11affiliationtext: Departamento de Ingeniería Industrial, Facultad de
Ingeniería, Universidad de Concepción22affiliationtext: Northeastern
University London33affiliationtext: Center for Collective Learning, ANITI,
Université de Toulouse44affiliationtext: Department of Mathematics and
Statistics, Faculty of Economics – Skopje, Ss. Cyril and Methodius University
in Skopje, Macedonia55affiliationtext: Millennium Nucleus for the Study of
Politics, Public Opinion and Media, Chile66affiliationtext: Departamento de
Administración Pública y Ciencia Política, Universidad de Concepción

Abstract

The simplified hypothesis that an election is polarized as an explanation of
recent electoral outcomes worldwide is centered on perceptions of voting
patterns rather than ideological data from the electorate. While the
literature focuses on measuring polarization using ideological-like data from
electoral studies–which are limited to economically advantageous countries and
are representative mostly to national scales–we argue that, in fact, voting
patterns can lead to mapping effective proxies of citizen divisions on
election day. This paper perspectives two complementary concepts, Election
Polarization (EP) and Election Competitiveness (EC), as a means to understand
voting patterns on Election Day. We present an agnostic approach that relies
solely on election data and validate it using synthetic and real-world
election data across 13 countries in the Eurozone, North America, Latin
America, and New Zealand. Overall, we find that we can label and distinguish
expectations of polarized and competitive elections in these countries, and we
report that EP positively correlates with a metric of political polarization
in the U.S., unlocking opportunities for studies of polarization at the
regional level and for lower/middle-income countries where electoral studies
are available, but surveys are limited.

##  1 Introduction

Was the election polarized? This timely question, frequently overused by
politicians and journalists after unforeseen electoral results, such as the
two Chilean constitution referendums, the 2016-2020 U.S. presidential
election, or the 2023 Argentinian election, has inspired recent work studying
links between election results and polarization [144, 145]. The salience of
studying polarization relies upon its pivotal role in political processes. It
can restrict the government’s ability to respond to threats and overcome
conflicts, potentially leading to democratic breakdowns [160, 153]. Although,
it might help to mobilize the voters [125]. Recently, a strand of literature
argues that some neglected locations by globalization “take revenge” on big
cities by voting for anti-establishment candidates [163, 155, 154] emerging a
polarization that might not be related to preferences related to political
proposals and competencies.

Throughout the last decade, political polarization has increased both in
developed and developing countries. However, the literature lacks to capture
modern polarization with empirical measures using open data [106]. While
several definitions of polarization have been proposed (e.g.,
iyengar2012affect, fiorina2008political), reliable data related to forms of
political polarization, especially at the regional level, are expensive and
challenging to collect. Overall, existing work depends on one-time electoral
survey collected from a small amount of the population, which can only be done
by a few established democracies as the United States [150, 152, 157, 134,
106], or by self-funded surveys [120, 111]. This limitation is significant
because existing metrics, results, and implications are biased toward the
developed world and cannot be generalized to other contexts. Hence, there is a
need for cost-effective and simple metrics that can capture political
polarization (or its proxies) also for less economically advantaged regions
worldwide.

This paper proposes two complementary and agnostic metrics for mapping
political polarization and electoral competition on Election Day using only
observed electoral data. We call them Election Polarization (EP) and Election
Competitiveness (EC). These measures complement each other and allow us to
enhance our understanding of elections and polarization by disaggregating the
footprint of voting patterns. In order to provide a simple metric that can be
applied to virtually any election, we do not explicitly account for ideology
in its conceptualization. Throughout this paper, we demonstrated that it is
feasible to obtain a proxy for polarization relying only on voting patterns.
Our definitions of the measures are inspired by the esteban1994measurement’s
identification-alienation framework and the social identity theory [161]. We
argue that political candidates and parties in democratic countries encompass
a pool of different ideological perspectives on society. Citizens who vote for
the same candidate or political party feel more similar to each other than
those with a different preference. For us, Election Polarization is a
manifestation of a latent form of political polarization in the electorate.
Political parties and candidates take positions and strategies to achieve
their goals. This mobilization is neither spontaneous nor arbitrary: parties
concentrate their efforts on places where they are more likely to succeed in
their goals. These movements are reflected through voting patterns, which is a
rich and openly available input that is accessible for virtually all elections
in democratic countries.

Once the EP and EC measures are constructed, we compare their properties with
traditional polarization and competitiveness measures in the literature. To
validate these metrics towards a diverse overview of multiple electoral
systems worldwide, we perform an extensive empirical analysis by collecting
presidential and parliamentary election data for 13 countries. We collect data
from the U.S., France, and Latin American countries such as Argentina, Brazil,
Chile, Mexico, and Peru for presidential elections. We consider EU countries
such as Belgium, Italy, Germany, and Spain for parliamentary elections. We
also include Canada and New Zealand to consider systems that stem from British
tradition. We do this by evaluating whether we can distinguish theoretical
expectations of politically polarized or competitive elections and regions
using the collected data. We then move to statistically assess the correlation
across the EP and EC and the existent measures. Our analysis suggests that we
can define a polarized election by only knowing whether candidates’
preferences are geographically clusterized. To further validate the metrics,
we perform a robustness check, demonstrating that our results can be
generalized to any aggregation scale (e.g., city, county, state, or country),
can be used for virtually any election type and electoral system (e.g.,
presidential or senate, majoritarian or proportional systems), are robust
towards the number of candidates, the (effective) number of parties, and the
level of abstentions. Altogether, these results provide an interesting setting
for analyzing voters’ behaviors across different nations. To our knowledge,
this is one of the first efforts aiming to develop a comparable measure of
polarization at regional and country levels.

The rest of the paper is organized as follows. The next section discusses
current measures of polarization and competitiveness. Section 3 presents the
theoretical definition of the proposed measures on Election Polarization and
Election Competitiveness. Section 4 presents theoretical and empirical
properties of the measures. Section 5 perspectives on the usefulness of our
metrics for future research, and, finally, Section 6 concludes.

##  2 Theoretical Discussion

Is the population more polarized than in the past? Likely, no other
contemporary debate gets more attention from such diverse disciplines than
this question. Nevertheless, defining polarization is not simple. Broadly, we
define polarization as the division into two or more contrasting groups in a
society. Related literature makes the distinction into several branches, such
as issues [123, 152], geographical [119], socioeconomic [151, 131, 126, 168],
mass & elite [128, 130], public opinion [140], partisan [124], or affective
[121, 117, 135, 136], to name a few. The salience of capturing its temporal
and spatial dynamics relies on their effects on our society.

Polarization is not a simple concept to model, as it is both a state and a
process [123]. Moreover, polarization affects groups differently [114].
Regarding the causes of polarization, there are still gaps in the literature
[148]. Empirical evidence indicates that an increase in inequality at the
local level can be associated with an increase in the probability of
supporting an extreme ideologically political party [167]. Partisan
polarization is associated with greater importance for substantive policy
considerations for citizens’ vote choice in competitive elections [143]. More
recently, affective and elite polarization has been associated with lower
attitudes towards COVID-19 [116]. Nonetheless, not all associations for
polarization have a bad connotation. Previous work has established that
political polarization is positively associated with control of corruption
perception [162, 112] and shi2019wisdom found evidence that more ideologically
polarized editorial teams produce articles of a higher quality than
homogeneous teams.

To measure forms of polarization, related literature has relied on self-
reported surveys [105, 166], social media data [108], election data [106, 137,
103], socio-demographic characteristics [158], network analysis [165, 133],
individual preferences [127], to name a few. For instance, a line of argument
in economics proposes measurements for polarization inspired by the shrinking
of the middle-class’ phenomenon, contrasting with existing metrics in
inequality. A notable example is the seminal contribution of
esteban1994measurement. These authors aim to generalize polarization metrics
upon a common framework, suggesting that society is polarized whether there
are individuals who identify themselves with those having a similar feature
(e.g., income) and are alienated from individuals in other groups
(identification-alienation framework).

Despite the interest it is tracked in academic circles and has far-reaching
real-life implications, there is an important missing puzzle piece in this
literature: existing measures of polarization are not easily accessible and
comparable to understanding its causes, conditions, and effects worldwide and
at a regional scale. Data availability is biased toward the developed world,
focused on recent years, and usually obtained via one-time data collected
through expensive electoral studies and surveys (e.g., diermeier2023dynamics,
dias2022nature,bubeck2022states,lee2021more). While these studies offer a
comprehensive overview of polarization, their economic cost makes it
challenging to gather, at the same time, reliable individual ideological data
and a representative view at regional levels. Furthermore, some simplified
hypotheses that arise during electoral times, such as “the population is more
polarized than in the past” and “the elections are polarized,” do not fit the
traditional metrics of affective, partisan, or political polarization.
Instead, these statements are derived from the voting patterns in the election
itself and are grounded exclusively on the electoral results. Our approach
addresses these limitations and contributes a more globally applicable
perspective on polarization in electoral times.

