Local Diversity of Condorcet Domains

TL;DR

This paper introduces a new measure of preference diversity within well-behaved voting domains, analyzing the maximum local diversity possible in Condorcet domains and establishing optimality results.

Contribution

It develops an egalitarian measure of preference diversity, generalizes key concepts, and characterizes the maximum local diversity in Condorcet domains, showing Black's single-peaked domain is optimal.

Findings

Black's single-peaked domain is optimal for local diversity.

Condorcet domains can have maximum local diversity without maximum order.

The paper establishes an upper bound for local diversity in large sets of alternatives.

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.