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

TL;DR

This work develops a metric geometry for ranking-based voting by extending Kendall tau and Spearman footrule to incomplete ballots through two coordinate embeddings, the Borda embedding and the head-to-head embedding, producing distances $d_B$ and $d_H$ that capture global ranking structure. It introduces ballot graphs and generalized ballot graphs to realize these distances as path metrics, enabling both coordinate- and graph-based analyses for partial ballots and slate-level preferences. The authors show how to identify voter blocs and candidate slates via clustering in the induced spaces, provide synthetic validation and real-world Scottish election results demonstrating robust, interpretable structure, and connect polarization and proportionality metrics to the learned clusters. The framework supports practical analysis and visualization of electoral structure, with robust performance across embeddings and methods, and is supported by open data and code for replication.

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.

Figures (12)

• Figure 1: Under the pessimistic convention, the total shifts to convert $(A,B,D,F)$ to $(B,C,A)$ sum to 12, while they sum to 10 in the averaged convention; the Borda distances would be 6 and 5, respectively.
• Figure 2: The basic ballot graph $\mathcal{G}_3$ and shortcut ballot graph $\mathcal{G}_3^+$. Swaps of the first and last place correspond to new edges of length 2, providing shortcuts between nodes that would otherwise be 3 apart.
• Figure 3: An illustrative portion of the shortcut ballot graph $\mathcal{G}_4^+$, showing the connections between ballots headed by candidates 4 and 1. The construction of this picture shows the recursion $|\Omega_m|=m\cdot|\Omega_{m-1}|+m$.
• Figure 4: For ballots $\sigma=(A,C,B)$ and $\tau=(D,C,A,E,B)$, we first complete $\sigma$ to $\sigma'$ and then make swaps as indicated (in red) for a total Borda distance of $d_B(\sigma',\tau')=1+2+1=4$. This correctly matches half the $L^1$ difference of $\mathfrak{b}(\sigma')=(4,2,3,1,0)$ and $\mathfrak{b}(\tau')=(2,0,3,4,1)$.
• Figure 5: MDS plot of the ballots in a synthetic election $\mathcal{E}$. Ballots marked green were hit from both centers.

Theorems & Definitions (28)

• Definition 2.1: Borda and head-to-head distance
• Example 2.2
• Definition 2.3: Ballot graphs
• Example 2.4: Illustrating ballot graph construction
• Proposition 2.5
• proof
• Example 2.6
• Theorem 2.7: Graphs match $L^1$ distances
• proof
• Remark 2.8