Estimation of the unimodal strong stochastic transitivity model and its application in Premier League rankings

Pairwise comparison problems, traditionally prevalent in sports competitions, have become increasingly important in recommendation systems and online decision-making with the rise of Internet applications. The strong stochastic transitivity (SST) model has gained attention for its flexibility, broad applicability, and efficiency in extracting complex structures compared to parametric models. However, the limited assumptions of the SST model make estimating the probability matrix challenging, raising questions about its effectiveness. In this paper, we introduce the unimodal strong stochastic transitivity (USST) model, which retains the SST model's flexibility and includes many parametric models as special cases. We show that the constrained least squares estimator in the USST model achieves rate optimality up to a polynomial factor of łog(łog n). Additionally, we develop an efficient algorithm for estimating the probability matrix, achieving rate optimality up to a polynomial factor of łog n. Our algorithm enjoys optimality, up to a logarithmic factor, compared to existing SST-related methods. We illustrate the superiority of our proposed algorithm through numerical simulations and validate its effectiveness using real data from English Premier League matches.