Note on level r consensus

TL;DR

This paper investigates the hierarchy of level r consensus in preference profiles and demonstrates that higher levels of consensus can collapse to lower levels, simplifying the understanding of consensus structures.

Contribution

The paper reveals that the hierarchy of level r consensus partially collapses, showing that high-level consensus implies lower-level consensus around the same preference.

Findings

Hierarchy of level r consensus partially collapses

Profiles with high-level consensus also exhibit low-level consensus

Simplifies the analysis of preference consensus structures

Abstract

We show that the hierarchy of level $r$ consensus partially collapses. In particular, any profile $\pi\in \mathcal{P}$ that exhibits consensus of level $(K-1)!$ around $\succ_0$ in fact exhibits consensus of level $1$ around $\succ_0$.