  For nonbalanced 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
$\varepsilon$ singular perturbation proposed by Conner and Grant (2000), to
address this problem and 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-$\varepsilon$ models are
derived. Numerical studies show that the ranking is robust to the different
$\varepsilon$. We apply the proposed methods to the data of the 2008 NFL
regular season.
