# Computations of volumes and Ehrhart series in four candidates elections

## TL;DR

This paper presents experimental computations of election outcome probabilities and paradoxes using polytope volume and Ehrhart series calculations, providing insights into voting system behaviors.

## Contribution

It introduces a method using Normaliz to compute election-related probabilities through polytope volume and Ehrhart series analysis, applied to four-candidate social choice scenarios.

## Key findings

- Quantifies probabilities of Condorcet and Borda paradoxes.
- Analyzes Condorcet efficiency of plurality voting with runoff.
- Provides precise computational results for election outcome scenarios.

## Abstract

We describe several experimental results obtained in four candidates social choice elections. These include the Condorcet and Borda paradoxes, as well as the Condorcet efficiency of plurality voting with runoff. The computations are done by Normaliz. It finds precise probabilities as volumes of polytopes and counting functions encoded as Ehrhart series of polytopes.