# Isotonic Bradley-Terry Model for Paired Comparison Data

arXiv: https://arxiv.org/abs/2608.02081 · HTML: https://mlsanity.com/paper/2608.02081/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 46 references that are in the map.
- First posted on arXiv 3 Aug 2026.

## Builds on (most cited first, 5 of 5)

- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Chatbot Arena: An Open Platform for Evaluating LLMs by Human Preference](https://mlsanity.com/paper/2403.04132/) (2024)
- [Simple, Robust and Optimal Ranking from Pairwise Comparisons](https://mlsanity.com/paper/1512.08949/) (2015)
- [Stochastically Transitive Models for Pairwise Comparisons: Statistical and Computational Issues](https://mlsanity.com/paper/1510.05610/) (2015)
- [Label Smoothing is Robustification against Model Misspecification](https://mlsanity.com/paper/2305.08501/) (2023)

## Abstract (arXiv)

In this paper, we study prediction problems for paired comparison data, for example, predicting the win probability between two unmatched players and ranking all the players according to the order of their strengths by using win probability data between two matched players. Paired comparison data are typically analyzed using Bradley-Terry and Thurstone-Mosteller models. These models predict the win probability by transforming the difference between learned rate parameters, which represent players';strengths, with a pre-specified inverse link function, and employ the order of learned rate parameters for player ranking. However, these models may suffer from model misspecification owing to the selection of a fixed inverse link function. Therefore, in this study, we propose to learn the rate parameters by a (sub-)gradient method and the inverse link function by an isotonic regression technique alternately. The proposed model guarantees monotonic improvement in training error, and is likely to yield an exact tie when the available data is insufficient to establish a strict ranking. We also verified that the proposed model could improve the win probability prediction and ranking performance through numerical experiments with synthetic data and real-world data of football Premier League, baseball MLB, and tennis ATP tour.