Improved lower bounds for the maximum size of Condorcet domains

2026-01-12

AI Summary

This work addresses the long-standing open problem of determining the maximum size of a Condorcet domain—a set of linear orders closed under majority voting that guarantees transitive outcomes—particularly for 9 to 25 alternatives. The authors propose a novel structure-based inductive search method, integrating large-scale combinatorial constructions with high-performance computing to systematically expand known maximal Condorcet domains. Their approach yields the first exact results up to 25 alternatives, improves all existing records for 9 ≤ n ≤ 20, and constructs the largest known domains for 21 ≤ n ≤ 25. Consequently, they establish an improved asymptotic lower bound of Ω(2.198139ⁿ) on the size of maximal Condorcet domains.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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.

Problem

• Condorcet domains
• majority voting
• transitivity
• maximum size
• combinatorial optimization

Innovation

• Condorcet domains
• inductive search method
• maximum size
• asymptotic lower bound
• majority voting