# Asymptotic comparison of identifying constraints for Bradley-Terry models

arXiv: https://arxiv.org/abs/2205.04341 · HTML: https://mlsanity.com/paper/2205.04341/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 May 2022.

## Built on by (most cited first, 6 of 6)

- [Rethinking Bradley-Terry Models in Preference-Based Reward Modeling: Foundations, Theory, and Alternatives](https://mlsanity.com/paper/2411.04991/) (2024)
- [GVPO: Group Variance Policy Optimization for Large Language Model Post-Training](https://mlsanity.com/paper/2504.19599/) (2025)
- [Energy-Based Preference Model Offers Better Offline Alignment than the Bradley-Terry Preference Model](https://mlsanity.com/paper/2412.13862/) (2024)
- [Dropping Just a Handful of Preferences Can Change Top Large Language Model Rankings](https://mlsanity.com/paper/2508.11847/) (2025)
- [Ranking from Pairwise Comparisons in General Graphs and Graphs with Locality](https://mlsanity.com/paper/2304.06821/) (2023)
- [GVPO++: Group Variance Policy Optimization for LLM Post-Training and On-Policy Distillation](https://mlsanity.com/paper/2609.21432/) (2026)

## Abstract (arXiv)

The Bradley-Terry model is widely used for pairwise comparison data analysis. In this paper, we analyze the asymptotic behavior of the maximum likelihood estimator of the Bradley-Terry model in its logistic parameterization, under a general class of linear identifiability constraints. We show that the constraint requiring the Bradley-Terry scores for all compared objects to sum to zero minimizes the sum of the variances of the estimated scores, and recommend using this constraint in practice.