeddings, whose PCC values are superior to the averages reported in Table 2).

## B.2. Monotonicity

Our next evaluation metric is what we call *monotonicity*. The intuition is that if the original distance between elections X and Y is larger than the original distance between elections X and Z, then we expect that the same will hold for the embedded distances.

Formally, for a given embedding summary $Q = (\mathcal{E}, d_M, d_{Euc})$ and a given election $X \in \mathcal{E}$, we define the average monotonicity of this election in the embedding summary to be:
<figure><img src="image_30.png" /><figcaption>Fig. 15. Monotonicity coloring for the three embeddings from Fig. 5. Each election X (i.e., each point X) has a color reflecting its value μ(X) for the respective embedding.</figcaption></figure>

$$ \mu_Q(X) = \frac{1}{l(|X| - 1)} \sum_{Y,Z \in \mathcal{E} \setminus X} \Delta_X(Y, Z), $$

where Δ_X(Y, Z) is equal to 1, if

$$ \operatorname{sgn}(d_{\text{Euc}}(X, Y) - d_{\text{Euc}}(X, Z)) = \operatorname{sgn}(d_{\mathcal{M}}(X, Y) - d_{\mathcal{M}}(X, Z)), $$

and is equal to 0 otherwise. Positive (negative) signs in the above equation mean that both the original and the embedded distances between X and Y were larger (smaller) than the distances between X and Z. The larger is the average monotonicity, the better.

In Fig. 15 we present the three maps from Fig. 5 where each point (election) is colored accordingly to its monotonicity value. The larger (the closer to green) the value, the better, and the lower (the closer to black) the value, the worse. Monotonicity equal to 1 means that all inequalities are maintained after the embedding. For all three maps, the main message is the same: Elections from the IC, SPOC, Mallows, Walsh, and multidimensional Euclidean models are nicely embedded. Then, elections from the single-crossing and group-separable models are still fine, but on average worse than the previously mentioned models. Finally, we have elections from the 1D-Interval, Conitzer, and urn models, which are the worst embedded ones (not counting some caterpillar group-separable elections for MDS embedding, which are obviously wrong).

In Table 10 we provide the total average monotonicity values for our collections of dataset. We see that with respect to monotonicity, KK and MDS embeddings perform best, followed by FR. This is in sync with the results regarding the Pearson correlation coefficient.

We mention that instead of defining monotonicity for triples of elections, we could have defined it for quadruples. Given elections X and Y and elections A and B, we could ask if the fact that the original distance between X and Y is smaller than the original distance between A and B implies the same relation between the embedded distances. This would have lead to a somewhat more
<caption>Table 10</caption>

Total average monotonicity values for collections of datasets of various sizes. After the ± signs we report the standard deviations.

<table><thead><tr><th rowspan="2">dataset</th><th colspan="3">total average monotonicity values</th></tr><tr><th>FR</th><th>MDS</th><th>KK</th></tr></thead><tbody><tr><td>4 × 100</td><td>0.8684 ± 0.0051</td><td>0.8936 ± 0.0044</td><td>0.8968 ± 0.0068</td></tr><tr><td>10 × 100</td><td>0.8637 ± 0.0112</td><td>0.9045 ± 0.0054</td><td>0.9130 ± 0.0065</td></tr><tr><td>20 × 100</td><td>0.8649 ± 0.0105</td><td>0.9091 ± 0.0062</td><td>0.9207 ± 0.0040</td></tr><tr><td>100 × 100</td><td>0.8689 ± 0.0065</td><td>0.9224 ± 0.0076</td><td>0.9225 ± 0.0096</td></tr></tbody></table>

<figure><img src="image_31.png" /><figcaption>Fig. 16. Distortion coloring for the three embeddings from Fig. 5. Each election *X* (i.e., each point *X*) has a color reflecting its value AMR(*X*) for the respective embedding.</figcaption></figure>

accurate analysis, but would require significant amount of computation without providing a clear advantage (in each of our datasets, we would have to analyze nearly $480^4 \approx 53 \cdot 10^9$ quadruples of elections).

