The Bradley-Terry Stochastic Block Model

TL;DR

This paper introduces a Bayesian stochastic block model extension of the Bradley-Terry model to cluster items in pairwise comparison data, demonstrated on men's tennis rankings, revealing evolving competitive tiers.

Contribution

It develops a fully Bayesian Bradley-Terry stochastic block model with a Gibbs sampler for joint inference of clusters and item rankings, providing interpretable results.

Findings

Men's tennis players cluster into 3-4 tiers per season.

The top tier size decreased mid-2000s and increased after 2018.

Tennis competitiveness has increased in recent years.

Abstract

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 analyzes men's tennis results from ATP tournaments over the seasons 2000-2022. We find that the top 100 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 and has increased since, providing evidence that men's tennis has become more competitive in recent years.