##  3 Our Measure

Inspired by existing work using election results and building upon the
identification-alienation framework of Esteban and Ray, we propose a simple
and agnostic metric of polarization applicable worldwide only using election
data. The main reason for pursuing this cost-effective approach lies in the
fact that elections in democratic countries represent a rich and reliable data
source on society’s preferences. As transparency guarantees reliability in
democracy, election data is often openly shared at disaggregated levels, and
some widely-known democracy indexes such as the “Democracy Index” [138] use
election outcomes to provide measures of democracy wealth.

Conceptually, Election Polarization (EP) manifests latent citizen divisions on
election day. While we recognize that this conceptualization contrasts with
the traditional view that polarization measures explicitly rely on individual
ideological data, we argue that empirically, this data is challenging and
expensive to collect, especially if we are interested in studying polarization
dynamics at local levels and, most importantly, in economically disadvantaged
countries without electoral studies tradition. At the core of our
conceptualization lies the concept of geographical clustering (e.g.,
kim2003spatial) and the principle of dispersion dimaggio1996have. This
dimension refers to the phenomenon in which the population with similar
political preferences is concentrated in specific geographical areas. This
involves the existence of electoral strongholds and targeted strategies of
political candidates and parties. For us, grounded in the social identity
theory and the identification-alienation framework, it is reasonable to argue
that citizens who choose the same candidate (or party) should be, on average,
more similar than those who prefer a different candidate, and an effective
form of mapping this polarization is through the voting dispersion within a
territory.

To construct our metric for election polarization, we account for the
electoral competition as a complementary characteristic. While polarization
and competitiveness are traditionally studied as different phenomena, we
suggest that their interplay is essential to understanding the levels of
polarization depicted in voting patterns. For instance, a two-candidate
election in which each one holds 50% of the preferences is not necessarily
polarized. Both candidates could be just very competitive or undistinguished
to voters. In this strand of literature, blais2009general define a competitive
election as the one in which its outcome is uncertain. Traditionally, the
conventional measure of electoral competition is the margin of victory: the
distance between the two leading candidates or parties. Existing literature
proposes metrics of election competition based on district-level data taking
into account the characteristics of the electoral system, voting share, and
number of votes [129, 109], and they implicitly rely on assumptions concerning
parties mobilizing power efforts to translate into votes [118]. For us,
competitiveness must map the distance between all candidacies, no matter the
electoral rule.

Our work implicitly assumes that voters identify themselves as similar to
those who choose the same candidate and feel alienated from those with a
different candidate or party preference. To capture polarization and
competitiveness, we introduce a new concept applied to political candidates or
parties: their antagonism.

###  3.1 Definition of Antagonism

We define the antagonism of a candidate $i$ as the divisiveness level that she
or he generates in the electorate. This antagonism can be reflected by three
characteristics: i) The competitiveness between candidates, ii) the voting
dispersion of a candidate within a geography, and iii) her relative relevance
measured by the number of votes. To better distinguish them, we decompose the
antagonism by in-group (or between-) and out-group (or within-) components. We
subsequently show that between-antagonism captures competitiveness and within-
antagonism captures polarization.

First, we introduce some notation. Let $N$ be the number of (effective)
candidates or parties, and let $M$ be the number of electoral units (e.g.,
district, polling stations). For a candidate $i$, let $share_{i,k}$ and
$votes_{i,k}$ be her percent and number of votes in $k$-unit, respectively,
and let $\overline{share_{i}}$ be the share of $i$ in the election. We now
explore each component of antagonism.

###  3.2 Within-Antagonism

This dimension explores the individual performance of a candidate $i$ within a
geography. In other terms, the existence of geographical clustering in
preferences. Consider a city calling for candidacies to elect a future mayor.
The city is divided into two districts: Alpha and Beta, composed of 300 voters
each. In turn, the districts divide the electors into three precincts
equitably. Let {$A,B,C$} be three candidates running for the position. By
exploring the outcome per district (Alpha: $A$=$30,29,31$; $B$=$20,19,21$;
$C$=$50,52,48$ and Beta: $A$=$5,0,85$; $B$=$5,40,15$; $C$=$90,60,0$–Each
number represents a precinct within the district), we observe that candidates
obtained the same votes ($C=150,A=90,B=60$). However, we observe that the
voting dispersion for each candidate at the precinct level was almost null in
Alpha in comparison to Beta, where candidate $C$ even obtained 0 votes in a
precinct of Beta. This idea leads to the definition of Within-Antagonism as a
form to capture the voting dispersion of a candidate within a geography.

$\text{Within-A}_{i}=\frac{\sum\limits_{k=1}^{M}{votes_{i,k}|share_{i,k}-{\overline{share_{i}}}|}}{\left(N-1\right)\sum\limits_{k=1}^{M}{votes_{i,k}}}$
(1)

The Within-Antagonism ranges from 0 to $1/N$, where values closer to $1/N$
indicate that the distribution of the votes on candidate $i$ is less balanced.
By construction, as this dimension of antagonism grounds on the dispersion of
votes, we subsequently demonstrate that this can be equivalent to a proxy for
political polarization.

###  3.3 Between-Antagonism

This dimension relies on the competitiveness between candidates. In line with
the literature on electoral competition, we consider that a population divided
50-50 in an election of two candidates must be more competitive than a 99-1
outcome since the voters are far away from a unanimous consensus over the
winner. The same rule can be generalized for elections with more than two
candidates (33.3% for three, 25% for four, etc.). Furthermore, we point out
that a result of 50-49-1 between three candidates should be labeled as
competitive since the third candidate is irrelevant to the outcome.

We formally define the Between-Antagonism of candidate $i$ as the distance
(measured by the percent of votes) with respect to all the other candidates in
an election as follows:

$\text{Between-A}_{i}=\frac{\sum\limits_{j=1}^{N}{\sum\limits_{k=1}^{M}{votes_{i,k}(1-|share_{i,k}-{share_{j,k}}|)}}}{N(N-1)\sum\limits_{k=1}^{M}{votes_{i,k}}}$
(2)

To capture the distance between groups, we subtract the distance between two
candidates from the unit. That is, whether the distance between two candidates
is minimum (two candidates with the same percent of votes), the antagonism is
maximum (or 1). Conversely, whether the distance between two candidates is
maximal (one candidate reaches the unanimity of votes), the antagonism is
minimal (or 0). The values range from 0 and $1/N$, when values closer to $1/N$
mean that the candidate has a greater between-antagonism. Thus, Between-
Antagonism of a candidate $i$ represents how much this candidate was preferred
over the others.

###  3.4 Definition of Election Polarization (EP)

We motivated the idea that each candidate’s antagonism represents a fraction
of information over the level of polarization and competition of an election,
and the sum of all antagonisms present in society must represent metrics of
competitiveness and polarization.

First, we conceptualize Election Polarization (EP) as a proxy for political
polarization, and it empirically measures the dispersion of votes across a
geography. This metric is formally defined as the sum of within-antagonisms
from all (effective) candidates on Election Day, i.e.,

EP $\displaystyle=\sum_{i=1}^{N}{\text{Within-A}_{i}}.$ (3)

The EP ranges from 0 to 1, with values closer to the unit indicating greater
polarization.

###  3.5 Definition of Election Competitiveness (EC)

Second, analogous to our definition of EP, we conceptualize Election
Competitiveness (EC) as a metric of closeness from all candidates competing in
the election. This metric is formally defined as the sum of between-
antagonisms from all (effective) candidates on Election Day, i.e.,

EC $\displaystyle=\sum_{i=1}^{N}{\text{Between-A}_{i}},$ (4)

The EC ranges from 0 to 1, with values closer to the unit indicating greater
competitiveness.

Both metrics can be applied to elections under any electoral system,
regardless of the number of candidates or precincts/aggregation units.
Nevertheless, these metrics can be only comparable to elections with the same
number of candidates (See Appendix). As both metrics complement each other,
their summation is at most 1. This is achieved in elections in which all
candidates obtain the same voting percentage.

##  4 Empirical analysis

###  4.1 Properties of Election Polarization and Election Competitiveness

We start exploring EP and EC numerically in order to understand their
properties and draw comparisons with traditional measures of polarization and
competitiveness. Figure 1 illustrates eight synthetic elections of (a-c) two
and (d-f) three candidates with six precincts each–sorted from North to South.
To exemplify edge cases of tied election results, we compare our EC and EP
values with the ones calculated using Esteban-Ray (ER) and Dispersion (DP) for
polarization and Margin of Victory (MV) for electoral competition to showcase
their similarities and differences. While traditional measures for
competitiveness and polarization treat all such elections as equals, our
approach reveals hidden nuance by distinguishing whether the ties reflect
close elections or rather geographical clustering of the electorate.

