Rank consistent Bradley-Terry models for repeated tournaments

Abstract

The primary objective of this paper is to model the win-loss records of matches in a repeated tournament using the ranks of the teams. The work proposes modifications of Bradley-Terry (BT) model to make the estimation consistent with the ranks of the participating teams. The BT model with restricted maximum likelihood estimation involves too many parameters and the estimates typically lack strict monotonicity. A proposed class of rank-percentile BT models based on different parametric distribution resolves both the issues. Parameter estimation, goodness-of-fit using suitably framed test statistic and its null distribution, change point analysis in a nested model framework, as well as other estimation aspects are discussed in this article. Adaptive variations of the model that allow estimates to alter are also discussed. For demonstration, National Collegiate Athletic Association (NCAA) men and women basketball tournament data are considered. The discussed models provide excellent fit to the historical data using only a few parameters. The fit validates the ranking procedure implemented by the NCAA. The models can be extended in more general tournament structures, as shown through an analysis of results from the Indian Premiere League. The work done has potential for application in the wider domain of paired comparisons.