# Expected Frequency Matrices of Elections: Computation, Geometry, and Preference Learning

arXiv: https://arxiv.org/abs/2205.07831 · HTML: https://mlsanity.com/paper/2205.07831/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 48 references that are in the map.
- First posted on arXiv 16 May 2022.

## Builds on (most cited first, 8 of 8)

- [Eliciting Single-Peaked Preferences Using Comparison Queries](https://mlsanity.com/paper/1401.3449/) (2014)
- [What Do Multiwinner Voting Rules Do? An Experiment Over the Two-Dimensional Euclidean Domain](https://mlsanity.com/paper/1901.09217/) (2019)
- [Putting a Compass on the Map of Elections](https://mlsanity.com/paper/2105.07815/) (2021)
- [Understanding Distance Measures Among Elections](https://mlsanity.com/paper/2205.00492/) (2022)
- [Generating Single Peaked Votes](https://mlsanity.com/paper/1503.02766/) (2015)
- [Recognising Multidimensional Euclidean Preferences](https://mlsanity.com/paper/1602.08109/) (2016)
- [Practical Algorithms for Multi-Stage Voting Rules with Parallel Universes Tiebreaking](https://mlsanity.com/paper/1901.09791/) (2019)
- [Collecting, Classifying, Analyzing, and Using Real-World Elections](https://mlsanity.com/paper/2204.03589/) (2022)

## Built on by (most cited first, 5 of 5)

- [DeepVoting: Learning and Fine-Tuning Voting Rules with Canonical Embeddings](https://mlsanity.com/paper/2408.13630/) (2024)
- [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)
- [A Map of Diverse Synthetic Stable Roommates Instances](https://mlsanity.com/paper/2208.04041/) (2022)
- [Fairness in the Multi-Secretary Problem](https://mlsanity.com/paper/2511.23097/) (2025)

## Abstract (arXiv)

We use the ``map of elections'' approach of Szufa et al. (AAMAS-2020) to analyze several well-known vote distributions. For each of them, we give an explicit formula or an efficient algorithm for computing its frequency matrix, which captures the probability that a given candidate appears in a given position in a sampled vote. We use these matrices to draw the ``skeleton map'' of distributions, evaluate its robustness, and analyze its properties. Finally, we develop a general and unified framework for learning the distribution of real-world preferences using the frequency matrices of established vote distributions.