An empirical Bayes model for time-varying paired comparisons ratings: Who is the greatest women's tennis player?

We present a methodology for fitting a time-varying paired comparisons model using an empirical Bayes approach. The model simultaneously avoids two problems that typically arise with paired comparisons data: first, that extreme values of estimated strengths can occur for competitors appearing in and winning a small number of games, producing absurd rankings, and second, that the time-varying strengths 'balloon' over time. The empirical Bayes approach automatically shrinks the strength estimates towards the mean, thus avoiding both issues. We present our model and demonstrate its use in the setting of tennis in search of an answer to the question: who is the greatest women's player of all time. Our results suggest that Steffi Graf is a strong candidate, but, using confidence intervals on the rankings themselves, others cannot be ruled out.