An additional advantage of our metrics can become apparent by looking at the
correlation between EP and DP. While both metrics exhibit similar trends and
successfully identify the most polarized election, as our metric ranges from 0
to 1, this feature enhances its interpretation.

We then expand the analysis by increasing the number of candidates. Expanding
this observation through simulations (See Appendix), we find that in an
election between $N$ candidates, maximal EP + EC is reached when the outcome
of each candidate is closer to $1/N$ regardless of voting dispersion nor
closeness, and minimal EP + EC is reached for unanimous elections.

Figure 1: Edge-cases of eight fictional tied elections for two (a-c) and three
(d-f) candidates. Each row represents a precinct, and each color represents a
candidate. Comparison of traditional competitiveness (b, e) and polarization
measurements (c, f) with our approaches for two (b-c) and three (e-f)
candidates.

###  4.2 Election Polarization and Election Competitiveness Worldwide

Next, we assess whether our metrics for EC and EP can identify and quantify
theoretical expectations of politically polarized or competitive elections and
regions. To this end, we perform an extensive empirical analysis using real-
world election data by collecting presidential election data from Chile
(2013-2021), Peru (2021), Brazil (2018-2022), Argentina (2023), Mexico (2018),
France (2002-2022), and the United States (2000-2020) and parliamentary
election data from Belgium (2014-2019), New Zealand (2023), Germany (2021),
Italy (2022), Canada (2019), and Spain (2019) at the most disaggregated
available level.

Figure 2 reports the findings from polarized and competitive elections from 13
countries and 553 regions. To facilitate cross-regional comparisons, we
normalize the values by subtracting the mean and dividing by the standard
deviation per country. Furthermore, we report the empirical relationship of EC
and EP in these regions. Figure 2 a-b shows the election polarization and
competitiveness, respectively. To highlight the potential of our metrics for
successfully mapping theoretical expectations of polarized elections, we
briefly mention some examples. We find that Catalonia is one of the most
polarized regions in Spain, reflecting the tensions surrounding the Catalan
independence movement [107]. Other notable cases reported in the figure are
the polarization in the Southeast in the U.S., the clear cleavage between
North and South in Brazil, and the peaks in regions with indigenous conflicts
in the South of Chile. Next, Figure 2 c reports the observed relationship of
EC and EP. The correlation of both measures is rather low ($\rho=-0.253$,
$P<0.001$). We find that, in general, most elections follow two main trends:
the election can be classified as polarized, or rather it can be classified as
competitive.

Figure 2: The geography of Election Polarization and Competitiveness
worldwide. (a) Election Polarization and (b) Election Competitiveness. We
calculated the regional-level values in Chile (2021), Argentina (2023), Brazil
(2022), Peru (2021), Mexico (2018), the United States (2020), Canada (2021),
France (2022), Spain (20), Belgium (2019), Germany (2021), Italy (2022) and
New Zealand (2023). Values are standardized by subtracting their mean and
dividing their standard deviation. (c) Relationship between standardized EP
and EC.

Then, we aggregate our analysis to explore the national-level dynamics of EP
and EC in France and the United States over a 20-year period. We selected
these countries since it is documented in previous work the existence of
recent polarized elections. In the United States, Figure 3 a shows that this
country experienced its most polarized election in 2016, followed by 2020. In
France, we report the most polarized election in 2020, followed by 2017–with a
26% increase among periods. Both results are aligned with the literature and
the political climate in these countries. The Yellow Vest Movement in France
[164, 115] and the irruption of Donald Trump in the United States politics
[102] are well-known cases that have motivated recent polarization discussion
in both countries. Following the previous results, we conducted an analogous
analysis for competitiveness, with results shown in Figure 3 c-d. We found
that the most competitive elections in the United States and France occurred
in 2000 and 2002, respectively. Anecdotally, these peaks could be attributed
to extremely competitive presidential elections. On the one hand, the 2000
election in the U.S. is well-known for the close result between Al Gore and
George Bush (even less than 1000 votes difference in Florida) [101], and the
2002 election in France was the first time that a far-right candidate made it
into the run-off after a virtually close election among the three main
majorities [147].

Figure 3: Temporal dynamics of Election Polarization in (a) France, and (b)
the United States. Analogously, temporal dynamics of Election Competitiveness
in (c) France and (d) the United States. In France, we used presidential
election data from 2002 to 2022. In the United States, we used presidential
election data from 2000 to 2020.

Having established the feasibility and ability of our metrics to label
theoretical expectations of polarization worldwide, we compare the EP and EC
of Swing and Partisan states in the United States (Figure 4). Swing states are
considered crucial in the battle for the White House. In order to get elected,
presidential candidates invest most of their campaign efforts in these
battlegrounds [104, 132], converting them into an excellent natural experiment
to provide construct validity to our metrics (See Appendix). Our assumption
considers that the elections must be competitive on average, but these
battlegrounds should not be more polarized than other states. Figure 4 a finds
that Swing states presented greater EC than Partisan states across time, and
Figure 4 b finds these differences are explained mainly by EC rather than EP.
In other words, we found more evidence to suggest that our methodological
proposition for EC and EP captures different phenomena of elections.

Figure 4: Comparison of EP and EP in Swing and Partisan States in the United
States. (a) Election Polarization (b) Election Competitiveness. Each dot
represents the value for a state.

###  4.3 Election Polarization and Election Competitiveness and its
Correlation with Mass Polarization and Electoral Turnout

Sections 4.1 and 4.2 offer insights demonstrating that our metrics for EP and
EC discern levels of polarization from competitiveness in society. To further
establish their usefulness, we now estimate regional-level political
polarization and electoral turnout from traditional instruments to analyze
their relationship with EP and EC. Our assumptions include that the political
polarization around an election should be reflected on Election Day, and
turnout must be positively associated with EC.

We focus on a state-level analysis in the United States since we ground on
fine-grained data for a more extended study period. We use data from the
Cooperative Election Study (CES) [191] and the U.S. Election Project [193].
Our interest is to investigate the combined characteristics that encompass
political polarization–the ideological distance between Democrats and
Republicans–and electoral turnout–proxied by the voting turnout on Election
Day and to explore the correlation. Our analysis does not target causation.
Rather, we focus on understanding whether the dimensions of EP are related to
political dimensions or whether socio-demographic features capture its
effects. To ensure that socio-demographic characteristics and no other factors
cause any increase in political polarization or turnout in the U.S., we
control our analysis population, political, and economic characteristics. We
also hold those unobserved factors in an Election Year by using year-fixed
effects since an increase in polarization could result from multiple factors
other than socio-demographic variables. For instance, the Cambridge Analytica
scandal might play a role in the polarization identified in the 2016 Election.

We set up panel regressions as follows:

$\displaystyle y_{g,t}$
$\displaystyle=\beta_{0}\mathrm{EP}_{g,t}+\beta_{1}\mathrm{EC}_{g,t}+\beta_{2}^{T}\mathrm{X}_{g,t}+\beta_{3}\mathrm{S}_{g}+\mu_{t}+\epsilon_{g,t},$

where $y_{g,t}$ is the dependent variable (political polarization and
electoral turnout) for the geography $g$ in year $t$), $\mathrm{EP}_{g,t}$ is
the EP and $\mathrm{EC}_{g,t}$ is the EC. $\mathrm{X}_{g,t}$ is a vector of
control variables that account for other factors addressed in the literature.
We control by inequality level (Gini coefficient), population density,
education attainment, annual unemployment rate, and GDP per capita.
$\mathrm{S}_{g}$ is a dummy to control by swing states, $\mu_{t}$ is the
period fixed effect, and $\epsilon_{g,t}$ is the error term. The variables
have been normalized by subtracting the average and dividing by the standard
deviation before running the regressions. Standard errors are clustered at the
state level. The main results are presented in Figure 5. Additional models are
presented in the Appendix.

##### Political Polarization

In Figure 5 (red line), we report correlations for political polarization with
our EP and EC metrics, controlled by economic, socio-demographic, and
political conditions. We find a positive relationship between EP and political
polarization (0.31, $P=0.001$), though slightly less pronounced than
educational attainment (-0.36, $P<0.001$). No significant association is
identified for EC. Grounded on the results, we interpret that EP serves as a
cost-effective proxy of political polarization, opening an avenue for more
nuanced research on polarization at more granular levels, such as counties or
districts. This is particularly salient for lower and middle-income countries
lacking national surveys to characterize this phenomenon.

##### Electoral Turnout

In Figure 5 (green line), we report correlations for electoral turnout with
the EP and EC, conditional on economic, socio-demographic, and political
conditions. We find that voting turnout positively correlates with EC (0.55,
$P=0.003$), and it is the most important variable in terms of size effect in
the model. A less significant association is observed for EP (0.29,
$P=0.036$), which, despite is not the main factor behind electoral turnout,
its statistical significance provides valuable insights suggesting that more
polarized elections lead to higher turnout. Since we additionally rely on
voting turnout data for France and Chile, we perform a univariate correlation
effect in both countries (France (NUTS-2):$\rho_{2022}=0.25$, Chile (region-
level): $\rho_{2021}=0.19$), in line with theoretical expectations for a
measure of electoral competition.

