Distances Between Top-Truncated Elections of Different Sizes

TL;DR

This paper extends the map of elections framework to handle elections of differing sizes and top-truncated votes, enabling direct visualization of large, real-world datasets such as Preflib. It introduces a feature-based distance, DAP, built from diversity $D$, agreement $A$, and polarization $P$, and shows it correlates with the isomorphic swap distance while natively handling truncation; it also analyzes the limitations of extending swap and positionwise distances and proposes a UN-consistent positionwise extension for cross-size comparisons. Through synthetic experiments and maps, the authors demonstrate that DAP yields stable, interpretable embeddings that resemble known statistical cultures (IC, Mallows, urn, Euclidean) and traces of Preflib data in relation to these models. The Map of Preflib illustrates practical applicability: real elections largely occupy regions corresponding to high-dimensional Euclidean, Mallows, or urn-generated elections, validating the approach for visual analytics and comparison of heterogeneous election data. Overall, the work provides a scalable toolkit for visualizing and comparing diverse election datasets without preprocessing, with clear guidance on when to use DAP or frequency-matrix-based distances depending on truncation and size characteristics.

Abstract

The map of elections framework is a methodology for visualizing and analyzing election datasets. So far, the framework was restricted to elections that have equal numbers of candidates, equal numbers of voters, and where all the (ordinal) votes rank all the candidates. We extend it to the case of elections of different sizes, where the votes can be top-truncated. We use our results to present a visualization of a large fragment of the Preflib database.

Figures (13)

• Figure 1: Maps of elections created using the (a) isomorphic swap and (b) positionwise distances. Each point corresponds to an election; the color represents the statistical culture it comes from: $\mathrm{ID}\xspace$, $\mathrm{UN}\xspace$, and $\mathrm{AN}\xspace$ refer to identity, uniformity, and antagonism elections; IC, Mallows, and urn mean impartial culture, the normalized Mallows model, and the Pólya-Eggenberger urn model, respectively; Interval, Square, Cube, 5D/10D Cube, Circle, and Sphere refer to Euclidean models; SP stands for single-peaked (in the Conitzer or Walsh models), SPOC for single-peaked on a circle, and GS for group-separable (caterpillar or balanced).
• Figure 2: Maps of elections created using the positionwise and DAP distances, for the truncation-oriented datasets. Top-$k$ truncated elections are marked with triangles, random-cut truncated ones with crosses, and complete ones with circles.
• Figure 3: Plots (a) show average distance from size-16 elections to different-sized, complete elections from the same culture, as a fraction of the maximum distance in our dataset (positionwise distance on the left, DAP distance on the right). Plots (b) show average distance from a complete election to its top-$k$ truncation, as a fraction of the maximum distance in our dataset (positionwise distance on the left, DAP distance on the right).
• Figure 4: Map of Preflib elections (black dots) in addition to the synthetic ones (large pale discs), obtained using the DAP distance.
• Figure 5: Illustration of the proof that $W(\vec{a},\vec{b}) \leq \frac{1}{m}\mathrm{emd}(\vec{a},\vec{b})$.

Theorems & Definitions (20)

• Proposition 2.1
• Proposition 3.1
• Proposition 3.2
• Proposition 3.3
• Theorem 3.4
• Proposition 3.5
• Proposition 3.6
• Proposition 3.7
• Proposition A.1
• proof