Whole-History Rating: A Bayesian Rating System for Players of Time-Varying Strength

International audienceWhole-History Rating (WHR) is a new method to estimate the time-varying strengths of players involved in paired comparisons. Like many variations of the Elo rating system, the whole-history approach is based on the dynamic Bradley-Terry model. But, instead of using incremental approximations, WHR directly computes the exact maximum a posteriori over the whole rating history of all players. This additional accuracy comes at a higher computational cost than traditional methods, but computation is still fast enough to be easily applied in real time to large-scale game servers (a new game is added in less than 0.001 second). Experiments demonstrate that, in comparison to Elo, Glicko, TrueSkill, and decayed-history algorithms, WHR produces better predictions

Dynamics of tournaments: the soccer case

A random walk-like model is considered to discuss statistical aspects of tournaments. The model is applied to soccer leagues with emphasis on the scores. This competitive system was computationally simulated and the results are compared with empirical data from the English, the German and the Spanish leagues and showed a good agreement with them. The present approach enabled us to characterize a diffusion where the scores are not normally distributed, having a short and asymmetric tail extending towards more positive values. We argue that this non-Gaussian behavior is related with the difference between the teams and with the asymmetry of the scores system. In addition, we compared two tournament systems: the all-play-all and the elimination tournaments.Comment: To appear in EPJ

Score-Based Bayesian Skill Learning

We extend the Bayesian skill rating system of TrueSkill to accommodate score-based match outcomes. TrueSkill has proven to be a very effective algorithm for matchmaking - the process of pairing competitors based on similar skill-level - in competitive online gaming. However, for the case of two teams/players, TrueSkill only learns from win, lose, or draw outcomes and cannot use additional match outcome information such as scores. To address this deficiency, we propose novel Bayesian graphical models as extensions of TrueSkill that (1) model player's offence and defence skills separately and (2) model how these offence and defence skills interact to generate score-based match outcomes. We derive efficient (approximate) Bayesian inference methods for inferring latent skills in these new models and evaluate them on three real data sets including Halo 2 XBox Live matches. Empirical evaluations demonstrate that the new score-based models (a) provide more accurate win/loss probability estimates than TrueSkill when training data is limited, (b) provide competitive and often better win/loss classification performance than TrueSkill, and (c) provide reasonable score outcome predictions with an appropriate choice of likelihood - prediction for which TrueSkill was not designed, but which can be useful in many applications. © 2012 Springer-Verlag