Going back to our initial question, our results suggest that election
polarization can be a simple and suitable proxy for political polarization,
and we set up our metric for election competitiveness as its complement.

Figure 5: State-level Association of Mass Polarization and Electoral Turnout
in the United States (2008-2020). Each model represents a multivariate
regression, with year-fixed effects. Standard errors are in brackets and are
robustly clustered by state. We standardized both independent and dependent
variables to have a mean of zero and a standard deviation of one. Coefficients
represent the p-values. $R^{2}_{\text{Mass Polarization}}=60\%$ and
$R^{2}_{\text{Voting Turnout}}=62\%$. Additional Models can be found in the
Appendix.

###  4.4 Robustness Check

To conclude, we perform a robustness analysis to further validate and extend
the reliability of our metrics by testing scenarios that could potentially
bias its interpretation: i) the aggregation level, ii) abstentions and spoilt
votes, iii) election type (i.e., two elections held on the same day), iv) the
use of first round vs. runoff data, and v) the effective number of candidates
(or parties). To this end, we compared variations of EP and EC calculated via
these five approaches, and this analysis is reported in Figure 6. We focus our
analysis on Chile, France, and the United States, since we ground on fine-
grained data to test these scenarios over a larger period.

##### Aggregation Levels

First, we assess the use of different levels of granularity of data by
comparing the regional-level EP and EC in France (department-level, 2022), the
United States (state-level, 2020), and Chile (region-level, 2021). Discarding
a potential bias toward the use of a specific aggregation level is relevant
for elections in which data can not be obtained at polling station level.
Specifically, we used precincts and county levels as our minimum aggregation
unit. Figure 6 a-c report a strong positive correlation for both indices
(France: EP$=0.925$, $P<0.01$, EC$=0.988$, $P<0.01$; United States:
EP$=0.877$, $P<0.01$, EC$=0.857$, $P<0.01$; Chile: EP$=0.912$, $P<0.01$,
EC$=0.971$, $P<0.01$), discarding the hypothesis that a specific aggregation
level for our datasets altered the trends of our main results.

##### Abstentions and Spoilt Votes

Second, we assess the level of abstentions, blanks, and spoilt votes in the
election. To this end, we compared EP and EC computed using all the candidates
with the one computed by including abstentions, spoilt, and blank votes
treated as additional antagonists. We report a positive correlation for EP in
Chile (Chile${}_{2017}=0.945,P<0.001$; Chile${}_{2021}=0.818,P<0.001$) and
France (0.44, $P<0.001$). Nevertheless, this correlation is not statistically
significant for EC (Chile${}_{2017}=0.276,P=0.301$;
Chile${}_{2021}=0.774,P<0.001$, France${}_{2022}=0.176,P=0.086$). Our analysis
shows that overall, EP seems to be more robust than EC concerning levels of
participation in elections, suggesting that abstentions might not bias the EP
in most cases. We point out here that the statistical power is not maintained
for all the cases, so this should be a concern while selecting the proper data
to analyze. We infer that EC only reflects special election conditions, and
additional analyses should be required to interpret it under scenarios with
high abstention and spoilt votes. A potential non-causal explanation could be
related to the fact that EC reflects a measure of competition exclusively in
the election, but EP reflects a more profound condition of geographies, such
as its polarization, as suggested in this study.

##### Election Type

Third, we assess the EP and EC obtained from two elections held on the same
day. In instances where multiple elections are celebrated simultaneously, we
consider that citizens are subject to similar influences, such as discussions
with friends, ad campaigns, TV, and social media. Consequently, those
elections should reach a similar polarization level. To test this assumption,
we use election data from general, senate, and representative elections in the
United States in 2020 (precinct level) and senate and presidential elections
in Chile in 2021. Figure 6 c-d shows a strong positive correlation between the
EP computed for both elections (USRepresentatives: EP$=0.831$, $P<0.01$,
EC$=0.688$, $P<0.01$; USSenate: EP$=0.859$, $P<0.01$, EC$=0.871$, $P<0.01$;
Chile: EP$=0.736$, $P=0.024$, EC$=-0.073$, $P=0.851$). This finding enriches
our metric for EP by suggesting its robustness and providing more evidence
that EC can be interpreted as a measure applicable only to an election.

##### First round vs. Runoff

Fourth, we compare the EP and EC obtained using first round and runoff
presidential election data. Given that the time gap between both rounds is
usually short, for example, two weeks in France and four weeks in Chile, it
seems reasonable to think that changes in polarization would be marginal.
Nevertheless, we know that candidates moderate their political positions in
the second rounds [110] and citizens might be forced to vote by the “lesser
evil” in some contests and not for their real preferences. Figure 6 reports
the correlation of both indices for France (2002 and 2022) and Chile (2021)
(More analyses in the Appendix). We systematically observe that the values
from the first and second rounds are uncorrelated or exhibit a slight
correlation. For instance, in the 2002 election in France, where it is
documented that citizens voted against the far-right candidate, the
correlation for EC was -0.347 ($P<0.001$), and the one for EP was 0.309
($P=0.02$). To sum up, we conclude that the use of runoff data is not suitable
for comparing the polarization or real competitiveness between rounds.
Nevertheless, there is still a window of opportunity to study polarization in
the same round from different years.

##### Effective Number of Candidates

Finally, we evaluate whether our metrics calculated by excluding protest
candidacies–or only focusing on the effective candidates [142]–can reach
similar values to the ones using all candidates. This is relevant since the
indices can be comparable only for elections with the same number of
candidates over time. We perform two approaches: On the one hand, we compute
EP and EC by relying on the top-2 candidates, then top-3, and so on, until
completing all of them (See Appendix) by grouping the excluded candidates
under a synthetic candidate called “other”. On the other hand, we compare our
metrics using all candidates and the one using the effective number of
candidates. In both cases, we use presidential election data from France
(2002-2022) and Chile (2013-2021), which feature more than two effective
candidates per election. Figure 6 m-n reports the Pearson correlation of our
indices computed from all candidates and the one calculated using the top-$N$
candidates, and Figure 6 o reports the Pearson correlation with respect to the
second approach. In both analyses, we observe a similar pattern: We find a
strong positive correlation (greater than 0.8) for any number of candidates
that represent at least 80 % of votes for both metrics, and surprisingly, EP
can be mapped by only using the two main majorities in presidential systems,
being robust to this finding for all elections analyzed. This analysis,
however, could not be replicated for proportional systems, claiming for
further analysis in this direction.

These results computationally validate the usefulness of applying our metrics
using election data under multiple assumptions (e.g., temporally, spatially)
and relying on incomplete and noisy data, contributing to the reliability of
this approach. From these analyses, we learn that EP reflects geographical
phenomena, transcending the nature of election data and type. EC,
nevertheless, can not be generalized to infer competitiveness in society
beyond the scope of a particular election.

Figure 6:  Robustness Check using Real Data from Chile, France, and the United
States. Aggregation Level in the (a) United States, 2020; (b) Chile, 2021; and
(c) France, 2022. Abstentions and Spoilt Votes in (d) France, 2022; (e) Chile,
2017; and (f) Chile, 2021. Election Type in (g) the United States, 2020, by
comparing presidential and senate elections; (h) the United States, 2020, by
comparing presidential and house elections; and (i) Chile, 2021, by comparing
presidential and senate elections. First Round vs Runoff in (j) France, 2002;
(k) France, 2017; (l) Chile, 2021. Effective Number of Candidates in (m)
Chile, 2013-2021; (n) France (2002-2022); and (o) both countries. Values in
$\rho$ represent the Pearson correlation. The variables have been normalized
by subtracting the average and dividing by the standard deviation. The dashed
lines from (a-l) represent $y=x$, and in (m-o) represent a correlation equal
to 0.8.

##  5 New Measures’ Usefulness in Future Research

The finding that election polarization proxies political polarization has
important implications for scholars. Our metrics can be applied to virtually
any aggregation level, which highlights their potential to characterize
polarization even at local levels, which was, thus far, virtually impossible
to achieve. This means that EP can become an index of polarization of
geographies, which is especially relevant for comparative studies worldwide.

Furthermore, having demonstrated the feasibility of mapping our metrics using
citizens’ voting patterns on election day exclusively and its robustness under
several scenarios and assumptions that could bias its interpretations, our
perspective accounts for their usefulness as a starting point for studies in
political science and economic geography. On the one hand, this study aligns
with the active and timely debate on the determinants, conditions, and
consequences of polarization worldwide, allowing us to leverage our knowledge
even toward historical levels of polarization. On the other hand, scholars
from economic geography interested in studying factors that predict the
development of territories could use the EP to characterize how polarized
regions could be associated with economic growth levels.

