# Bradley-Terry model under general comparison graphs

arXiv: https://arxiv.org/abs/2610.06237 · HTML: https://mlsanity.com/paper/2610.06237/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 48 references that are in the map.
- First posted on arXiv 5 Oct 2026.

## Placed in

- LLM alignment / RLHF / preference optimisation

## Builds on (most cited first, 19 of 19)

- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Deep reinforcement learning from human preferences](https://mlsanity.com/paper/1706.03741/) (2017)
- [Estimation from Pairwise Comparisons: Sharp Minimax Bounds with Topology Dependence](https://mlsanity.com/paper/1505.01462/) (2015)
- [Spectral Method and Regularized MLE Are Both Optimal for Top-K Ranking](https://mlsanity.com/paper/1707.09971/) (2017)
- [Uncertainty quantification in the Bradley-Terry-Luce model](https://mlsanity.com/paper/2110.03874/) (2021)
- [Item Response Theory -- A Statistical Framework for Educational and Psychological Measurement](https://mlsanity.com/paper/2108.08604/) (2021)
- [An application of incomplete pairwise comparison matrices for ranking top tennis players](https://mlsanity.com/paper/1611.00538/) (2016)
- [Spectral Ranking Inferences based on General Multiway Comparisons](https://mlsanity.com/paper/2308.02918/) (2023)
- [ℓ∞-Bounds of the MLE in the BTL Model under General Comparison Graphs](https://mlsanity.com/paper/2110.10825/) (2021)
- [A unified analysis of likelihood-based estimators in the Plackett--Luce model](https://mlsanity.com/paper/2306.02821/) (2023)
- [A General Pairwise Comparison Model for Extremely Sparse Networks](https://mlsanity.com/paper/2002.08853/) (2020)
- [Ranking from Pairwise Comparisons in General Graphs and Graphs with Locality](https://mlsanity.com/paper/2304.06821/) (2023)
- [Statistical ranking with dynamic covariates](https://mlsanity.com/paper/2406.16507/) (2024)
- [Recent advances in the Bradley--Terry Model: theory, algorithms, and applications](https://mlsanity.com/paper/2601.14727/) (2026)
- [Graph Resistance and Learning from Pairwise Comparisons](https://mlsanity.com/paper/1902.00141/) (2019)
- [Statistical inference for pairwise comparison models](https://mlsanity.com/paper/2401.08463/) (2024)
- [Top-K ranking with a monotone adversary](https://mlsanity.com/paper/2402.07445/) (2024)
- [Entrywise Error Bounds for Spectral Ranking with Semi-Random Adversaries](https://mlsanity.com/paper/2605.23854/) (2026)
- [Convergence analysis of a family of Zermelo-type iterations for the Bradley--Terry model](https://mlsanity.com/paper/2607.22221/) (2026)

## Abstract (arXiv)

The Bradley-Terry model is a parametric model for ranking from pairwise comparisons. Existing asymptotic theory for the maximum likelihood estimator (MLE) in regimes where the number of objects grows often requires homogeneity assumptions or compatibility conditions on comparison graphs, which limits its applicability to many practical settings. In this work, we establish uniform consistency of the MLE under general deterministic comparison designs. Our pairwise error bound consists of a pair-specific term governed by the effective resistance between the corresponding objects and a global term governed by the spectral gap of the unnormalized graph Laplacian. The bound is tight up to logarithmic factors for certain graph sequences. For any prespecified sequence of pairs satisfying an additional balancing condition, we further establish asymptotic normality of the corresponding estimated utility differences. As an application, we obtain uniform consistency for independent-edge random graph designs whenever the minimum edge probability exceeds the ErdHos-R'enyi connectivity threshold by a logarithmic factor, as well as asymptotic normality under additional balancing conditions.