MSE-optimal K-factor of the Elo rating system for round-robin tournament

Chan Victor ()
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Chan Victor: Department of Mathematics, Western Washington University, Bellingham 98225, WA, USA

Journal of Quantitative Analysis in Sports, 2022, vol. 18, issue 1, 59-72

Abstract: The Elo rating system contains a coefficient called the K-factor which governs the amount of change to the updated ratings and is often determined by empirical or heuristic means. Theoretical studies on the K-factor have been sparse and not much is known about the pertinent factors that impact its appropriate values in applications. This paper has two main goals: to present a new formulation of the K-factor that is optimal with respect to the mean-squared-error (MSE) criterion in a round-robin tournament setting and to investigate the effects of the relevant variables, including the number of tournament participants n, on the optimal K-factor (based on the model-averaged MSE). It is found that n and the variability of the deviation between the true rating and the pre-tournament rating have a strong influence on the optimal K-factor. Comparisons between the MSE-optimal K-factor and the K-factors from Elo and from the US Chess Federation as a function of n are also provided. Although the results are applicable to other sports in similar settings, the study focuses on chess and makes use of the rating data and the K-factor values from the chess world.

Keywords: chess; Monte Carlo methods; paired-comparison models; win probability (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1515/jqas-2021-0079

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