The Bradley–Terry stochastic block model

The Bradley–Terry model is widely used for the analysis of pairwise comparison data and, in essence, produces a ranking of the items under comparison. We embed the Bradley–Terry model within a stochastic block model, allowing items to cluster. The resulting Bradley–Terry SBM (BT–SBM) ranks clusters so that items within a cluster share the same tied rank. We develop a fully Bayesian specification in which all quantities—the number of blocks, their strengths, and item assignments—are jointly learned via a fast Gibbs sampler derived through a Thurstonian data augmentation. Despite its efficiency, the sampler yields coherent and interpretable posterior summaries for all model components. Our motivating application analyses men’s tennis results from ATP tournaments from the 2000 season up to the 2025 season. We find that the top 105 players can be broadly partitioned into three or four tiers in most seasons. Moreover, the size of the strongest tier was small from the mid-2000s to 2018. Between 2019 and 2022, we observe a transition period characterised by a gradual widening of the top tier, while in more recent seasons (2023–2025) the structure appears to revert to a more elite configuration, coinciding with the rise of dominant players such as Carlos Alcaraz and Jannik Sinner.