The single-crossing property on a tree

TL;DR

This paper extends the single-crossing property to tree structures, enabling new constructions of Condorcet domains, and provides algorithms for recognition and winner determination in such profiles.

Contribution

It introduces the concept of single-crossing on trees, proves existence and minimality results, and offers polynomial algorithms for recognition and voting rules.

Findings

Existence of profiles single-crossing on any tree

Polynomial-time recognition algorithm for such profiles

Efficient winner determination for Chamberlin-Courant rule

Abstract

We generalize the classical single-crossing property to single-crossing property on trees and obtain new ways to construct Condorcet domains which are sets of linear orders which possess the property that every profile composed from those orders have transitive majority relation. We prove that for any tree there exist profiles that are single-crossing on that tree; moreover, that tree is minimal in this respect for at least one such profile. Finally, we provide a polynomial-time algorithm to recognize whether or not a given profile is single-crossing with respect to some tree. We also show that finding winners for Chamberlin-Courant rule is polynomial for profiles that are single-crossing on trees.