### B.3. Distortion

In Fig. 16, we present the map from Fig. 14 with elections colored according to their distortion (i.e., according to their AMR vaults; recall Section 4.3.2). The best distortion is witnessed by the KK embedding, followed by those for FR and MDS. In the cases of KK and FR, urn elections, as well as 1D-Interval and Conitzer ones, have the highest average distortion. This is in sync with the results regarding monotonicity. In case of MDS, we additionally see elections with high average distortion in the middle of the impartial culture cluster and within the highly dimensional Euclidean clusters.

In Table 11 we present the average values of the average distortions for the three embedding methods and datasets with different numbers of candidates. For each dataset, we compute the average AMR value for its elections, as well as the standard deviation. We observe two patterns. The first one is related to the embedding methods: KK is always the best, followed by FR, with MDS being the worst (except for the 4 × 100 dataset). The second pattern is that for KK the more candidates there are, the lower is the distortion value. For MDS and FR this relation is not as clear. Overall, for each method the distances in the embedding are, on average, off by 20 – 30%.
<caption>**Table 11**</caption>

Total average distortion values (i.e., AMR values) for collections of datasets of various sizes. After the ± signs we report the standard deviations.

<table><caption>**Table 12**</caption><thead><tr><th rowspan="2">dataset</th><th colspan="3">total average distortion values</th></tr><tr><th>FR</th><th>MDS</th><th>KK</th></tr></thead><tbody><tr><td>4 × 100</td><td>1.3213 ± 0.0157</td><td>1.3099 ± 0.0076</td><td>1.2612 ± 0.0158</td></tr><tr><td>10 × 100</td><td>1.3119 ± 0.0194</td><td>1.3531 ± 0.0108</td><td>1.2625 ± 0.0125</td></tr><tr><td>20 × 100</td><td>1.2979 ± 0.0195</td><td>1.3545 ± 0.0126</td><td>1.2406 ± 0.0060</td></tr><tr><td>100 × 100</td><td>1.3006 ± 0.0256</td><td>1.3225 ± 0.0194</td><td>1.2119 ± 0.0123</td></tr></tbody></table>

Overview of all election datasets that are part of the Preflib database (as of the end of 2020). “Inst.” stands for the number of elections in the dataset, “m” for the number of candidates and “Type” for the type of the votes in the dataset (“soc” means that all votes are strict complete orders; “soi” means that all votes are strict incomplete orders; “toc” means that all votes are weak incomplete orders).

