Outer Diversity of Condorcet Domains

2026-09-30

AI Summary

This study addresses the absence of a quantitative metric for the external diversity of Condorcet domains, defining it as the expected swap distance between a uniformly random vote and its nearest vote within the domain. By integrating combinatorial optimization, probability theory, numerical simulation, and asymptotic analysis, this work systematically establishes the first theoretical framework for this metric. The proposed approach reveals the diversity patterns of maximal domains under small candidate sets and derives asymptotic bounds for specific domains. Ultimately, these contributions provide a novel paradigm for the structural analysis of preference domains.

Abstract

A Condorcet domain is a set of rankings over a given candidate set, such that every election that consists only of (an odd number of) votes from the domain has a transitive majority relation. We study outer diversity of Condorcet domains, i.e., a measure that quantifies expected swap distance from a random vote to a closest one in the domain. We numerically analyze outer diversity for maximal Condorcet domains with few candidates, and then we establish its asymptotic behavior for several special domains, mostly obtaining theoretical results.

Problem

• Condorcet domains
• outer diversity
• swap distance
• maximal Condorcet domains
• asymptotic behavior

Innovation

• Condorcet domains
• outer diversity
• swap distance
• asymptotic behavior
• maximal Condorcet domains