Computations of volumes in five candidates elections

TL;DR

This paper presents analytical computations of election outcome probabilities in five-candidate social choice scenarios under the Impartial Anonymous Culture, using volume calculations of high-dimensional polytopes.

Contribution

It provides exact probability calculations for various voting paradoxes and efficiencies in five-candidate elections using advanced polytope volume algorithms.

Findings

Precise probabilities for Condorcet and Borda paradoxes.

Efficiency metrics for plurality, negative plurality, and Borda voting.

Analysis includes runoff voting systems.

Abstract

We describe several analytical results obtained in five candidates social choice elections under the assumption of the Impartial Anonymous Culture. These include the Condorcet and Borda paradoxes, as well as the Condorcet efficiency of plurality, negative plurality and Borda voting, including their runoff versions. The computations are done by Normaliz. It finds precise probabilities as volumes of polytopes in dimension 119, using its recent implementation of the Lawrence algorithm.