A comprehensive analysis of the Elo rating algorithm: Stochastic model, convergence characteristics, design guidelines, and experimental results. (arXiv:2212.12015v1 [cs.LG])

The Elo algorithm, due to its simplicity, is widely used for rating in sports
competitions as well as in other applications where the rating/ranking is a
useful tool for predicting future results. However, despite its widespread use,
a detailed understanding of the convergence properties 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
(one-on-one) competitions. Specifically, analytical expressions are derived
characterizing the behavior/evolution of the skills and of important
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, some design guidelines as well as discussions about the
performance of the algorithm are provided. To illustrate the applicability of
the theoretical findings, experimental results are shown, corroborating the
very good match between analytical predictions and those obtained from the
algorithm using real-world data (from the Italian SuperLega, Volleyball
League).