A combinatorial representation of Arrow's single-peaked domains

TL;DR

This paper introduces a new combinatorial representation for Arrow's single-peaked domains using generalized arrangements of pseudolines, expanding the understanding of Condorcet domains beyond maximal width cases.

Contribution

It proposes a novel combinatorial model for Arrow's single-peaked domains, extending existing representations to non-maximal width cases.

Findings

Representation via generalized pseudoline arrangements

Connection between single-peaked domains and combinatorial objects

Extension of combinatorial models for Condorcet domains

Abstract

The most studied class of Condorcet domains (acyclic sets of linear orders) is the class of peak-pit domains of maximal width. It has a number of combinatorial representations by such familiar combinatorial objects like rhombus tilings and arrangements of pseudolines. Arrow's single-peaked domains are peak-pit but do not have maximal width. We suggest how to represent them by means of generalised arrangements of pseudolines.