<table><thead><tr><th>Preflib ID</th><th>Name</th><th>Inst.</th><th>m</th><th>Type</th><th>Selected</th><th>Reason to reject</th></tr></thead><tbody><tr><td>1</td><td>Irish</td><td>3</td><td>9,12,14</td><td>soi</td><td>yes</td><td>-</td></tr><tr><td>2</td><td>Debian</td><td>8</td><td>4-9</td><td>toc</td><td>no</td><td>Too few candidates</td></tr><tr><td>3</td><td>Mariner</td><td>1</td><td>32</td><td>toc</td><td>no</td><td>Too many ties</td></tr><tr><td>4</td><td>Netflix</td><td>200</td><td>3,4</td><td>soc</td><td>no</td><td>Too few candidates</td></tr><tr><td>5</td><td>Burlington</td><td>2</td><td>6</td><td>toi</td><td>no</td><td>Too few candidates</td></tr><tr><td>6</td><td>Skate</td><td>48</td><td>14-30</td><td>toc</td><td>yes</td><td>-</td></tr><tr><td>7</td><td>ERS</td><td>87</td><td>3-29</td><td>soi</td><td>yes</td><td>-</td></tr><tr><td>8</td><td>Glasgow</td><td>21</td><td>8-13</td><td>soi</td><td>yes</td><td>-</td></tr><tr><td>9</td><td>AGH</td><td>2</td><td>7,9</td><td>soc</td><td>no</td><td>Too few candidates</td></tr><tr><td>10.1</td><td>Formula</td><td>48</td><td>22-62</td><td>soi</td><td>no</td><td>Incomplete and few votes</td></tr><tr><td>10.2</td><td>Skiing</td><td>2</td><td>~50</td><td>toc</td><td>no</td><td>Few votes and many ties</td></tr><tr><td>11.1</td><td>Webimpact</td><td>3</td><td>103, 240, 242</td><td>soc</td><td>no</td><td>Too many candidates and too few votes (~5)</td></tr><tr><td>11.2</td><td>Websearch</td><td>74</td><td>100-200, ~2000</td><td>soi</td><td>no</td><td>Too few votes (~4)</td></tr><tr><td>12</td><td>T-shrit</td><td>1</td><td>11</td><td>soc</td><td>yes</td><td>-</td></tr><tr><td>13</td><td>Anes</td><td>19</td><td>3-12</td><td>toc</td><td>no</td><td>Too many ties</td></tr><tr><td>14</td><td>Sushi</td><td>1</td><td>10</td><td>soc</td><td>yes</td><td>-</td></tr><tr><td>15</td><td>Clean Web</td><td>79</td><td>10-50, ~200</td><td>soc</td><td>no</td><td>Too few votes (~4)</td></tr><tr><td>16</td><td>Aspen</td><td>2</td><td>5,11</td><td>toc</td><td>yes</td><td>-</td></tr><tr><td>17</td><td>Berkeley</td><td>1</td><td>4</td><td>toc</td><td>no</td><td>Too few candidates</td></tr><tr><td>18</td><td>Minneapolis</td><td>4</td><td>7,9,379,477</td><td>soi</td><td>no</td><td>Incomplete votes</td></tr><tr><td>19</td><td>Oakland</td><td>7</td><td>4-11</td><td>toc</td><td>no</td><td>Incorrect data (votes like: 1,1,1)</td></tr><tr><td>20</td><td>Pierce</td><td>4</td><td>4,5,7</td><td>toc</td><td>no</td><td>Too few candidates</td></tr><tr><td>21</td><td>San Francisco</td><td>14</td><td>4-25</td><td>toc</td><td>no</td><td>Incorrect data (votes like: 1,1,1)</td></tr><tr><td>22</td><td>San Leonardo</td><td>3</td><td>4,5,7</td><td>toc</td><td>no</td><td>Too few candidates</td></tr><tr><td>23</td><td>Takoma</td><td>1</td><td>4</td><td>toc</td><td>no</td><td>Too few candidates</td></tr><tr><td>24</td><td>MT Dots</td><td>4</td><td>4</td><td>soc</td><td>no</td><td>Too few candidates</td></tr><tr><td>25</td><td>MT Puzzles</td><td>4</td><td>4</td><td>soc</td><td>no</td><td>Too few candidates</td></tr><tr><td>26</td><td>Fench Presidential</td><td>6</td><td>16</td><td>toc</td><td>no</td><td>Approval ballots</td></tr><tr><td>27</td><td>Proto French</td><td>1</td><td>15</td><td>toc</td><td>no</td><td>Approval ballots</td></tr><tr><td>28</td><td>APA</td><td>12</td><td>5</td><td>soi</td><td>no</td><td>Too few candidates</td></tr><tr><td>29</td><td>Netflix NCW</td><td>12</td><td>3,4</td><td>soc</td><td>no</td><td>Too few candidates</td></tr><tr><td>30</td><td>UK labor party</td><td>1</td><td>5</td><td>soi</td><td>no</td><td>Too few candidates</td></tr><tr><td>31</td><td>Vermont</td><td>15</td><td>3-6</td><td>toc</td><td>no</td><td>Approval ballots</td></tr><tr><td>32</td><td>Cujae</td><td>7</td><td>6,32</td><td>soc/soi/toc</td><td>no</td><td>Multiple issues</td></tr><tr><td>33</td><td>San Sebastian Poster</td><td>2</td><td>17</td><td>toc</td><td>no</td><td>Approval ballots</td></tr><tr><td>34</td><td>Cities survey</td><td>2</td><td>36, 48</td><td>soi</td><td>yes</td><td>-</td></tr></tbody></table>

## Appendix C. Selection of preflib datasets

In Table 12 we list the datasets that were available on Preflib when conducting our experiment on real-life elections (at the end of 2020), including information whether we chose to include it in our study and the reason for rejection (in case we deemed a given dataset unsuitable).

### Data availability

The code and data are available. The link is available on the paper's first page (https://github.com/Project-PRAGMA/Journal-Drawing-a-Map-of-Elections)
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