Improved lower bounds for the maximum size of Condorcet domains

TL;DR

This work tackles the long-standing problem of determining the maximum size of Condorcet domains (CDs) by developing an inductive construction that builds larger CDs on $n{+}1$ alternatives from large CDs on $n$ alternatives. The authors implement a three-step inductive algorithm—instantiation, overlap compatibility, and completion with pruning—along with a deterministic shortcut, enabling computation of the largest known CDs for $9\le n\le 20$ and enabling improved constructions for $21\le n\le 25$ via existing methods. As a key theoretical contribution, they obtain an improved asymptotic lower bound on the maximum CD size: $\Omega(2.198139^n)$, by combining their domains with Fishburn’s replacement scheme and product constructions that yield $e(n+m)\ge 2 e(n) e(m)$. The results also yield structural insights into the new domains, including abundance properties (peak-pit, copiousness), poset structure of restrictions, and nontrivial core behavior, while raising open problems about exact values for certain classes and larger-$n$ behavior. Overall, the paper advances both computational and combinatorial understanding of Condorcet domains and provides concrete, scalable methods for constructing larger CDs with controlled structural properties.

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 $Ω(2.198139^n)$. Finally, we discuss properties of the domains found and state several open problems and conjectures.

Theorems & Definitions (10)

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