A stochastic rank ordered logit model for rating multi-competitor games and sports

Mark E. Glickman; Jonathan Hennessy

doi:10.1515/jqas-2015-0012

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A stochastic rank ordered logit model for rating multi-competitor games and sports

Mark E. Glickman and Jonathan Hennessy

Published/Copyright: June 16, 2015

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From the journal Journal of Quantitative Analysis in Sports Volume 11 Issue 3

Abstract

Many games and sports, including races, involve outcomes in which competitors are rank ordered. In some sports, competitors may play in multiple events over long periods of time, and it is natural to assume that their abilities change over time. We propose a Bayesian state-space framework for rank ordered logit models to rate competitor abilities over time from the results of multi-competitor games. Our approach assumes competitors’ performances follow independent extreme value distributions, with each competitor’s ability evolving over time as a Gaussian random walk. The model accounts for the possibility of ties, an occurrence that is not atypical in races in which some of the competitors may not finish and therefore tie for last place. Inference can be performed through Markov chain Monte Carlo (MCMC) simulation from the posterior distribution. We also develop a filtering algorithm that is an approximation to the full Bayesian computations. The approximate Bayesian filter can be used for updating competitor abilities on an ongoing basis. We demonstrate our approach to measuring abilities of 268 women from the results of women’s Alpine downhill skiing competitions recorded over the period 2002–2013.

Keywords: dynamic model; exploded logit; order statistics; Plackett-Luce; ranking

Corresponding author: Mark E. Glickman, Center for Healthcare Organization and Implementation Research, Edith Nourse Rogers Memorial Hospital (152), Bldg 70, 200 Springs Road, Bedford, MA 01730, USA, Tel.: +781 687-2875, Fax: +781 687-3106, e-mail: mg@bu.edu; and Department of Health Policy and Management, Boston University School of Public Health, 715 Albany Street, Boston, MA 02118, USA

Acknowledgments

We thank Peter Vint and Steven Powderly at the U.S. Olympic Committee for providing the data for the this work. This research was supported in part by a research contract from the U.S. Olympic Committee.

Appendix A

Newton-Raphson algorithm for optimizing log-posterior

We outline the steps for implementing the Newton-Raphson algorithm to find the posterior mode of θ_t in Equation (13). Let the first and second derivatives of Equation (13) as functions of θ_t be

D1(θt)=(D1.1(θt), …, D1.n(θt))D2(θt)=(D2.11(θt)⋯D2.1n(θt)⋮⋱⋮D2.n1(θt)⋯D2.nn(θt).)

For event k at time t, let

pik(θt)=exp(θit)∑ℓ=1mk−1(Xk)ℓi(Xkηt)′mk−1

where η_t=exp(θ_t). Then

(23)

D1.i(θt)=−(θit−μitσit2)+∑k=1Kt∑ℓ=1mk(Wk)ℓi−∑k=1Ktpik(θt) (23)

(24)

D2.ii(θt)=−1σit2−∑k=1Ktpik(θt)(1−pik(θt)) (24)

(25)

D2.ih(θt)=∑k=1Ktpik(θt)phk(θt) (25)

The Newton-Raphson algorithm proceeds in the following manner.

Select starting vector of posterior means, μt∗0=(μ1t∗0,…,μnt∗0). We have found that a good choice is to perform the following sequence of calculations.
1. Calculate πit∗0=∑k=1Kt(∑ℓ=1mk−1(Xk)ℓi)∑k=1Kt(mk−1), the proportion of the times competitor i is outperformed by his/her opponents during period t. Note that 1−πit∗0 is therefore the proportion of times competitor i outperforms his/her opponents.
2. Let F be the cumulative distribution function (cdf) for a standard logistic distribution, and F^–1 the inverse cdf. Let qit∗0=F−1(0.01+0.981−πit∗0)) be the quantile of the standard logistic distribution evaluated at the out-performance probability scaled to stay between 0.01 and 0.99. The scaling ensures that the quantiles are not infinite if the player always outperforms his/her opponents.
3. Let μit∗0=qit∗0+μit/σit21+1/σit2, a weighted average of qit∗0 with the prior mean μ_it.
At iteration j, j=1, 2, …, let
(26)μt∗j=μt∗j−1−D2−1(μt∗j−1)D1(μt∗j−1). (26)
The iteration is repeated until μt∗j changes by a negligible amount. The final estimated posterior means and standard deviations at iteration J are given by
(27)μt∗=μt∗J (27)
(28)σt∗=diag (D2−1(μt∗J)). (28)

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Article note:

The views expressed in this article are those of the authors and do not necessarily reflect the views of the Department of Veterans Affairs.

Published Online: 2015-6-16

Published in Print: 2015-9-1

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https://doi.org/10.1515/jqas-2015-0012

Keywords for this article

dynamic model; exploded logit; order statistics; Plackett-Luce; ranking