Exploiting Polyhedral Symmetries in Social Choice

TL;DR

This paper demonstrates how exploiting polyhedral symmetries can make complex probability calculations in social choice theory feasible, especially for election scenarios with many candidates.

Contribution

It introduces a symmetry-based approach to simplify polyhedral computations in social choice, enabling analysis of larger and more complex voting models.

Findings

Symmetry exploitation reduces computational complexity.

Applied method to Condorcet's paradox and voting efficiency cases.

Enabled analysis of previously infeasible election scenarios.

Abstract

A large amount of literature in social choice theory deals with quantifying the probability of certain election outcomes. One way of computing the probability of a specific voting situation under the Impartial Anonymous Culture assumption is via counting integral points in polyhedra. Here, Ehrhart theory can help, but unfortunately the dimension and complexity of the involved polyhedra grows rapidly with the number of candidates. However, if we exploit available polyhedral symmetries, some computations become possible that previously were infeasible. We show this in three well known examples: Condorcet's paradox, Condorcet efficiency of plurality voting and in Plurality voting vs Plurality Runoff.