Fast computation of rankings from pairwise comparisons

We study the ranking of individuals, teams, or objects on the basis of pairwise comparisons using the Bradley-Terry model. Maximum-likelihood estimates of rankings within this model are commonly made using a simple iterative algorithm first introduced by Zermelo almost a century ago. Here we describe an alternative and similarly simple iteration that solves the same problem much faster -- over a hundred times faster in some cases. We demonstrate this algorithm with applications to a range of example data sets and derive some results regarding its convergence.