Isotonic Bradley-Terry Model for Paired Comparison Data
2026-08-03 • Machine Learning
Machine Learning
AI summaryⓘ
The authors look at how to predict which player will win in a matchup based on past game results and how to rank players by skill. Traditional methods use fixed mathematical functions to guess win chances, but these may not always fit the data well. They propose a new approach that learns both player strengths and the best way to model win probabilities simultaneously. Their method improves prediction accuracy and can handle ties when there isn't enough data to decide a clear ranking. They tested this on simulated data and real sports data like soccer, baseball, and tennis.
Paired Comparison DataBradley-Terry ModelThurstone-Mosteller ModelWin ProbabilityInverse Link FunctionIsotonic RegressionGradient MethodPlayer RankingModel Misspecification
Authors
Ryoya Yamasaki
Abstract
In this paper, we study prediction problems for paired comparison data, for example, predicting the win probability between two unmatched players and ranking all the players according to the order of their strengths by using win probability data between two matched players. Paired comparison data are typically analyzed using Bradley-Terry and Thurstone-Mosteller models. These models predict the win probability by transforming the difference between learned rate parameters, which represent players'\;strengths, with a pre-specified inverse link function, and employ the order of learned rate parameters for player ranking. However, these models may suffer from model misspecification owing to the selection of a fixed inverse link function. Therefore, in this study, we propose to learn the rate parameters by a (sub-)gradient method and the inverse link function by an isotonic regression technique alternately. The proposed model guarantees monotonic improvement in training error, and is likely to yield an exact tie when the available data is insufficient to establish a strict ranking. We also verified that the proposed model could improve the win probability prediction and ranking performance through numerical experiments with synthetic data and real-world data of football Premier League, baseball MLB, and tennis ATP tour.