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.

翻译：本文发展了排序统计量的度量几何理论，证明了统计学文献中的两种主要排列距离——Kendall tau距离与Spearman footrule距离——能够通过坐标嵌入和图实现自然地推广至不完全排序。这为我们提供了一个统一框架，用以连接计算社会选择中的若干热点议题：度量偏好（及度量失真）、极化现象与比例代表性。作为一项重要应用，该度量结构能够高效识别选民集团及其偏好候选人组合。由于相关定义适用于部分选票，我们不仅能在合成选举中实施这些方法，还能将其应用于一系列真实世界选举。由此获得的鲁棒聚类方法常能产生完全一致的选民分组结果——尽管其中一类方法基于孔多塞相容排序规则，而另一类则否。