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

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.

The Evaluation of Rating Systems in Team-based Battle Royale Games

“…Since the outcome of free-for-all games is a ranked list of players or teams, to evaluate rank predictions, most of the previous works focused on the ordinal association between the predicted and observed ranks. Rank correlation coefficients and their derivatives [4,[17][18][19][20] are among the metrics commonly used for this purpose. A recent study compared several metrics based on how they explain the predictive power of rating systems [21].…”

Understanding and Pushing the Limits of the Elo Rating Algorithm

“…Some approaches in the literature go around this problem by ignoring the draws, other count them as partial wins/losses with fractional score (Langville and Meyer, 2012, Ch. 11) (Glickman and Hennessy, 2015). Such heuristics, while potentially useful, do not show explicitly how to predict the results of the games from the rating levels.…”

Athlete rating in multi-competitor games with scored outcomes via monotone transformations

“…For multi-competitor games, approaches to rating competitors typically extend the Plackett-Luce model (Plackett, 1975) through the evolution of the latent ability parameters. For example, Glickman and Hennessy (2015) models the evolution as a discrete stochastic process, Caron and Teh (2012) uses a nonparametric stochastic process, Baker and McHale (2015a) interpolates abilities between discrete time points, and McKeough (2020) considers parametric growth curves over time. Dynamic models have also been proposed for head-to-head games with win/loss outcomes, in both team (Herbrich, Minka and Graepel, 2006) and individual (Glickman, 1999(Glickman, , 2001Cattelan, Varin and Firth, 2013;Baker and McHale, 2015b) game settings.…”