Improved lower bounds for the maximum size of Condorcet domains

TL;DR

This paper advances the understanding of Condorcet domains by computationally identifying larger domains for 9 to 25 alternatives, thereby improving lower bounds on their maximum size and contributing to voting theory.

Contribution

The authors develop a new inductive search method and use supercomputing to find larger Condorcet domains, improving known bounds for 9 to 25 alternatives.

Findings

Largest known domains for 9 to 20 alternatives improved

New lower bound for maximum size: Ω(2.198139^n)

Properties and open problems of the domains discussed

Abstract

Condorcet domains are sets of linear orders with the property that, whenever voters' preferences are restricted to the domain, the pairwise majority relation (for an odd number of voters) is transitive and hence a linear order. Determining the maximum size of a Condorcet domain, sometimes under additional constraints, has been a longstanding problem in the mathematical theory of majority voting. The exact maximum is only known for $n\leq 8$ alternatives. In this paper we use a structural analysis of the largest domains for small $n$ to design a new inductive search method. Using an implementation of this method on a supercomputer, together with existing algorithms, we improve the size of the largest known domains for all $9 \leq n \leq 20$. These domains are then used in a separate construction to obtain the currently largest known domains for $21 \leq n \leq 25$, and to improve the best asymptotic lower bound for the maximum size of a Condorcet domain to $\Omega(2.198139^n)$. Finally, we discuss properties of the domains found and state several open problems and conjectures.