Map of Elections

TL;DR

The thesis introduces a comprehensive framework, Map of Elections, to study and visualize the space of ordinal elections by combining datasets, computable distance measures, and 2D embeddings. It formalizes both isomorphic and nonisomorphic distance notions, analyzes their computational properties, and develops embedding-based maps using methods like Kamada-Kawai. A rich set of statistical cultures, compass elections, and Euclidean models underpin synthetic data generation, while aggregate representations and ILP formulations enable practical distance computation. The work demonstrates how maps reveal structure across models, supports experimental design for evaluating voting rules, and offers pathways for real-life data analysis, including map-based comparisons and visualization of complex election spaces. Overall, it provides a principled, scalable toolkit for exploring the geometry of elections and their algorithmic properties.

Abstract

Our main contribution is the introduction of the map of elections framework. A map of elections consists of three main elements: (1) a dataset of elections (i.e., collections of ordinal votes over given sets of candidates), (2) a way of measuring similarities between these elections, and (3) a representation of the elections in the 2D Euclidean space as points, so that the more similar two elections are, the closer are their points. In our maps, we mostly focus on datasets of synthetic elections, but we also show an example of a map over real-life ones. To measure similarities, we would have preferred to use, e.g., the isomorphic swap distance, but this is infeasible due to its high computational complexity. Hence, we propose polynomial-time computable positionwise distance and use it instead. Regarding the representations in 2D Euclidean space, we mostly use the Kamada-Kawai algorithm, but we also show two alternatives. We develop the necessary theoretical results to form our maps and argue experimentally that they are accurate and credible. Further, we show how coloring the elections in a map according to various criteria helps in analyzing results of a number of experiments. In particular, we show colorings according to the scores of winning candidates or committees, running times of ILP-based winner determination algorithms, and approximation ratios achieved by particular algorithms.

Figures (11)

• Figure 1: An example of Map of Elections.
• Figure 2: ILPs runtime (in seconds) for Harmonic-Borda voting rule.
• Figure 3: Examples of PCC values. In each picture we visualize two vectors $(x_1,\dots,x_n)$ and $(y_1,\dots,y_n)$ as points $(p_1,\dots,p_n)$, where each point $p_i$ has coordinates $x_i, y_i$.
• Figure 4: Toy example with matrix distances (left) and the map (right).
• Figure 5: Three examples of clone decomposition trees.

Theorems & Definitions (35)

• Example 2.1
• Definition 3.1
• Example 3.1
• Definition 3.2: mir:j:single-crossing, rob:j:tax
• Example 3.2
• Definition 3.3
• Example 3.3
• Definition 3.4
• Proposition 3.1
• proof