This study did not analyze the antagonism of political candidates and parties.
Analogous to the interpretation for EP and EC, we are able to obtain a metric
of the political candidates’ antagonism (or divisiveness) at virtually any
aggregation level, allowing us to differentiate how divisive a candidate is
throughout a territory. For instance, we observed that the antagonism of
political candidates varies within a country; however, we did not explore the
determinants and consequences of this unequal pattern. Our metric for
antagonism can help study the chances of divisive candidates in runoff
elections and also for the literature on median voters.

Altogether, these examples represent just a few avenues wherein our metrics
can enrich our understanding of geographic dynamics and the extent of
divisiveness and polarization within them. We expect to continue providing
responses to fully understand the results of our methodological contribution.

##  6 Conclusion

This paper presents two complementary measures of Election Polarization and
Competitiveness based on Election Day outcomes. These metrics make a valuable
contribution to the existing literature by offering a straightforward and
cost-effective proxy for delineating polarization. To construct the measures,
we conceptualize the antagonism of political candidates and parties, which is
represented by the closeness between candidates and voting dispersion across a
territory.

Importantly, this paper provides a diverse overview of worldwide electoral
systems and political contexts. Using synthetic and empirical data from 13
countries, we demonstrated that our approach successfully labeled polarizing
elections and growing trends presented in the literature, such as France, the
United States, Chile, Spain, and Brazil. We demonstrated that EC and EP are
straightforward metrics for understanding and comprehending polarization from
a spatial perspective. Our approach has the potential to become an index of
polarization in geographies, opening the possibility of extending the study of
polarization beyond the U.S. borders and towards more granular scales.
Nevertheless, we call for further focus on causation mechanisms between forms
of ideological polarization and the voting patterns revealed on Election Day.
A second stage could be linking the ideological position of candidates with
the antagonism level and exploring the before and after election polls and
their relationship with the EP.

Our analysis reveals that Election Polarization is positively associated with
a measure of political polarization, and Election Competitiveness correlates
to voting turnout. These results indicate that we can define a polarized
election by simply knowing whether two conditions are satisfied: i) if there
is total participation of the electorate (or close to that) in the election,
and ii) if voters’ preferences are highly clustered geographically. However,
our findings do not claim causation, highlighting the need for additional
behavioral studies to confirm that Election Polarization could be defined as a
proxy for political polarization.

Demonstrating that election data can be a reliable proxy for mapping political
polarization is especially relevant to the literature. Evidence from the 2018
Brazilian presidential election suggested that Jair Bolsonaro exploited
demographic segregation to win the election [145]. Similar findings were
observed in the 2016 United States presidential election, showing that
unusually explicit appeals to racial and ethnic resentment and anti-
establishment feelings attracted strong support from white working-class
voters in favor of Donald Trump [102]. Measuring and distinguishing
polarization from election competition is essential to study their impacts on
our society.

This study has some limitations that can be taken into consideration for
further work. On the one hand, while our approach does not account for
ideology in its conceptualization, we emphasized that if we want to provide a
simple and representative form of mapping polarization at local scales, it is
almost impossible to fulfill these conditions by including ideology in the
equation. While some instruments provide ideological scores for political
candidates or parties, we argue that this is not an attribute that can be
obtained for all political parties/candidates worldwide. On the other hand, as
the relationship between electorate sorting and political polarization has
been established using electoral studies data from the United States, we call
for further studies exploring the conditions and determinants of election
polarization in different countries and its associations with sociodemographic
conditions. One possibility is studying how the EP correlates to theoretical
expectations of polarization, such as corruption control, inequality, or
rurality, to name a few.

To conclude, we set up some considerations. First, the geographical hierarchy
of an election must represent the most extensive aggregation level to
calculate our metrics. The reason is that local candidates’ quality could bias
the party’s support in a location [156]. Second, the EC is only valid in its
corresponding election and can not be generalized to infer global levels of
competition of parties/candidates. Third, our method is not valid for non-
free-and-fair elections. To exemplify this, a referendum with 99% approval in
a dictatorship does not mean that society is not polarized at all. Finally,
our metrics are unsuitable for comparing values between the first round vs
runoff in the same electoral year since we find that, in some cases, both
values are uncorrelated.

## Data Availability Statement

Upon the final acceptance of this paper, the data and source code used to
perform this study will be publicly deposited on Harvard Dataverse.

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# Mapping Election Polarization and Competitiveness using Election Results

Carlos Navarrete Corresponding author<EMAIL_ADDRESS>Mariana Macedo
Viktor Stojkoski  Marcela Parada-Contzen  Christopher Martínez

# Appendix for “Mapping Election Polarization and Competitiveness using
Election Results”

Carlos Navarrete Corresponding author<EMAIL_ADDRESS>Mariana Macedo
Viktor Stojkoski  Marcela Parada-Contzen  Christopher Martínez

###### Contents

  1. 1 Introduction
  2. 2 Theoretical Discussion
  3. 3 Our Measure
    1. 3.1 Definition of Antagonism
    2. 3.2 Within-Antagonism
    3. 3.3 Between-Antagonism
    4. 3.4 Definition of Election Polarization (EP)
    5. 3.5 Definition of Election Competitiveness (EC)
  4. 4 Empirical analysis
    1. 4.1 Properties of Election Polarization and Election Competitiveness
    2. 4.2 Election Polarization and Election Competitiveness Worldwide
    3. 4.3 Election Polarization and Election Competitiveness and its Correlation with Mass Polarization and Electoral Turnout
    4. 4.4 Robustness Check
  5. 5 New Measures’ Usefulness in Future Research
  6. 6 Conclusion
  7. G Supplementary Methods
    1. G.1 Data Collection
      1. G.1.1 Argentina
      2. G.1.2 Belgium
      3. G.1.3 Brazil
      4. G.1.4 Canada
      5. G.1.5 Chile
      6. G.1.6 France
      7. G.1.7 Germany
      8. G.1.8 Italy
      9. G.1.9 Mexico
      10. G.1.10 New Zealand
      11. G.1.11 Peru
      12. G.1.12 Spain
      13. G.1.13 United States
    2. G.2 Data Curation
      1. G.2.1 Data Files
      2. G.2.2 Pipelines
    3. G.3 Sociodemographic Characteristics
      1. G.3.1 United States
      2. G.3.2 Special considerations in the U.S.
    4. G.4 Political Interest
    5. G.5 Synthetic data
    6. G.6 Data Analysis
    7. G.7 Comparison of Polarization Measures
      1. G.7.1 Esteban and Ray, 1994
      2. G.7.2 Dispersion
      3. G.7.3 Polarization: Esteban-Ray, Wang-Tsui, Dispersion
    8. G.8 Definition of Swing States
    9. G.9 Definition of Mass Polarization
    10. G.10 Method Validation
      1. G.10.1 Robustness by Voting Percentage
  8. H Supplementary Results
    1. H.1 Regression Analysis
  9. I Supplementary Tables
  10. J Supplementary Figures

##  G Supplementary Methods

###  G.1 Data Collection

This section presents our procedure for downloading, curating, and
standardizing elections from each country included in this study.

####  G.1.1 Argentina

##### Data Source

Datos Argentina, the official website of the Argentinian government to share
open data [184]. Data is deposited by Dirección Nacional Electoral.

##### Period

2019 and 2023 First Round’ Presidential Elections.

##### Aggregation Level

Data obtained at polling station level. We use provinces as our territorial
division.

####  G.1.2 Belgium

##### Data Source

Direction of Elections (Direction des Elections in French) [185].

##### Period

2019 General Election.

##### Aggregation Level

Data obtained at the commune level. We use NUTS-2 as our territorial division.

####  G.1.3 Brazil

##### Data Source

Tribunal Superior Eleitoral [199]

##### Period

2018 and 2022 First Round’ Presidential Elections.

##### Aggregation Level

We use states as our territorial division.

####  G.1.4 Canada

##### Data Source

Elections Canada [186].

##### Period

2021 Federal Election (Lower Chamber).

##### Aggregation Level

Data obtained at polling station level. We use states as our territorial
division.

####  G.1.5 Chile

##### Data Source

The Servicio Electoral (SERVEL in its acronym in Spanish) [197]
111https://www.servel.cl/resultados-en-excel-por-mesa-a-partir-del-ano-2012/.

##### Period

2013-2021 First Round and Runoff Presidential Elections.

##### Aggregation Level

Data obtained at polling station level. We use administrative structure
(regions and provinces) as our territorial division. The datasets were curated
considering the most recent administrative structure established in 2017
[175]. Thus, we perform the analysis using the administrative structure as of
April 2023 of 16 regions (sorted from North to South): Arica y Parinacota,
Tarapacá, Antofagasta, Atacama, Coquimbo, Valparaíso, Metropolitana,
O’Higgins, Maule, Ñuble, Biobío, Araucanía, Los Ríos, Los Lagos, Aysen, and
Magallanes.

