Prediction Accuracy of Linear Models for Paired Comparisons in Sports

Linear models for paired comparisons, the Bradley-Terry model and the Thurstone-Mosteller model in particular, are widely used in sports for ranking and rating purposes. By their formulation, these models predict the probability that a player or team defeats another if the playing strengths of the players or teams are known.In this paper, we investigate the prediction accuracy of the two linear models by using them to describe three simple theoretical games which mimic actual sports and whose winning probability, given the playing strength of each player, can be expressed explicitly. A theoretical result is presented, which provides the basis of a linearization method that enables these games to be represented by linear models. The predicted winning probabilities from the linear models are then compared to the actual ones. Comparisons are also made in prediction accuracy between the Bradley-Terry model and the Thurstone-Mosteller model.