  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.