####  G.1.6 France

##### Data Source

Le ministère de l’Interieur et des Outre-mer [183].

##### Period

2002-2022 First Round and Runoff Presidential Elections.

##### Aggregation Level

Data obtained at the polling station level. We use NUTS-2 and NUTS-3 as our
territorial divisions.

##### Special Considerations

The NUTS-3-level analysis encompassed 96 territories from Metropolitan France,
after removing overseas territories. In Section “Robustness Check” of the main
manuscript, we converted the 18 circumscriptions from the department of Paris
(75) into communes. This fact is due to the fact that department 75 is just
composed of one commune (75056), making it impossible to aggregate data for
Paris at the commune level.

####  G.1.7 Germany

##### Data Source

Die Bundeswahlleiterin [172].

##### Period

2021 General (Lower Chamber) Election.

##### Aggregation Level

Data obtained at polling station level. We use NUTS-2 v2021 as our territorial
division.

####  G.1.8 Italy

##### Data Source

Ministero dell’Interno [176].

##### Period

2022 General Election (Chamber of Deputies), held on September 25th.

##### Aggregation Level

Data obtained at the commune level. We use NUTS-2 v2021 as our territorial
division.

####  G.1.9 Mexico

##### Data Source

National Electoral Institute (Instituto Nacional Electoral in Spanish) [190].

##### Period

2018 Presidential Election.

##### Aggregation Level

Data obtained at polling station level. We use Federative Entity (state) as
our territorial division.

####  G.1.10 New Zealand

##### Data Source

Electoral Commission [187].

##### Period

2023 General Election (held on 14th October 2023)

##### Aggregation Level

Data obtained at voting place level. We use regions as our territorial
division.

##### Special Considerations

We excluded the Māori electorate not associated with a particular region.

####  G.1.11 Peru

##### Data Source

National Office of Electoral Processes (Oficina Nacional de Procesos
Electorales in Spanish) [196].

##### Period

2021 Presidential Election.

##### Aggregation Level

Data obtained at polling station level. We use regions as our territorial
division.

####  G.1.12 Spain

##### Data Source

Ministerio del Interior [189].

##### Period

2019 General Election (held in November).

##### Aggregation Level

Data obtained at polling station level. We use NUTS-2 v2021 as our territorial
division.

##### Special Considerations

Considering that the form of government in Spain is a parliamentary monarchy,
we focus our analysis and download procedure to get data from congress
elections. In this case, the political parties act as the equivalent of
candidates.

####  G.1.13 United States

##### Data Source

The MIT Election Data + Science Lab (MEDSL) supports research on elections in
the United States by sharing election outcome data for the Presidential,
Senate, and House elections at different aggregation levels.

##### Period

2000-2020 Presidential Elections and 2016-2020 House and Senate Elections.

##### Aggregation Level

Data obtained at the county level for presidential elections from 2000-2020
[177] and at the precinct level for elections from 2016 to 2020 [178, 179,
182, 180, 181]. We use states as our aggregation unit.

##### Special Considerations

The robustness analyses were conducted using the precinct-level data for the
2016 and 2020 House, Senate, and Presidential elections. Each state was
labeled as Swing or Partisan. The District of Columbia was not considered in
the robustness validation neither regression analysis. Due to the structure of
the U.S. electoral system, we conduct all the analysis by just considering the
two main majorities per election. Even though, in most cases, the main two
majorities correspond to a candidate from the Republican and Democrat parties,
there are some exceptions, as in the 2016 Senate election in California. We
point out that we do not present all precinct-level findings since we do not
have data available from a broader period than 2016-2020. Nevertheless, as
demonstrated in the paper, our method is robust to use different aggregation
levels.

###  G.2 Data Curation

The pipelines were coded in Python, mainly using Jupyter Notebook, Pandas
v1.5.1 [194], SciPy v1.9.3 [200], and NumPy v1.23.4 [188]. The regression
analyses were performed in R [192]. The visualizations were created using
Seaborn v0.12.1, Matplotlib v3.6.2, and ggplot2. Each Notebook includes a
step-by-step explanation of both data cleaning, standardization, and analysis.
We will publicly deposit the source code and curated datasets into a
repository on GitHub.

####  G.2.1 Data Files

The data is deposited in the folder data_curated. The filename structure
follows the convention: {country}_{year}_{round}.csv.gz, where the parameter
country represents the country (Chile, France, United States), year represents
the election year, and round represents the electoral round (values accepted
are first_round, runoff, general, senate, and house). Each file is compressed
in GZIP (Table S1 for details).

We point out that we create a unique identifier for each unit called
polling_id (e.g., the precinct in the United States for the general election).
This feature concatenates hierarchical aggregate levels according to the data
origin. For example, the polling_id joins state and county FIPS in the United
States (2000), and the polling_id joins region, province, commune, and polling
station name in Chile (2021).

We created a complementary file named
{country}_{year}_{round}_location.csv.gz, that contains metadata regarding the
voting unit. Both files can be concatenated using polling_id feature.

####  G.2.2 Pipelines

The pipelines aim to allow researchers to replicate the findings presented in
this paper. Roughly, we divide the pipelines in three main folders, as
follows:

  * •

Pipeline: Details the data cleaning procedure for each country (or election in
some cases).

  * •

Maps: Creates a set of maps or visualizations.

  * •

Robustness: Presents a set of robustness tests and validations. Values
accepted for type are Abstentions, Multicandidate, Validation, and Runoff.

###  G.3 Sociodemographic Characteristics

Sociodemographic characteristics included in our regression analyses, such as
inequality level, annual unemployment rate, or median household income, were
downloaded from official public institutions (e.g., Census Bureau) or
repositories maintained by scholars or private institutions. Here, we briefly
describe the data origin and curation procedure.

####  G.3.1 United States

##### Census Bureau

  * •

Land area in Sq. Km. from State Area Measurements [174].

  * •

Population (Estimation) (2000-2020)

##### American Community Survey (ACS)

This survey is the premier source for detailed population and housing
information in the U.S. [169]. The Census Bureau releases every year a version
of ACS, providing detailed household and population information.

  * •

Income inequality (Gini coefficient) (2010-2020) [173]

##### State Health Access Data Assistance Center (SHADAC)

This institution at the University of Minnesota provides a collection of
statistics in order to help bridge the gap between policy and health issues
[198]. From SHADAC, we downloaded the following data sets:

  * •

Annual unemployment rate (2000-2020)

##### Bureau of Economic Analysis (BEA)

The Bureau of Economic Analysis (BEA) is the federal agency responsible for
producing statistics and data about the U.S. economy [171]. We downloaded the
following state-level data sets from BEA:

  * •

Personal Income per capita (in Dollars) (2000-2020).

  * •

Real Gross Domestic Product (Millions of chained 2012 dollars) (2000-2020).

  * •

Personal Expenditure per capita (2000-2020).

####  G.3.2 Special considerations in the U.S.

##### Income inequality

We tested our models in a shorter period using the Gini from the Census and
imputing 2010 as the value for 2008.

##### Income inequality

We relied on the region-level Gini coefficient (using 2017 data) computed by
Brevis (2020) [195].

###  G.4 Political Interest

We proxy political interest as the participation rate on Election Day. From
France (2022) and Chile (2017, 2021), election data already includes the
number of abstentions per polling station. In the United States, we downloaded
voting statistics from the U.S. Elections Project (2000-2020) [193], and used
the Voting Elegible Population (VEP) turnout rate. The univariate correlations
can be found in Figure S8.

###  G.5 Synthetic data

Throughout the manuscript, we explored the numerical properties of EP using
synthetic data. Algorithms 1 and 2 summarize our approach for elections of two
and three candidates, respectively. Here we sampled
$\mu\in\\{0.5,0.66,0.75,0.8333,1\\}$, $\sigma\in\\{0.0025,0.05,0.10,0.25\\}$,
and we set the number of units $M$ in 100. It should be noted that we consider
as constant the number of votes per unit, in our case, 100.

Algorithm 1 Data sampling algorithm for a given election between two
candidates.

