# Convergence analysis of a family of Zermelo-type iterations for the Bradley--Terry model

arXiv: https://arxiv.org/abs/2607.22221 · HTML: https://mlsanity.com/paper/2607.22221/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 50 references that are in the map.
- First posted on arXiv 24 Jul 2026.

## Builds on (most cited first, 15 of 15)

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- [Efficient computation of rankings from pairwise comparisons](https://mlsanity.com/paper/2207.00076/) (2022)
- [Uncertainty quantification in the Bradley-Terry-Luce model](https://mlsanity.com/paper/2110.03874/) (2021)
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- [Optimal Full Ranking from Pairwise Comparisons](https://mlsanity.com/paper/2101.08421/) (2021)
- [An application of incomplete pairwise comparison matrices for ranking top tennis players](https://mlsanity.com/paper/1611.00538/) (2016)
- [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)
- [On Sinkhorn's Algorithm and Choice Modeling](https://mlsanity.com/paper/2310.00260/) (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)
- [Statistical Modelling of Citation Exchange Between Statistics Journals](https://mlsanity.com/paper/1312.1794/) (2013)

## Built on by (most cited first, 1 of 1)

- [Bradley-Terry model under general comparison graphs](https://mlsanity.com/paper/2610.06237/) (2026)

## Abstract (arXiv)

Zermelo's algorithm is a classical method for computing the maximum likelihood estimator in the Bradley--Terry (BT) model, but its convergence can be slow in practice. To accelerate computation, Newman introduced a family of Zermelo-type fixed-point iterations parameterized by α, with Zermelo's algorithm recovered at α=1. Empirical evidence suggests that the choice α=0 often converges substantially faster, making it a promising alternative, yet the mechanism underlying this acceleration remains elusive. This paper provides theoretical insight into this phenomenon through a systematic local convergence analysis. We derive closed-form expressions for local convergence factors under synchronous and asynchronous updates and analyze their dependence on α via spectral analysis of the associated Jacobian matrices. For synchronous updates, we show that the algorithm may fail to converge when α<1, and its local convergence factor is quasi-convex in α under the population BT model. In contrast, asynchronous updates are always locally convergent, and their local convergence factor is provably monotonically increasing in α under the population BT model of consistently ordered bipartite comparison graphs, establishing the optimality of α=0 in this setting. We further establish asymptotic approximation results for the population convergence factors under the BT model, justifying their practical relevance. Numerical experiments on synthetic and real-world datasets confirm the theory. Our analysis complements existing convergence results and shows that the acceleration of α=0 arises not only from the parameter choice but, more importantly, from the use of asynchronous updates.