Stochastic analysis of the Elo rating algorithm in round-robin tournaments

TL;DR

This paper provides a stochastic analysis of the Elo rating algorithm in round-robin tournaments, deriving analytical expressions for skill evolution and offering insights into its convergence and performance.

Contribution

It introduces a detailed stochastic framework for understanding Elo's convergence in round-robin settings, including analytical expressions and practical guidelines.

Findings

Analytical expressions describe skill evolution over time.

The step-size hyperparameter influences convergence behavior.

Experimental results validate the theoretical analysis with real-world data.

Abstract

The Elo algorithm, renowned for its simplicity, is widely used for rating in sports tournaments and other applications. However, despite its widespread use, a detailed understanding of the convergence characteristics of the Elo algorithm is still lacking. Aiming to fill this gap, this paper presents a comprehensive (stochastic) analysis of the Elo algorithm, considering round-robin tournaments. Specifically, analytical expressions are derived describing the evolution of the skills and performance metrics. Then, taking into account the relationship between the behavior of the algorithm and the step-size value, which is a hyperparameter that can be controlled, design guidelines and discussions about the performance of the algorithm are provided. Experimental results are shown confirming the accuracy of the analysis and illustrating the applicability of the theoretical findings using real-world data obtained from SuperLega, the Italian volleyball league.