$\mu\in[0,1],\sigma\in[0,0.25],m>0,N\leftarrow 2$

$\sum\limits_{k=0}^{1}r_{i,k}=1\And\sum\limits_{i=0}^{m-1}\sum\limits_{j=0}^{1}v_{i,k}=100m,\forall
i\in[0,m-1]$

$a\leftarrow\mu$ $\triangleright$ Mean

$s\leftarrow\sigma$ $\triangleright$ Standard deviation

$M\leftarrow m$ $\triangleright$ Number of voting units

for $i\leftarrow 0$ to $M-1$ do

$r_{0,i}\leftarrow\mathcal{N}(a,\,s)$

if $r_{0,i}>1$ then

$r_{0,i}\leftarrow 1$

else if $r_{0,i}<0$ then

$r_{0,i}\leftarrow 0$

end if

$r_{0,i}\leftarrow r_{0,i}-r_{0,i}\mod{0.01}$

$r_{1,i}\leftarrow 1-r_{0,i}$

end for

for $i\leftarrow 0$ to $N-1$ do

for $j\leftarrow 0$ to $M-1$ do

$v_{i,k}\leftarrow 100r_{i,k}$

end for

end for

Algorithm 2 Data sampling algorithm for a given election between three
candidates.

$\mu_{1},\mu_{2}\in[0,1],\sigma_{1},\sigma_{2}\in[0,0.25],m>0,N\leftarrow 3$

$\sum\limits_{j=0}^{2}r_{i,k}=1\And\sum\limits_{i=0}^{m-1}\sum\limits_{j=0}^{2}v_{i,k}=100m,\forall
i\in[0,m-1]$

$a_{1}\leftarrow\mu_{1}$ $\triangleright$ Mean (1)

$a_{2}\leftarrow\mu_{2}$ $\triangleright$ Mean (2)

$s_{1}\leftarrow\sigma_{1}$ $\triangleright$ Standard deviation (1)

$s_{2}\leftarrow\sigma_{2}$ $\triangleright$ Standard deviation (2)

$M\leftarrow m$ $\triangleright$ Number of voting units

for $i\leftarrow 0$ to $M-1$ do

$r_{0,i}\leftarrow\mathcal{N}(a_{1},\,s_{1})$

$r_{1,i}\leftarrow\mathcal{N}(a_{2},\,s_{2})$

for $j\leftarrow 0$ to $1$ do

if $r_{i,k}>1$ then

$r_{i,k}\leftarrow 1$

else if $r_{0,i}<0$ then

$r_{i,k}\leftarrow 0$

end if

end for

if $(r_{0,i}+r_{1,i})>1$ then

$r_{1,i}\leftarrow 1-r_{0,i}$

end if

$r_{2,i}\leftarrow 1-(r_{0,i}+r_{1,i})$

end for

for $i\leftarrow 0$ to $M-1$ do

for $j\leftarrow 0$ to $N-1$ do

$v_{i,k}\leftarrow 100r_{i,k}$

end for

end for

Algorithm 3 An algorithm with caption

$n\geq 0$

$y=x^{n}$

$y\leftarrow 1$

$X\leftarrow x$

$N\leftarrow n$

while $N\neq 0$ do

if $N$ is even then

$X\leftarrow X\times X$

$N\leftarrow\frac{N}{2}$ $\triangleright$ This is a comment

else if $N$ is odd then

$y\leftarrow y\times X$

$N\leftarrow N-1$

end if

end while

###  G.6 Data Analysis

We detail our data analysis procedure. Roughly, the file pipeline_files.py
contains a function that converts the preprocessed election data into a
measure of polarization and pipeline.py is a simplified version to calculate
EP and EC. This procedure can be summarized as follows:

  * •

We select an aggregation level.

  * •

In case of existence in the data, we exclude blank votes, spoilt votes, and
abstentions.

  * •

To compare elections with a different number of candidates over time, we
maintain the top-2 majorities in the United States, four majorities in Chile,
and eight majorities in France. The rest of the candidates were excluded from
the data.

  * •

We re-calculate candidates’ voting percentages in each aggregation unit.

  * •

We calculate Within- and Between- antagonism of $i$.

###  G.7 Comparison of Polarization Measures

We adapted traditional polarization measurements, such as Esteban-Ray and
Dispersion, in order to compute alternative forms of antagonism level for
candidate $i$ by considering the $M$ voting units. The algorithmic
implementations of each method can be found in
Comparison_Polarization_Measures.ipynb.

####  G.7.1 Esteban and Ray, 1994

$ER_{i}=K\sum\limits_{k=1}^{M}\sum\limits_{j=1}^{N}v_{i,k}^{1+\alpha}v_{j,k}|r_{i,k}-r_{j,k}|$
(6)

Where $v_{i,k}$ is the number of votes of $i$ in $k$, $r_{i}$ the percent of
votes of $i$ in $k$, and $K$ and $\alpha$ represents freedom degrees
introduced by the authors. We set $\alpha=0.25$ and $\alpha=1$. In all cases,
we set $K=\frac{1}{\left(\sum\limits_{k=1}^{M}v_{i,k}\right)^{(2+\alpha)}}$

####  G.7.2 Dispersion

$\text{Dispersion}_{i}=\sqrt{\frac{\sum\limits_{k=1}^{M}\left(r_{i,k}-\mu_{i}\right)^{2}}{M-1}}$
(7)

Where $M$ is the number of units, $r_{i,k}$ the percent of votes of $i$ in
$k$, and $\mu_{i}$ is the voting percentage of $i$ in the election.

####  G.7.3 Polarization: Esteban-Ray, Wang-Tsui, Dispersion

Let $A^{\prime}$ be the antagonism of a candidate $i$ computed with a
classical approach. As introduced in the manuscript, the total election
polarization of a society is represented as follows:

$EP^{\prime}=\sum\limits_{i=1}^{N}A^{\prime}_{i}$ (8)

We compare those measures both by simulations (Figure S7) and using data from
the last presidential election in Chile (2021) (Table S6), France (2022)
(Table S7), and the United States (2020) (Table S8).

###  G.8 Definition of Swing States

We define a swing state as one in which there has been at least one winning
candidate from both parties (Republican and Democratic) in the last four
presidential elections (2008-2020). We label Swing states: Arizona, Georgia,
Indiana, North Carolina, Florida, Iowa, Ohio, Wisconsin, Michigan, and
Pennsylvania. Analogously, a Republican state is one in which the Republican
Party’s candidate won in all four elections; similarly, a Democratic state is
one in which the Democratic Party’s candidate won in all four elections.

###  G.9 Definition of Mass Polarization

We use the Cooperative Election Study (CES) [191] to estimate Mass
Polarization in the United States. The advantage of using CES compared to the
American National Election Studies (ANES) is that the first has been
previously used to estimate a measure of polarization at the state level. To
calculate the ideological score of party $p$ in a state $g$ at a time $t$ as
follows:

$\displaystyle\text{Ideology}_{p,g,t}=\sum\limits_{a=1}^{A}S_{a,p,g,t}\cdot
w_{a,g,t}$ (9)

he parameters include $S_{a,g,t}$ as the ideological score of participant $a$
(1=Lean, 2=Not very strong, 3=Strong) and $w_{a,p,g,t}$ as is the weight of
$a$.

Then, we formally define Mass Polarization (PP) as the perceived ideological
distance between the Democratic ($p=1$) and Republican ($p=2$) parties [170].

$\displaystyle\text{PP}_{g,t}=|\text{Ideology}_{1,g,t}-\text{Ideology}_{2,g,t}|$
(10)

From the survey documentation, we use the feature pid7, which collects the
7-point self-reported partisan identity. We numbered that scale in the
following form: “Strong Democrat”: -3, “Not Very Strong Democrat”: -2, “Lean
Democrat”: -1, “Independent”: 0, “Lean Republican”: 1, “Not Very Strong
Republican”: 2, “Strong Republican”: 3. Values for Not sure were considered as
Null.

###  G.10 Method Validation

####  G.10.1 Robustness by Voting Percentage

As mentioned in the main manuscript, we use data from presidential elections
in France (2002-2022) and Chile (2013-2021) to test robustness by only using a
fraction of the data or only the effective number of candidates given in the
election.

Going back to the manuscript, we first calculate each candidate’s voting
percentage in the general election. Then we compute the EP by relying on the
top-2 candidates, then three candidates, and so on, grouping the excluded
candidates under the category “other”. To perform the analysis, we calculate
the similarity (using Pearson correlation) between the EP computed for all
candidates and the one calculated for the top-$N$.

##  H Supplementary Results

###  H.1 Regression Analysis

We further explore the relationship between control variables with respect to
EC and EC to analyze whether our measure of polarization satisfies theoretical
associations of sociodemographic characteristics with measures of polarization
and discards that just mirror inequality or macroeconomic conditions. We set
up the regression model as follows:

$\displaystyle y_{g,t}$
$\displaystyle=\beta_{0}^{T}\mathrm{X}_{g,t}+\mu_{t}+\gamma_{g}+\epsilon_{g,t}$
(11)

Where $y_{g,t}$ is the dependent variable for the geography $g$ in year $t$
(in our case, the EP and EC), $\mathrm{X}_{g,t}$ is a vector of control
variables in $t$, $\mu_{t}$ is the period fixed effect, and $\epsilon_{g,t}$
is the error term. We standardized both independent and dependent variables to
have a mean of zero and a standard deviation of one. Standard errors are
clustered at the state level. Tables S4 and S3 use EC and EP as dependent
variables, respectively. To complement, Table S5 reports the full coefficient
and statistics of models presented in the main manuscript.

