Boltzmann and Fokker–Planck Equations Modelling the Elo Rating System with Learning Effects

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, non-local 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.