  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
\'{E}l\H{o} 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 \'{E}l\H{o}
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 \'{E}l\H{o}'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.