##  I Supplementary Tables

Feature |  Description | Type
---|---|---
polling_id |  Unique identifier for the minimum aggregation unit | string
candidate |  Candidate name | string
value |  Candidate’s votes in the unit. | integer
rank |  Candidate’s ranking in the unit. | integer
flag_candidates |  Whether to consider candidates are null, abstentions, and blank votes. | boolean
rate |  Candidate’s percent of votes in the unit | float
Table S1: Features description of election data sets curated in this manuscript. Each data point represents the results of a candidate in the minimum aggregation unit (e.g., precinct, polling place, county). Each data set is openly shared and deposited in a public repository. Feature |  Description | type
---|---|---
polling_id |  Unique identifier of the minimum aggregation unit. | string
value |  Number of votes of a candidate or party (depending on the election) | integer
rate |  Voting percentage of a candidate in the aggregation unit. | float
Table S2: Features in Data Shared for Location. | Model 1 | Model 2 | Model 3
---|---|---|---
Inequality level (t) | $-0.01$ | $0.03$ | $0.35^{**}$
| $(0.22)$ | $(0.24)$ | $(0.17)$
Unemployment (t) |  | $-0.10$ | $-0.14$
|  | $(0.15)$ | $(0.13)$
Ed. Attainment (t) |  |  | $-0.11$
|  |  | $(0.21)$
GDP PC (t) |  |  | $0.24$
|  |  | $(0.23)$
Pop. Density (t) |  |  | $-0.67^{***}$
|  |  | $(0.23)$
Num. obs. | $204$ | $204$ | $204$
R2 (full model) | $0.09$ | $0.10$ | $0.31$
R2 (proj model) | $0.00$ | $0.00$ | $0.24$
Adj. R2 (full model) | $0.07$ | $0.07$ | $0.28$
Adj. R2 (proj model) | $-0.03$ | $-0.03$ | $0.21$
Num. groups: year | $4$ | $4$ | $4$
Num. groups: swing | $2$ | $2$ | $2$
${}^{***}p<0.01$; ${}^{**}p<0.05$; ${}^{*}p<0.1$
Table S3: State-level association of EP in the United States (2008-2020). Each model represents a multivariate regression, with year-fixed effects. Standard errors are in brackets and are robustly clustered by state. We standardized both independent and dependent variables to have a mean of zero and a standard deviation of one. | Model 1 | Model 2 | Model 3
---|---|---|---
Inequality level (t) | $-0.38$ | $-0.44$ | $-0.05$
| $(0.29)$ | $(0.31)$ | $(0.10)$
Unemployment (t) |  | $0.18$ | $0.13$
|  | $(0.11)$ | $(0.10)$
Ed. Attainment (t) |  |  | $0.31^{**}$
|  |  | $(0.13)$
GDP PC (t) |  |  | $-0.19$
|  |  | $(0.17)$
Pop. Density (t) |  |  | $-0.68^{***}$
|  |  | $(0.12)$
Num. obs. | $204$ | $204$ | $204$
R2 (full model) | $0.23$ | $0.24$ | $0.64$
R2 (proj model) | $0.16$ | $0.17$ | $0.61$
Adj. R2 (full model) | $0.21$ | $0.22$ | $0.63$
Adj. R2 (proj model) | $0.13$ | $0.15$ | $0.59$
Num. groups: year | $4$ | $4$ | $4$
Num. groups: swing | $2$ | $2$ | $2$
${}^{***}p<0.01$; ${}^{**}p<0.05$; ${}^{*}p<0.1$
Table S4: State-level association of EC in the United States (2008-2020). Each model represents a multivariate regression, with year-fixed effects. Standard errors are in brackets and are robustly clustered by state. We standardized both independent and dependent variables to have a mean of zero and a standard deviation of one. | MP (1) | MP (2) | VT (1) | VT (2)
---|---|---|---|---
EP (t) | $0.33^{***}$ | $0.31^{***}$ | $-0.01$ | $0.29^{**}$
| $(0.07)$ | $(0.09)$ | $(0.13)$ | $(0.14)$
EC (t) | $-0.11^{*}$ | $0.04$ | $0.21^{**}$ | $0.55^{***}$
| $(0.06)$ | $(0.10)$ | $(0.10)$ | $(0.18)$
Inequality level (t) |  | $0.26^{**}$ |  | $-0.27^{**}$
|  | $(0.10)$ |  | $(0.11)$
Unemployment (t) |  | $0.01$ |  | $-0.05$
|  | $(0.11)$ |  | $(0.06)$
Ed. Attainment (t) |  | $-0.36^{***}$ |  | $0.53^{***}$
|  | $(0.13)$ |  | $(0.11)$
GDP PC (t) |  | $-0.10$ |  | $0.03$
|  | $(0.19)$ |  | $(0.16)$
Pop. Density (t) |  | $0.28$ |  | $0.31$
|  | $(0.17)$ |  | $(0.23)$
Num. obs. | $204$ | $204$ | $303$ | $201$
R2 (full model) | $0.47$ | $0.60$ | $0.30$ | $0.62$
R2 (proj model) | $0.17$ | $0.38$ | $0.05$ | $0.51$
Adj. R2 (full model) | $0.45$ | $0.58$ | $0.28$ | $0.60$
Adj. R2 (proj model) | $0.15$ | $0.34$ | $0.03$ | $0.48$
Num. groups: year | $4$ | $4$ | $6$ | $4$
Num. groups: swing | $2$ | $2$ | $2$ | $2$
${}^{***}p<0.01$; ${}^{**}p<0.05$; ${}^{*}p<0.1$
Table S5: State-level association for Mass Polarization (MP) and Voting Turnout (VT) in the United States (2008-2020). Each model represents a multivariate regression, with year-fixed effects. These models represent the ones presented in the main manuscript. Standard errors are in brackets and are robustly clustered by state. We standardized both independent and dependent variables to have a mean of zero and a standard deviation of one. Method | (1) | (2) | (3) | (4) | (5)
---|---|---|---|---|---
(1) EP | 1.0 | -0.24 | 0.03 | -0.29 | 0.99
(2) EC | - | 1.0 | -0.21 | -0.1 | -0.25
(3) Esteban-Ray (0.25) | - | - | 1.0 | 0.85 | 0.12
(4) Esteban-Ray (1) | - | - | - | 1.0 | -0.21
(5) Dispersion | - | - | - | - | 1.0
Table S6: Correlation matrix for measures of polarization in Chile (2021) Method | (1) | (2) | (3) | (4) | (5)
---|---|---|---|---|---
(1) EP | 1.0 | 0.38 | -0.04 | -0.33 | 0.67
(2) EC | - | 1.0 | -0.51 | -0.71 | 0.33
(3) Esteban-Ray (0.25) | - | - | 1.0 | 0.86 | 0.32
(4) Esteban-Ray (1) | - | - | - | 1.0 | -0.06
(5) Dispersion | - | - | - | - | 1.0
Table S7: Correlation matrix for measures of polarization in France (2022) Method | (1) | (2) | (3) | (4) | (5)
---|---|---|---|---|---
(1) EP | 1.0 | -0.55 | 0.82 | 0.03 | 0.5
(2) EC | - | 1.0 | -0.46 | 0.05 | -0.18
(3) Esteban-Ray (0.25) | - | - | 1.0 | 0.47 | 0.49
(4) Esteban-Ray (1) | - | - | - | 1.0 | 0.06
(5) Dispersion | - | - | - | - | 1.0
Table S8: Correlation matrix for measures of polarization in the United States
(2020)

##  J Supplementary Figures

Figure S7: Comparison of traditional polarization measures and EC/EP by using
synthetic data. The X-axis represents the sampled mean, and each bar
represents a dispersion level. Error bars are calculated by seaborn with 95%
confidence and are, in some places, thinner than the symbols in the figure
Figure S8: Univariate correlation between EC and Voting Turnout in (a) France,
2022 and (b) Chile, 2021. Both results correspond to the first round. Figure
S9: Region-level EC robustness in France (2022) by excluding candidates from
the analysis. Note: ***$p<0.01$, **$p<0.05$, *$p<0.1$. Figure S10: Region-
level EP robustness in France (2022) by excluding candidates from the
analysis. Note: ***$p<0.01$, **$p<0.05$, *$p<0.1$. Figure S11: Region-level EC
robustness in Chile (2021) by excluding candidates from the analysis. Note:
***$p<0.01$, **$p<0.05$, *$p<0.1$. Figure S12: Region-level EP robustness in
Chile (2021) by excluding candidates from the analysis. Note: ***$p<0.01$,
**$p<0.05$, *$p<0.1$.

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