  Democracies employ elections at various scales to select officials at the
corresponding levels of administration. The geographical distribution of
political opinion, the policy issues delegated to each level, and the
multilevel interactions between elections can all greatly impact the makeup of
these representative bodies. This perspective is not new: the adoption of
federal systems has been motivated by the idea that they possess desirable
traits not provided by democracies on a single scale. Yet most existing models
of polarization do not capture how nested local and national elections interact
with heterogeneous political geographies. We begin by developing a framework to
describe the multilevel distribution of opinions and analyze the flow of
variance among geographic scales, applying it to historical data in the United
States from 1912 to 2020. We describe how unstable elections can arise due to
the spatial distribution of opinions and how tradeoffs occur between national
and local elections. We also examine multi-dimensional spaces of political
opinion, for which we show that a decrease in local salience can constrain the
dimensions along which elections occur, preventing a federal system from
serving as an effective safeguard against polarization. These analyses, based
on the interactions between elections and opinion distributions at various
scales, offer insights into how democracies can be strengthened to mitigate
polarization and increase electoral representation.
