Convergence and stationary distribution of Elo rating systems

TL;DR

This paper analyzes the long-term behavior of the Elo rating system, proving convergence to a unique stationary distribution and exploring its properties through theoretical results and simulations.

Contribution

It establishes the convergence and properties of the Elo rating process, providing the first rigorous analysis of its stationary distribution and long-term behavior.

Findings

Converges exponentially to a unique equilibrium distribution.

Stationary distribution has finite exponential moments.

Ratings converge to true skills as K decreases at rate √K.

Abstract

The Elo rating system is a popular and widely adopted method for measuring the relative skill levels of players or teams in various sports and competitions. It assigns players numerical ratings and dynamically updates them based on game results and a model parameter $K$, which determines the sensitivity of rating changes. Assuming random games, this leads to a Markov chain for the evolution of the ratings of the $N$ players in the league. Despite its widespread use, little is known about the long-term behavior of this process. Aiming to fill this gap, in this article we prove that the process converges to its unique equilibrium distribution at an exponential rate in the 2-Wasserstein distance and almost surely. Moreover, we show important properties of the stationary distribution, such as the finiteness of an exponential moment, full support, and convergence to the players' true skills as $K$ decreases, at a rate of $\sqrt{K}$. We also provide Monte Carlo simulations that illustrate some of these properties and offer new insights.