Condorcet Domains of Degree at most Seven

TL;DR

This paper enumerates all maximal Condorcet domains with up to seven alternatives, providing new insights into their properties, symmetries, and connections to other domain types, supported by extensive computational data.

Contribution

It introduces a novel algorithm for constructing Condorcet domains, performs the first exhaustive enumeration for n ≤ 7, and analyzes their properties and symmetries.

Findings

Complete enumeration of maximal Condorcet domains for n ≤ 7

Resolution of open questions in the literature

New results on symmetry properties of Condorcet domains

Abstract

In this paper we give the first explicit enumeration of all maximal Condorcet domains on $n\leq 7$ alternatives. This has been accomplished by developing a new algorithm for constructing Condorcet domains, and an implementation of that algorithm which has been run on a supercomputer. We follow this up by the first survey of the properties of all maximal Condorcet domains up to degree 7, with respect to many properties studied in the social sciences and mathematical literature. We resolve several open questions posed by other authors, both by examples from our data and theorems. We give a new set of results on the symmetry properties of Condorcet domains which unify earlier works. Finally we discuss connections to other domain types such as non-dictatorial domains and generalisations of single-peaked domains. All our data is made freely available for other researches via a new website.