This paper develops a geometric way to compare and summarize ranked ballots. The authors show that two standard measures of difference between rankings — Kendall tau (which counts how many pairwise swaps separate two rankings) and Spearman’s footrule (which sums how far candidates shift in position) — can be extended to ballots that are incomplete or allow ties. They do this by turning ballots into points in a coordinate space or nodes in a sparse graph, so distances between ballots become ordinary geometric or path distances.

Concretely, the paper gives two coordinate embeddings. The “Borda” embedding turns a ranking into a vector of scores (a candidate ranked jth gets score m−j), and the L1 distance between these vectors (scaled) recovers a Spearman-type measure. The “head-to-head” embedding records each pairwise comparison as +1, −1, or 0, and the L1 distance there (scaled) recovers Kendall tau. The authors also build “ballot graphs” whose path distances reproduce the same metrics while keeping the graph sparse. In the head-to-head picture, the center that minimizes L1 distance is the Kemeny ranking (a consensus ranking based on pairwise wins); in the Borda picture the L2 minimizer recovers the usual Borda ranking.

Why this matters: turning ballots into points or graph nodes creates a common geometric language for topics in voting theory such as metric preferences (how voters and candidates might lie in an underlying space), polarization, and proportionality. The metric view gives practical methods to cluster similar voters into blocs and to identify slates of preferred candidates. Because the definitions handle partial ballots, the methods can be run on both synthetic data and real elections. The authors test the approach on simulated elections and on a suite of over 1,000 ranked‑choice local government elections in Scotland (2012–2022), which include five major parties and many minor ones. Importantly, the clustering methods do not use party labels and often produce stable groupings even when the clustering is driven by different ranking rules (one family of methods is Condorcet‑consistent while the other is not).