Metric geometry for ranking-based voting: Tools for learning electoral structure

TL;DR

This paper introduces a unified metric geometric framework for ranking statistics, extending Kendall tau and Spearman distances to incomplete rankings, enabling analysis of electoral preferences and voter clustering in real-world elections.

Contribution

It develops a novel geometric approach that unifies permutation distances for incomplete rankings, facilitating new methods for electoral analysis and voter clustering.

Findings

Effective clustering of voters in real elections

Robust voter grouping across different ranking methods

Extension of permutation distances to incomplete rankings

Abstract

In this paper, we develop the metric geometry of ranking statistics, proving that the two major permutation distances in the statistics literature -- Kendall tau and Spearman footrule -- extend naturally to incomplete rankings with both coordinate embeddings and graph realizations. This gives us a unifying framework that allows us to connect popular topics in computational social choice: metric preferences (and metric distortion), polarization, and proportionality. As an important application, the metric structure enables efficient identification of blocs of voters and slates of their preferred candidates. Since the definitions work for partial ballots, we can execute the methods not only on synthetic elections, but on a suite of real-world elections. This gives us robust clustering methods that often produce an identical grouping of voters -- even though one family of methods is based on a Condorcet-consistent ranking rule while the other is not.