A Discrete Evolutionary Model for Chess Players' Ratings

TL;DR

This paper introduces a stochastic evolutionary model for chess ratings that explains the normal distribution of ratings and their variance growth over time, validated with real-world data and accounting for slight skewness.

Contribution

It presents the first stochastic model for the evolution of chess ratings, deriving the distribution and variance growth, and incorporates skewness for better accuracy.

Findings

Ratings follow an approximately normal distribution with small negative skew.

Variance of ratings increases logarithmically over time.

Model parameters can be accurately recovered through simulations.

Abstract

The Elo system for rating chess players, also used in other games and sports, was adopted by the World Chess Federation over four decades ago. Although not without controversy, it is accepted as generally reliable and provides a method for assessing players' strengths and ranking them in official tournaments. It is generally accepted that the distribution of players' rating data is approximately normal but, to date, no stochastic model of how the distribution might have arisen has been proposed. We propose such an evolutionary stochastic model, which models the arrival of players into the rating pool, the games they play against each other, and how the results of these games affect their ratings. Using a continuous approximation to the discrete model, we derive the distribution for players' ratings at time $t$ as a normal distribution, where the variance increases in time as a logarithmic function of $t$. We validate the model using published rating data from 2007 to 2010, showing that the parameters obtained from the data can be recovered through simulations of the stochastic model. The distribution of players' ratings is only approximately normal and has been shown to have a small negative skew. We show how to modify our evolutionary stochastic model to take this skewness into account, and we validate the modified model using the published official rating data.