# Properties of Position Matrices and Their Elections

arXiv: https://arxiv.org/abs/2303.02538 · HTML: https://mlsanity.com/paper/2303.02538/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Mar 2023.

## Built on by (most cited first, 1 of 1)

- [Drawing a Map of Elections](https://mlsanity.com/paper/2504.03809/) (2025)

## Abstract (arXiv)

We study the properties of elections that have a given position matrix (in such elections each candidate is ranked on each position by a number of voters specified in the matrix). We show that counting elections that generate a given position matrix is #P-complete. Consequently, sampling such elections uniformly at random seems challenging and we propose a simpler algorithm, without hard guarantees. Next, we consider the problem of testing if a given matrix can be implemented by an election with a certain structure (such as single-peakedness or group-separability). Finally, we consider the problem of checking if a given position matrix can be implemented by an election with a Condorcet winner. We complement our theoretical findings with experiments.