Condorcet domains of tiling type

TL;DR

This paper introduces a novel method for constructing large Condorcet domains using rhombus tiling diagrams, unifies previous approaches, and addresses conjectures on their maximal sizes by providing a counterexample.

Contribution

It presents a new tiling-based construction method for Condorcet domains and demonstrates the equivalence of conjectures on their maximal sizes, including a counterexample.

Findings

The tiling method unifies earlier constructions of Condorcet domains.

Three conjectures on maximal sizes are shown to be equivalent.

A counterexample disproves the conjectures on maximal sizes.

Abstract

A Condorcet domain (CD) is a collection of linear orders on a set of candidates satisfying the following property: for any choice of preferences of voters from this collection, a simple majority rule does not yield cycles. We propose a method of constructing "large" CDs by use of rhombus tiling diagrams and explain that this method unifies several constructions of CDs known earlier. Finally, we show that three conjectures on the maximal sizes of those CDs are, in fact, equivalent and provide a counterexample to them.