An Analysis of Elo Rating Systems via Markov Chains

TL;DR

This paper provides a theoretical analysis of the Elo rating system using Markov chain techniques, demonstrating its effectiveness in learning skill parameters and its applications in tournament design.

Contribution

It introduces a Markov chain-based framework for analyzing Elo, showing its competitive learning rate and connecting it to optimal tournament and mixing strategies.

Findings

Elo learns skill parameters at a competitive rate.

A connection between Elo and fastest-mixing Markov chains is established.

Implications for efficient tournament design are discussed.

Abstract

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.