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Condorcet Domains of Degree at most Seven

Dolica Akello-Egwell, Charles Leedham-Green, Alastair Litterick, Klas Markström, Søren Riis

TL;DR

This work resolves the explicit enumeration and structural analysis of maximal Condorcet domains for up to seven alternatives using a novel generation algorithm implemented on high-performance computing infrastructure. It delivers complete MUCD catalogs, confirmsFishburn-type maximality results up to $n=7$, and provides extensive analysis of properties such as reducibility, copiousness, normality, self-duality, and connectivity. The study also clarifies relationships between Condorcet domains and other domain types (e.g., Arrow single-peaked, USP) and makes the data publicly available to support further research. Together, these contributions deepen understanding of the combinatorial structure of Condorcet domains and their potential applications in social choice and algorithm design.

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.

Condorcet Domains of Degree at most Seven

TL;DR

This work resolves the explicit enumeration and structural analysis of maximal Condorcet domains for up to seven alternatives using a novel generation algorithm implemented on high-performance computing infrastructure. It delivers complete MUCD catalogs, confirmsFishburn-type maximality results up to , and provides extensive analysis of properties such as reducibility, copiousness, normality, self-duality, and connectivity. The study also clarifies relationships between Condorcet domains and other domain types (e.g., Arrow single-peaked, USP) and makes the data publicly available to support further research. Together, these contributions deepen understanding of the combinatorial structure of Condorcet domains and their potential applications in social choice and algorithm design.

Abstract

In this paper we give the first explicit enumeration of all maximal Condorcet domains on 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.
Paper Structure (17 sections, 7 theorems, 1 equation, 3 figures, 12 tables)

This paper contains 17 sections, 7 theorems, 1 equation, 3 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Background material and Definitions
  4. Transformations and isomorphism of Condorcet domains
  5. The weak Bruhat Order and Condorcet domains as posets
  6. Bounds for the size of a MCD
  7. Closed CDs and sets of laws
  8. The Generation Algorithm
  9. Implied laws and redundancy
  10. Maximality
  11. Final reduction, parallelisation, and implementation
  12. The MUCDS of degree at most 7 and their properties.
  13. The sizes and numbers of MUCDs
  14. The structure of MUCDs
  15. Connectivity and Peak-Pit domains
  16. ...and 2 more sections

Key Result

Lemma 2.1

If two UCDs $A$ and $B$ are isomorphic then $Ag^{-1}=B$ for some $g$ in $A$.

Figures (3)

  • Figure 4.1: The number of MUCD classes as function of domains size for $n=7$
  • Figure 4.2: The smallest non-ample MUCD
  • Figure 4.3: A fixing MUCD of order 5 and size 4.

Theorems & Definitions (24)

  • Definition 2.1
  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • ...and 14 more