# Drawing a Map of Elections

arXiv: https://arxiv.org/abs/2504.03809 · HTML: https://mlsanity.com/paper/2504.03809/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 128 references that are in the map.
- First posted on arXiv 4 Apr 2025.

## Builds on (most cited first, 30 of 33)

- [Multi-Winner Voting with Approval Preferences](https://mlsanity.com/paper/2007.01795/) (2020)
- [On the Computation of Fully Proportional Representation](https://mlsanity.com/paper/1402.0580/) (2014)
- [Finding a Collective Set of Items: From Proportional Multirepresentation to Group Recommendation](https://mlsanity.com/paper/1402.3044/) (2014)
- [Eliciting Single-Peaked Preferences Using Comparison Queries](https://mlsanity.com/paper/1401.3449/) (2014)
- [Proportional Rankings](https://mlsanity.com/paper/1612.01434/) (2016)
- [What Do Multiwinner Voting Rules Do? An Experiment Over the Two-Dimensional Euclidean Domain](https://mlsanity.com/paper/1901.09217/) (2019)
- [Preference Restrictions in Computational Social Choice: A Survey](https://mlsanity.com/paper/2205.09092/) (2022)
- [Putting a Compass on the Map of Elections](https://mlsanity.com/paper/2105.07815/) (2021)
- [Achieving Fully Proportional Representation: Approximability Results](https://mlsanity.com/paper/1312.4026/) (2013)
- [Diversity, Agreement, and Polarization in Elections](https://mlsanity.com/paper/2305.09780/) (2023)
- [Guide to Numerical Experiments on Elections in Computational Social Choice](https://mlsanity.com/paper/2402.11765/) (2024)
- [Controlling Elections through Social Influence](https://mlsanity.com/paper/1711.08615/) (2017)
- [Understanding Distance Measures Among Elections](https://mlsanity.com/paper/2205.00492/) (2022)
- [Generating Single Peaked Votes](https://mlsanity.com/paper/1503.02766/) (2015)
- [Comparing Election Methods Where Each Voter Ranks Only Few Candidates](https://mlsanity.com/paper/1901.10848/) (2019)
- [How to Sample Approval Elections?](https://mlsanity.com/paper/2207.01140/) (2022)
- [A Quantitative and Qualitative Analysis of the Robustness of (Real-World) Election Winners](https://mlsanity.com/paper/2208.13760/) (2022)
- [Properties of the Mallows Model Depending on the Number of Alternatives: A Warning for an Experimentalist](https://mlsanity.com/paper/2401.14562/) (2024)
- [Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty](https://mlsanity.com/paper/1509.02424/) (2015)
- [Optimizing positional scoring rules for rank aggregation](https://mlsanity.com/paper/1609.07460/) (2016)
- [Proportional Approval Voting, Harmonic k-median, and Negative Association](https://mlsanity.com/paper/1704.02183/) (2017)
- [Practical Algorithms for Multi-Stage Voting Rules with Parallel Universes Tiebreaking](https://mlsanity.com/paper/1901.09791/) (2019)
- [Abductive and Contrastive Explanations for Scoring Rules in Voting](https://mlsanity.com/paper/2408.12927/) (2024)
- [Optimal Algorithms for Multiwinner Elections and the Chamberlin-Courant Rule](https://mlsanity.com/paper/2106.00091/) (2021)
- [An Experimental Comparison of Multiwinner Voting Rules on Approval Elections](https://mlsanity.com/paper/2401.11870/) (2024)
- [Condorcet Domains, Median Graphs and the Single Crossing Property](https://mlsanity.com/paper/1507.08219/) (2015)
- [Distance rationalization of anonymous and homogeneous voting rules](https://mlsanity.com/paper/1610.01900/) (2016)
- [Local Distance Constrained Bribery in Voting](https://mlsanity.com/paper/1901.08711/) (2019)
- [The Complexity of Subelection Isomorphism Problems](https://mlsanity.com/paper/2105.11923/) (2021)
- [A Map of Diverse Synthetic Stable Roommates Instances](https://mlsanity.com/paper/2208.04041/) (2022)

## Built on by (most cited first, 4 of 4)

- [Diversity of Structured Domains via k-Kemeny Scores](https://mlsanity.com/paper/2509.15812/) (2025)
- [Outer Diversity of Structured Domains](https://mlsanity.com/paper/2602.15708/) (2026)
- [Agreement, Diversity, and Polarization Indices for Approval Elections](https://mlsanity.com/paper/2605.14983/) (2026)
- [Outer Diversity of Condorcet Domains](https://mlsanity.com/paper/2610.00720/) (2026)

## Abstract (arXiv)

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.