Modelling Competitive Sports: Bradley-Terry-'ElHo Models for Supervised and On-Line Learning of Paired Competition Outcomes

Prediction and modelling of competitive sports outcomes has received much recent attention, especially from the Bayesian statistics and machine learning communities. In the real world setting of outcome prediction, the seminal 'ElHo update still remains, after more than 50 years, a valuable baseline which is difficult to improve upon, though in its original form it is a heuristic and not a proper statistical "model". Mathematically, the 'ElHo rating system is very closely related to the Bradley-Terry models, which are usually used in an explanatory fashion rather than in a predictive supervised or on-line learning setting. Exploiting this close link between these two model classes and some newly observed similarities, we propose a new supervised learning framework with close similarities to logistic regression, low-rank matrix completion and neural networks. Building on it, we formulate a class of structured log-odds models, unifying the desirable properties found in the above: supervised probabilistic prediction of scores and wins/draws/losses, batch/epoch and on-line learning, as well as the possibility to incorporate features in the prediction, without having to sacrifice simplicity, parsimony of the Bradley-Terry models, or computational efficiency of 'ElHo's original approach. We validate the structured log-odds modelling approach in synthetic experiments and English Premier League outcomes, where the added expressivity yields the best predictions reported in the state-of-art, close to the quality of contemporary betting odds.