Boltzmann and Fokker-Planck equations modelling the Elo rating system with learning effects

TL;DR

This paper introduces a kinetic model inspired by the Elo rating system, deriving Boltzmann and Fokker-Planck equations to analyze player ratings and strengths, including solution existence and long-term behavior.

Contribution

It presents a novel kinetic rating model with derived Boltzmann and Fokker-Planck equations, analyzing their solutions and dynamics in the context of Elo-like systems.

Findings

Existence of solutions to the Fokker-Planck equation

Analysis of long-term behavior of ratings

Numerical experiments illustrating dynamics

Abstract

In this paper we propose and study a new kinetic rating model for a large number of players, which is motivated by the well-known Elo rating system. Each player is characterised by an intrinsic strength and a rating, which are both updated after each game. We state and analyse the respective Boltzmann type equation and derive the corresponding nonlinear, nonlocal Fokker-Planck equation. We investigate the existence of solutions to the Fokker-Planck equation and discuss their behaviour in the long time limit. Furthermore, we illustrate the dynamics of the Boltzmann and Fokker-Planck equation with various numerical experiments.