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The Bayesian Intransitive Bradley-Terry Model via Combinatorial Hodge Theory

Hisaya Okahara, Tomoyuki Nakagawa, Shonosuke Sugasawa

TL;DR

This work addresses intransitivity in pairwise comparisons by embedding combinatorial Hodge theory into the Bradley-Terry framework. It decomposes the match-up function $M$ as $M=\mathrm{grad}\,s+\mathrm{curl}^*\Phi$, links $p_{ij}$ to $M$ via $p_{ij}=\sigma(M_{ij})$, and uses a Horseshoe prior on the curl component to adaptively capture cycle-induced structure while retaining a BT limit when intransitivity vanishes. Efficient Gibbs sampling with Pólya-Gamma augmentation enables scalable Bayesian inference and full uncertainty quantification for both global transitivity and local triad-level intransitivity, including a global measure $\mathcal{I}$ and local vorticity $\mathcal{C}_{ijk}$. The approach is demonstrated on simulations and Major League Baseball data, showing improved estimation accuracy and well-calibrated uncertainty, with substantial computational advantages over prior Bayesian methods for intransitivity and clear interpretability of gradient vs curl contributions. The framework provides a principled, scalable tool for uncertainty-aware ranking in the presence of cycle-induced effects, with practical impact for sports analytics and broader decision making.

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.

The Bayesian Intransitive Bradley-Terry Model via Combinatorial Hodge Theory

TL;DR

This work addresses intransitivity in pairwise comparisons by embedding combinatorial Hodge theory into the Bradley-Terry framework. It decomposes the match-up function as , links to via , and uses a Horseshoe prior on the curl component to adaptively capture cycle-induced structure while retaining a BT limit when intransitivity vanishes. Efficient Gibbs sampling with Pólya-Gamma augmentation enables scalable Bayesian inference and full uncertainty quantification for both global transitivity and local triad-level intransitivity, including a global measure and local vorticity . The approach is demonstrated on simulations and Major League Baseball data, showing improved estimation accuracy and well-calibrated uncertainty, with substantial computational advantages over prior Bayesian methods for intransitivity and clear interpretability of gradient vs curl contributions. The framework provides a principled, scalable tool for uncertainty-aware ranking in the presence of cycle-induced effects, with practical impact for sports analytics and broader decision making.

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.
Paper Structure (21 sections, 1 theorem, 33 equations, 16 figures, 4 tables)

This paper contains 21 sections, 1 theorem, 33 equations, 16 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Classical Transitive Models
  4. Existing Intransitive Models
  5. A Hodge-Theoretic Framework for Pairwise Comparisons
  6. Functions on a Simplicial Complex
  7. Operators on Functions on a Simplicial Complex
  8. Hodge Decomposition
  9. The Bayesian Intransitive Bradley-Terry Model
  10. Proposed model
  11. Bayesian prior specification
  12. Posterior computation algorithm
  13. Global Intransitivity Measure and Local Vorticity
  14. Simulation Studies
  15. Illustrative Comparison of Posterior Distributions
  16. ...and 6 more sections

Key Result

Theorem 1

Assume that the $3$-clique complex associated with the complete graph is used so that the harmonic subspace is trivial. Let $M = \mathrm{grad}\ s + \mathrm{curl}^\ast \Phi$ with $s\in L^2(\mathcal{V})$ and $\Phi\in L^2_\wedge(\mathcal{T})$. Suppose that (i) $s$ is orthogonalized by $\sum_{i=1}^N s_i

Figures (16)

  • Figure 1: Relations among the flow spaces and the linear operators $\mathrm{grad}, \mathrm{curl}$ and their adjoints.
  • Figure 2: Hodge decomposition of pairwise rankings
  • Figure 3: Posterior distributions of the intransitive parameters: $\theta_{ij}^\ast$ under the Intransitive Clustered Bradley-Terry (ICBT) model (left); the curl flow under the proposed BIBT model (right).
  • Figure 4: Mean squared error (MSE) as a function of sparsity for $N=10$. The panels display MSE$_M$ (left), MSE$_{\mathrm{grad}}$ (middle), and MSE$_{\mathrm{curl}}$ (right). Sparsity ranges from $0$, corresponding to perfectly intransitive, to $1$, corresponding to perfectly transitive.
  • Figure 5: Recovery accuracy as a function of sparsity for $N=10$. Sparsity ranges from $0$, corresponding to perfectly intransitive, to $1$, corresponding to perfectly transitive.
  • ...and 11 more figures

Theorems & Definitions (2)

  • Theorem 1: Identifiability of the BIBT model
  • proof