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The Wasserstein Bipolarization Index: A New Measure of Public Opinion Polarization, with an Application to Cross-Country Attitudes toward COVID-19 Vaccination Mandates

Hane Lee, Michael E. Sobel

TL;DR

The paper addresses the lack of principled measures for public opinion polarization by formulating an axiomatic framework inspired by income bipolarization and introducing a $p$-Wasserstein polarization index. The index quantifies polarization as the distance between the observed opinion distribution $\nu_X$ and a maximally separated endpoint distribution $\xi_{(\nu_X,c,\gamma)}$, using the $p$-Wasserstein distance and a center $c$ (often the median) with a mass parameter $\gamma$. It demonstrates that traditional metrics like variance and bimodality fail the proposed axioms, proves the $p$-Wasserstein measure satisfies Axioms A1–A5, and provides a rescaled, interpretable index $I_p(\nu_X, \xi_{(\nu_X,c,\gamma)})$ along with inference via asymptotic theory and confidence intervals. The empirical application to cross-country attitudes toward COVID-19 vaccination mandates shows US and UK as the most polarized and China, India, and France as the least, illustrating the method’s ability to yield nuanced, uncertainty-aware comparisons across populations. Overall, the work offers a principled, geometry-based tool for measuring polarization with broad applicability to public opinion data and policy analysis, and it provides software support via an R package for practitioners.

Abstract

Although the topic of opinion polarization receives much attention from the media, public opinion researchers and political scientists, the phenomenon itself has not been adequately characterized in either the lay or academic literature. To study opinion polarization among the public, researchers compare the distributions of respondents to survey questions or track the distribution of responses to a question over time using ad-hoc methods and measures such as visual comparisons, variances, and bimodality coefficients. To remedy this situation, we build on the axiomatic approach in the economics literature on income bipolarization, specifying key properties a measure of bipolarization should satisfy: in particular, it should increase as the distribution spreads away from a center toward the poles and/or as clustering below or above this center increases. We then show that measures of bipolarization used in public opinion research fail to satisfy one or more of these axioms. Next, we propose a $p$-Wasserstein polarization index that satisfies the axioms we set forth. Our index measures the dissimilarity between an observed distribution and a distribution with all the mass clustered on the lower and upper endpoints of the scale. We use our index to examine bipolarization in attitudes toward governmental COVID-19 vaccine mandates across 11 countries, finding the U.S and U.K are most polarized, China, France and India the least polarized, while the others (Brazil, Australia, Colombia, Canada, Italy, Spain) occupy an intermediate position.

The Wasserstein Bipolarization Index: A New Measure of Public Opinion Polarization, with an Application to Cross-Country Attitudes toward COVID-19 Vaccination Mandates

TL;DR

The paper addresses the lack of principled measures for public opinion polarization by formulating an axiomatic framework inspired by income bipolarization and introducing a -Wasserstein polarization index. The index quantifies polarization as the distance between the observed opinion distribution and a maximally separated endpoint distribution , using the -Wasserstein distance and a center (often the median) with a mass parameter . It demonstrates that traditional metrics like variance and bimodality fail the proposed axioms, proves the -Wasserstein measure satisfies Axioms A1–A5, and provides a rescaled, interpretable index along with inference via asymptotic theory and confidence intervals. The empirical application to cross-country attitudes toward COVID-19 vaccination mandates shows US and UK as the most polarized and China, India, and France as the least, illustrating the method’s ability to yield nuanced, uncertainty-aware comparisons across populations. Overall, the work offers a principled, geometry-based tool for measuring polarization with broad applicability to public opinion data and policy analysis, and it provides software support via an R package for practitioners.

Abstract

Although the topic of opinion polarization receives much attention from the media, public opinion researchers and political scientists, the phenomenon itself has not been adequately characterized in either the lay or academic literature. To study opinion polarization among the public, researchers compare the distributions of respondents to survey questions or track the distribution of responses to a question over time using ad-hoc methods and measures such as visual comparisons, variances, and bimodality coefficients. To remedy this situation, we build on the axiomatic approach in the economics literature on income bipolarization, specifying key properties a measure of bipolarization should satisfy: in particular, it should increase as the distribution spreads away from a center toward the poles and/or as clustering below or above this center increases. We then show that measures of bipolarization used in public opinion research fail to satisfy one or more of these axioms. Next, we propose a -Wasserstein polarization index that satisfies the axioms we set forth. Our index measures the dissimilarity between an observed distribution and a distribution with all the mass clustered on the lower and upper endpoints of the scale. We use our index to examine bipolarization in attitudes toward governmental COVID-19 vaccine mandates across 11 countries, finding the U.S and U.K are most polarized, China, France and India the least polarized, while the others (Brazil, Australia, Colombia, Canada, Italy, Spain) occupy an intermediate position.
Paper Structure (9 sections, 3 theorems, 13 equations, 2 figures)

This paper contains 9 sections, 3 theorems, 13 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Bipolarization: A New Measure
  3. Axioms
  4. Polarization Indices in Public Opinion Research: Variance and Bimodality
  5. $p$-Wasserstein measure of polarization
  6. A Cross-Country Comparison of Attitudes toward Government Mandates on Vaccination for COVID-19
  7. Rescaling $W_p(\nu_{X}, \xi_{(\nu_{X},c, \gamma)})$ to $[0,1]$: The index $I_p(\nu_{X}, \xi_{(\nu_{X},c, \gamma)})$.
  8. Results
  9. Discussion

Key Result

Proposition 1

Let $\nu_X$, with center $c$, be a probability measure defined on $\mathcal{X}=[\ell,L]$, and let $\xi_{(\nu_{X},c, \gamma)}$ denote the corresponding maximum separation measure. Then, the optimal $p$-Wasserstein coupling $\pi_\text{opt}$ maps mass $\nu_{X}[\ell,c) + \gamma\nu_{X}(c)$ to $\ell$, mas

Figures (2)

  • Figure 1: $\nu_Y$ is more polarized than $\nu_X$.
  • Figure 2: $p=1$ and $p=2$ Wasserstein polarization index point estimates with bars indicating 95% confidence intervals. Countries are ordered by increasing polarization for $p=2$.

Theorems & Definitions (8)

  • Definition
  • Definition
  • Definition
  • Definition
  • Definition
  • Proposition 1
  • Proposition 2
  • Proposition 3