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

Bertram Düring, Josephine Evans, Marie-Therese Wolfram

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.

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

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.
Paper Structure (14 sections, 22 theorems, 111 equations, 2 figures)

This paper contains 14 sections, 22 theorems, 111 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The kinetic Elo rating system with learning effects
  3. Modelling
  4. Numerical simulations
  5. Existence of steady states
  6. Main Results
  7. Discussion of the linear equation \ref{['eq:linearmain']}
  8. Topologies and spaces
  9. Schauder fixed point argument
  10. The function $G$ is well-defined
  11. Finding a set which is preserved by the function $G$
  12. Continuity of the function $G$
  13. Putting the steps together
  14. Conclusion and future research directions

Key Result

Theorem 3.1

Let Assumption assumptionb be satisfied. Then there exists a steady solution $f_\infty$ to eq:nlmain, which is a probability measure on $\mathbb{R}^2$. Furthermore, $f_\infty$ is a smooth function with a bounded exponential moment.

Figures (2)

  • Figure 1: Steady state player distribution $f_{\infty}$.
  • Figure 2: Decay of the relative energy \ref{['e:energy']} for different weights functions $\phi = \phi(\beta, \gamma)$ defined by \ref{['eq:definephi']} and for $\phi = f_{\infty}^{-1}$.

Theorems & Definitions (42)

  • Theorem 3.1
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.2: Prokhorov's Theorem
  • Lemma 3.3
  • proof
  • Definition 3.4
  • Definition 3.5
  • Proposition 3.1
  • ...and 32 more