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Drawing a Map of Elections

Stanisław Szufa, Niclas Boehmer, Robert Bredereck, Piotr Faliszewski, Rolf Niedermeier, Piotr Skowron, Arkadii Slinko, Nimrod Talmon

TL;DR

This work proposes polynomial-time computable positionwise distance and uses it to measure similarities between elections, and shows how coloring the elections in a map according to various criteria helps in analyzing results of a number of experiments.

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.

Drawing a Map of Elections

TL;DR

This work proposes polynomial-time computable positionwise distance and uses it to measure similarities between elections, and shows how coloring the elections in a map according to various criteria helps in analyzing results of a number of experiments.

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.

Paper Structure

This paper contains 19 sections, 5 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivating Example
  3. Maps of Elections
  4. Election Datasets and Compass.
  5. Similarities Between Elections.
  6. Embedding Algorithms.
  7. Use Cases for the Maps
  8. Related Work
  9. The Map Framework.
  10. Distances.
  11. Election Data.
  12. Embedding Algorithms.
  13. Preliminaries
  14. Elections
  15. Structured Preferences
  16. ...and 4 more sections

Figures (2)

  • Figure 1: Two examples of maps of elections. Each point on each of the maps represents an election with $10$ candidates and $100$ voters. Elections on the left were generated using various statistical cultures (as reflected by their colors), whereas those on the right were derived from real-life data (after appropriate preprocessing). We define various statistical cultures in \ref{['sec:cultures']}. Maps of synthetic elections are discussed in detail in \ref{['sec:map-cultures']}, whereas the map of real-life elections is discussed in \ref{['preflib_info']}; the maps will become fully comprehensible only after reading these sections. The maps include four special points (UN, ID, ST, AN), to which we refer as the "compass," and their connecting paths. The map on the right also includes pale blue and orange areas, which represent places where Mallows and urn elections would land, respectively.
  • Figure 2: Trees from Example \ref{['ex:g-s']}.

Theorems & Definitions (8)

  • Definition 2.1: Black bla:b:polsci:committees-elections, Peters and Lackner pet-lac:j:spoc
  • Example 2.1
  • Definition 2.2: Mirrlees mir:j:single-crossing, Roberts rob:j:tax
  • Example 2.2
  • Definition 2.3: ina:j:group-separableina:j:simple-majoritykar:j:group-separable
  • Example 2.3
  • Definition 2.4
  • Example 3.1