Ranking in the generalized Bradley–Terry models when the strong connection condition fails

Abstract

ABSTRACTFor non balanced paired comparisons, a wide variety of ranking methods have been proposed. One of the best popular methods is the Bradley–Terry model in which the ranking of a set of objects is decided by the maximum likelihood estimates (MLEs) of merits parameters. However, the existence of MLE for the Bradley–Terry model and its generalized models to allow for tied observation or home-field advantage or both to occur, crucially depends on the strong connection condition on the directed graph constructed by a win–loss matrix. When this condition fails, the MLE does not exist and hence there is no solution of ranking. In this paper, we propose an improved version of the ϵ singular perturbation proposed by Conner and Grant (2000), to address this problem and to extend it to the generalized Bradley–Terry models. Some necessary and sufficient conditions for the existence and uniqueness of the penalized MLEs for these generalized Bradley–Terry-ϵ models are derived. Numerical studies show that the ranking is robust to the different ϵ. We apply the proposed methods to the data of the 2008 NFL regular season.