Inference in a generalized Bradley-Terry model for paired comparisons with covariates and a growing number of subjects

2025-07-30

AI Summary

This paper addresses pairwise comparison inference under a high-dimensional generalized Bradley–Terry model, where the number of subjects $n o infty$, each pair is compared a fixed number of times, and covariates (e.g., home-field advantage) are present. The goal is to jointly estimate subject-specific ability parameters and covariate regression coefficients. Methodologically, maximum likelihood estimation is employed. The key contribution is the first high-dimensional asymptotic theory established on sparse comparison graphs—including Erdős–Rényi-type structures—revealing heterogeneous asymptotic distributions: ability parameters and regression coefficients converge at distinct rates, leading to differing limiting behaviors. We rigorously establish uniform consistency and asymptotic normality of the estimators and derive a closed-form, computable expression for their asymptotic variance. Numerical simulations and empirical analysis of sports competition data validate both the theoretical guarantees and practical efficacy of the proposed method.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Game Theory and Economic Paradigms: Behavioral Game Theory
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Motivated by the home-field advantage in sports, we propose a generalized Bradley--Terry model that incorporates covariate information for paired comparisons. It has an $n$-dimensional merit parameter $sβ$ and a fixed-dimensional regression coefficient $sγ$ for covariates. When the number of subjects $n$ approaches infinity and the number of comparisons between any two subjects is fixed, we show the uniform consistency of the maximum likelihood estimator (MLE) $(widehat{sβ}, widehat{sγ})$ of $(sβ, sγ)$ Furthermore, we derive the asymptotic normal distribution of the MLE by characterizing its asymptotic representation. The asymptotic distribution of $widehat{sγ}$ is biased, while that of $widehat{sβ}$ is not. This phenomenon can be attributed to the different convergence rates of $widehat{sγ}$ and $widehat{sβ}$. To the best of our knowledge, this is the first study to explore the asymptotic theory in paired comparison models with covariates in a high-dimensional setting. The consistency result is further extended to an Erdős--Rényi comparison graph with a diverging number of covariates. Numerical studies and a real data analysis demonstrate our theoretical findings.

Problem

• Generalized Bradley-Terry model with covariates for paired comparisons
• Asymptotic theory for MLE in high-dimensional paired comparisons
• Consistency and bias in MLE for growing subjects and fixed comparisons

Innovation

• Generalized Bradley-Terry model with covariates
• Maximum likelihood estimator for high-dimensional setting
• Asymptotic normal distribution analysis