A Note on Ranking in the Plackett-Luce Model for Multiple Comparisons

Ranking and rating individuals is a fundamental problem in multiple comparisons. One of the most well-known approaches is the Plackett-Luce model, in which the ordering is decided by the maximum likelihood estimator. However, the maximum likelihood estimate (MLE) does not exist when some individuals are never ranked lower than others or lose all their races. In this note, we proposed a penalized likelihood method to address this problem. As the penalized parameter goes to zero, the penalized MLE converges to the original MLE. Further, there exists a critical point in which the penalized likelihood ranking is independent of the choice of the penalized parameter. Several numerical examples are provided.