Markov degree of the Birkhoff model

TL;DR

This paper proves that the Markov degree of the Birkhoff model and its generalization is three, providing a key insight into the algebraic structure of ranking models and characterizing Markov bases for small r.

Contribution

It confirms the conjecture by Diaconis and Eriksson that the Markov degree is three and characterizes Markov bases for small values of r in the generalized model.

Findings

Markov degree of the Birkhoff model is three

Confirmed conjecture by Diaconis and Eriksson (2006)

Characterized Markov bases for small r

Abstract

We prove the conjecture by Diaconis and Eriksson (2006) that the Markov degree of the Birkhoff model is three. In fact, we prove the conjecture in a generalization of the Birkhoff model, where each voter is asked to rank a fixed number, say r, of candidates among all candidates. We also give an exhaustive characterization of Markov bases for small r.