The Bayesian Intransitive Bradley-Terry Model via Combinatorial Hodge Theory

TL;DR

This paper introduces a Bayesian extension of the Bradley-Terry model that incorporates intransitive preferences using combinatorial Hodge theory, improving estimation and uncertainty quantification in competitive networks.

Contribution

It develops a novel Bayesian intransitive Bradley-Terry model embedding Hodge theory, enabling explicit separation of transitive and intransitive effects with scalable inference.

Findings

Enhanced estimation accuracy over existing models

Well-calibrated uncertainty quantification

Efficient Gibbs sampler for scalable computation

Abstract

Pairwise comparison data are widely used to infer latent rankings in areas such as sports, social choice, and machine learning. The Bradley-Terry model provides a foundational probabilistic framework but inherently assumes transitive preferences, explaining all comparisons solely through subject-specific parameters. In many competitive networks, however, cycle-induced effects are intrinsic, and ignoring them can distort both estimation and uncertainty quantification. To address this limitation, we propose a Bayesian extension of the Bradley-Terry model that explicitly separates the transitive and intransitive components. The proposed Bayesian Intransitive Bradley-Terry model embeds combinatorial Hodge theory into a logistic framework, decomposing paired relationships into a gradient flow representing transitive strength and a curl flow capturing cycle-induced structure. We impose global-local shrinkage priors on the curl component, enabling data-adaptive regularization and ensuring a natural reduction to the classical Bradley-Terry model when intransitivity is absent. Posterior inference is performed using an efficient Gibbs sampler, providing scalable computation and full Bayesian uncertainty quantification. Simulation studies demonstrate improved estimation accuracy, well-calibrated uncertainty, and substantial computational advantages over existing Bayesian models for intransitivity. The proposed framework enables uncertainty-aware quantification of intransitivity at both the global and triad levels, while also characterizing cycle-induced competitive advantages among teams.