An iterative Markov rating method

Stephen Devlin; Thomas Treloar; Molly Creagar; Samuel Cassels

doi:10.1515/jqas-2019-0070

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An iterative Markov rating method

Stephen Devlin , Thomas Treloar , Molly Creagar und Samuel Cassels

Veröffentlicht/Copyright: 5. Oktober 2020

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Aus der Zeitschrift Journal of Quantitative Analysis in Sports Band 17 Heft 2

Abstract

We introduce a simple and natural iterative version of the well-known and widely studied Markov rating method. We show that this iterative Markov method converges to the usual global Markov rating, and shares a close relationship with the well-known Elo rating. Together with recent results on the relationship between the global Markov method and the maximum likelihood estimate of the rating vector in the Bradley–Terry (BT) model, we connect and explore the global and iterative Markov, Elo, and Bradley–Terry ratings on real and simulated data.

Keywords: Elo; Markov; ranking; rating

Corresponding author: Stephen Devlin, Mathematics, University of San Francisco, San Francisco, CA, USA, E-mail: smdevlin@usfca.edu

Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: None declared.
Conflict of interest statement: The authors declare no conflicts of interest.

Appendix

Theorem 1

Assume that teams are matched uniformly at random. Then the iterative Markov rating converges to the global Markov rating as the number of games grows.

Proof

Theorem 1 in Aldous (2017) establishes the convergence of the iterative Markov method. From equation (15), the change in mi is given by Δmi=C⋅mj if i beats j and Δmi=−C⋅mi if j beats i. Suppose pij denotes the probability that i beats j, so that pji=1−pij. At equilibrium, we have Δmi=0 for all i. The expected value of Δmi, E(Δmi), is given by

(29)E(Δmi)=pij⋅C⋅mj+pji(−C⋅mi)

(30)=pij(C⋅mj)+(1−pij) (−C⋅mi).

(31)=−C⋅mi+pij(C⋅mi+C⋅mj)

It follows that E(Δmi)=0 when pij=mimi+mj or when m satisfies the BT condition (6).

At the same time, Theorem 1 in Maystre and Grossglauser (2015) shows that the steady-state vector associated to the Global Markov chain described in equation (1) converges to the maximum likelihood estimate of the Bradley Terry vector. It thus follows that the Iterative Markov rating converges to the global Markov rating.□

This establishes the convergence of the iterative Markov method to the global Markov method indirectly. To build intuition into the nature of the convergence, however, consider the case of three teams with rating vector m=(m1,m2,m3)T. If Team 1 beats Team 2 then the iterative Markov updating rule in (14) is represented by multiplying m by the transition matrix I + C⋅E₁₂, where the only nonzero entries of Eij consist of a 1 in the i-th row, j-th column, and and −1 in the j-th row, j-th column. A sequence of games is given by a product of transition matrices, where the order of the games matters.

If Team 2 beats Team 3 followed by Team 1 beating Team 2, for instance, we have

(32)(I+C⋅E12)(I+C⋅E23)=I+C⋅(E12+E23)+C2⋅(E12 E23) ,

with E12 E23=[00100−1000]. In contrast, the same outcomes in reverse order (first Team 1 beats Team 2, then Team 2 beats Team 3) gives

(33)(I+CE23)(I+CE12)=I+C(E23+E12),

since (E₂₃E₁₂) = 0. The second order terms (multiplied by C²) in (32) and (33) can be understood as scheduling effects coming from the order in which games were played. More generally, the product EijErs is the zero matrix unless j = r, or i = r and j = s.

Suppose we have a sequence of games played and we wish to rate the teams using this iterative Markov method. The multiplication of the update matrices gives

(34)∏i≻j(I+CEij)=I+C⋅(∑Eij)+C2⋅(∑EijEmn+⋯)

Summing the Eij matrices over games-played, in the limit C → 0 we get

(35)∏i≻j(I+CEij)=I+C⋅(∑Eij)+O(C2)

(36)=I+C[−L1w12w13w21−L2w23w31w32−L3]+O(C2),

with wij and Li as in Section 2.1. Taking C=1N and using the fact that the Eij Emn products remain small as outlined above, the effect of the higher order terms goes to 0 and (36) gives the random walk matrix as in (1), assuming an equal number of games for all teams. Thus, the rating is given by the same eigenvector v as in (2).

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Received: 2019-07-02

Accepted: 2020-08-28

Published Online: 2020-10-05

Published in Print: 2021-06-25

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Artikel in diesem Heft

https://doi.org/10.1515/jqas-2019-0070

Schlagwörter für diesen Artikel

Elo; Markov; ranking; rating