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Convergence and stationary distribution of Elo rating systems

Roberto Cortez, Hagop Tossounian

TL;DR

This work analyzes the Elo rating process as a Markov chain on the zero-sum subspace $\mathcal{Z}_N$, driven by random matches and a fixed $K$-factor. It proves the existence and uniqueness of a stationary distribution $\pi$ and establishes exponential convergence in the $2$-Wasserstein distance $W_2$ (and almost-sure convergence via a natural coupling). It further shows that $\pi$ has finite exponential moments, full support on $\mathcal{Z}_N$, and that $\pi$ converges to the true skill vector $\rho$ as $K\to0$ at rate $\sqrt{K}$, with implications for bias and estimation. Numerical simulations illustrate density shapes, bias phenomena, and the $\sqrt{K}$ scaling for small $K$, providing practical insight into Elo dynamics. Overall, the results furnish a rigorous foundation for the long-time behavior of Elo systems and point to several avenues for extending the model and its analysis.

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.

Convergence and stationary distribution of Elo rating systems

TL;DR

This work analyzes the Elo rating process as a Markov chain on the zero-sum subspace , driven by random matches and a fixed -factor. It proves the existence and uniqueness of a stationary distribution and establishes exponential convergence in the -Wasserstein distance (and almost-sure convergence via a natural coupling). It further shows that has finite exponential moments, full support on , and that converges to the true skill vector as at rate , with implications for bias and estimation. Numerical simulations illustrate density shapes, bias phenomena, and the scaling for small , providing practical insight into Elo dynamics. Overall, the results furnish a rigorous foundation for the long-time behavior of Elo systems and point to several avenues for extending the model and its analysis.

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 , which determines the sensitivity of rating changes. Assuming random games, this leads to a Markov chain for the evolution of the ratings of the 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 decreases, at a rate of . We also provide Monte Carlo simulations that illustrate some of these properties and offer new insights.

Paper Structure

This paper contains 14 sections, 22 theorems, 75 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Elo rating
  3. Relevant literature, main results, and methodology
  4. Assumptions and definitions
  5. Plan of the paper
  6. The Elo process
  7. Convergence to the Elo stationary distribution
  8. Properties of the Elo stationary distribution
  9. Numerical results
  10. Density for different values of rho-1
  11. Density for different values of K
  12. Expected rating vs true skill
  13. Expected distance between rating and true skill vs K
  14. Conclusion and perspectives

Key Result

Theorem 1

There exist constants $\theta_0>0$ and $C>0$, such that for any $\theta \in (0,\theta_0]$, there exists $\kappa \in (0,1)$ depending on $\theta$, such that In particular, this yields exponential convergence to $\pi$ at rate $1-\kappa$ not depending on $\mathcal{L}(X_0)$, in the class of initial distributions satisfying $\mathbb{E}[\Vert X_0 \Vert^2 e^{\theta_0 \Vert X_0 \Vert}] < \infty$.

Figures (4)

  • Figure 1: Monte Carlo approximation of the probability density function of $X^1$ for five different values of $\rho^1$. Parameters: $K=0.4$, $L=0.5$. Samples per curve: $m=5 \times 10^7$.
  • Figure 2: Monte Carlo approximation of the probability density function of $X^1$ for five different values of $K$. Parameters: $\rho^1=0$, $L=0.5$. Samples per curve: $m=5 \times 10^7$.
  • Figure 3: Monte Carlo estimation of $b(2\mathbb{E}[X^1])$ and $\mathbb{E}[b(2X^1)]$ for 101 values of $\rho^1$, plotted as functions of $b(2\rho^1)$. Parameters: $K=1$, $L=0.5$. Samples per point: $m=5 \times 10^5$.
  • Figure 4: Monte Carlo estimation of $\mathbb{E}[|X^1-\rho^1|]$ for 100 values of $K$, together with the function $C\sqrt{K}$. Right plot is in log-log scale. Parameters: $\rho^1=0.5$, $L=0.5$. Samples per point: $m=5 \times 10^4$.

Theorems & Definitions (46)

  • Theorem 1: exponential convergence to the stationary distribution $\pi$
  • Theorem 2: convergence of $\pi$ to the true skills
  • Definition 3
  • Remark 4
  • Definition 5
  • Lemma 6
  • proof
  • Lemma 7
  • Lemma 8
  • proof
  • ...and 36 more