# An Analysis of Elo Rating Systems via Markov Chains

arXiv: https://arxiv.org/abs/2406.05869 · HTML: https://mlsanity.com/paper/2406.05869/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 42 references that are in the map.
- First posted on arXiv 9 Jun 2024.

## Builds on (most cited first, 8 of 8)

- [Bridging the Gap between Constant Step Size Stochastic Gradient Descent and Markov Chains](https://mlsanity.com/paper/1707.06386/) (2017)
- [Estimation from Pairwise Comparisons: Sharp Minimax Bounds with Topology Dependence](https://mlsanity.com/paper/1505.01462/) (2015)
- [Minimax-optimal Inference from Partial Rankings](https://mlsanity.com/paper/1406.5638/) (2014)
- [ℓ∞-Bounds of the MLE in the BTL Model under General Comparison Graphs](https://mlsanity.com/paper/2110.10825/) (2021)
- [Lifting Markov Chains To Mix Faster: Limits and Opportunities](https://mlsanity.com/paper/1705.08253/) (2017)
- [Learning to Identify Top Elo Ratings: A Dueling Bandits Approach](https://mlsanity.com/paper/2201.04480/) (2022)
- [A State-Space Perspective on Modelling and Inference for Online Skill Rating](https://mlsanity.com/paper/2308.02414/) (2023)
- [Zig-zag sampling for discrete structures and non-reversible phylogenetic MCMC](https://mlsanity.com/paper/2004.08807/) (2020)

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

- [Accelerating Skill Assessment in Chess: A Drift-Diffusion-Enhanced Elo Rating System](https://mlsanity.com/paper/2606.26267/) (2026)

## Abstract (arXiv)

We present a theoretical analysis of the Elo rating system, a popular method for ranking skills of players in an online setting. In particular, we study Elo under the Bradley--Terry--Luce model and, using techniques from Markov chain theory, show that Elo learns the model parameters at a rate competitive with the state of the art. We apply our results to the problem of efficient tournament design and discuss a connection with the fastest-mixing Markov chain problem.