Local Diversity of Condorcet Domains

TL;DR

This work introduces an egalitarian, abundance-based measure of local diversity for Condorcet domains, unifying and generalizing ampleness and copiousness. It proves a universal upper bound $2^{k-1}$ on local diversity for large numbers of alternatives and shows Black's single-peaked domain remains optimal under this measure, while also revealing high-diversity domains that are not of maximal size. The authors develop a constructive framework with $(k,s)$-abundant domains, explore inheritance of abundance across parameters, and analyze maximality and Ramsey properties to bound possible diversity. An extensive experimental analysis connects abundance to empirical and simulated preference data, illustrating how diversity can exceed that of constrained domains and how it concentrates with larger populations. Overall, the paper reframes domain design around local diversity, revealing new maximal-diversity domains and clarifying the relationship between diversity and domain size in social choice.

Abstract

Several of the classical results in social choice theory demonstrate that in order for many voting systems to be well-behaved the set domain of individual preferences must satisfy some kind of restriction, such as being single-peaked on a political axis. As a consequence it becomes interesting to measure how diverse the preferences in a well-behaved domain can be. In this paper we introduce an egalitarian approach to measuring preference diversity, focusing on the abundance of distinct suborders one subsets of the alternative. We provide a common generalisation of the frequently used concepts of ampleness and copiousness. We give a detailed investigation of the abundance for Condorcet domains. Our theorems imply a ceiling for the local diversity in domains on large sets of alternatives, which show that in this measure Black's single-peaked domain is in fact optimal. We also demonstrate that for some numbers of alternatives, there are Condorcet domains which have largest local diversity without having maximum order.

Figures (3)

• Figure 1: A $(3,4)$-abundant domain of size 4 for $n=8$
• Figure 2: The smallest discordant Condorcet domain
• Figure 3: Frequencies for given abundances. Dotted is Black's single-peaked domain, solid is 2004 AGH, dot-dashed is single-crossing, the thick solid line is Fishburn, the orange line is the symmetric/group-separable domain, red is the (5,18)-abundant domain, and dashed is the unrestricted domain/Impartial culture.

Theorems & Definitions (14)

• Definition 1
• Theorem 1
• proof
• Theorem 2
• proof
• Definition 2
• Proposition 1
• proof
• Theorem 3
• proof