Steady states of an Elo-type rating model for players of varying strength

TL;DR

This paper investigates the long-term behavior of a kinetic Elo-type rating model for players with varying strengths, establishing the existence of steady states through advanced mathematical techniques.

Contribution

It introduces a kinetic mean-field Fokker-Planck model for Elo ratings and proves steady state existence using Schauder fixed point and hypocoercivity methods.

Findings

Existence of steady states for the model

Application of hypocoercivity techniques

Use of Schauder fixed point argument

Abstract

In this paper we study the long-time behaviour of a kinetic formulation of an Elo-type rating model for a large number of interacting players with variable strength. The model results in a non-linear mean-field Fokker-Planck equation and we show the existence of steady states via a Schauder fixed point argument. Our proof relies on the study of a related linear equation using hypocoercivity techniques.