paint-brush
Fictitious Play for Mixed Strategy Equilibria in Mean Field Games: Convergence of Fictitious Playby@gamifications

Fictitious Play for Mixed Strategy Equilibria in Mean Field Games: Convergence of Fictitious Play

by Gamifications FTW PublicationsSeptember 25th, 2024
Read on Terminal Reader
Read this story w/o Javascript
tldt arrow

Too Long; Didn't Read

The study is based on the Peking University School of Mathematical Sciences. The authors claim that an optimal stopping mean field game is a potential game if there exist potential functions F, Ψ : C → R such that (https://cdn.hackernoon.com/images/fWZa4tUiBGemnqQfBGgCPf9594N2-c583z97.png)
featured image - Fictitious Play for Mixed Strategy Equilibria in Mean Field Games: Convergence of Fictitious Play
Gamifications FTW Publications HackerNoon profile picture

Authors:

(1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing;

(2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing;

(3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University.

Abstract and 1. Introduction

2 Model and 2.1 Optimal Stopping and Obstacle Problem

2.2 Mean Field Games with Optimal Stopping

2.3 Pure Strategy Equilibrium for OSMFG

2.4 Mixed Strategy Equilibrium for OSMFG

3 Algorithm Construction and 3.1 Fictitious Play

3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium

3.3 Algorithm Based on Fictitious Play

3.4 Numerical Analysis

4 Numerical Experiments and 4.1 A Non-local OSMFG Example

4.2 A Local OSMFG Example

5 Conclusion, Acknowledgement, and References

3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium

To prove the convergence of the generalized fictitious play (3.1), we first need to ensure that the iteration scheme is well-defined. We introduce the following definition:



Additionally, we require some additional technical assumptions on the functions f and ψ. A common assumption is that f and ψ are variations of potential functions. Games that satisfy this property are known as potential games, of which we present the formal definition as follows.


Definition 3.3 (potential games) We call an optimal stopping mean field game a potential game if there exist potential functions F, Ψ : C → R such that



Now we present the main convergence result. Without loss of generality, we assume the generator of the diffusion process is simply the Laplacian operator ∆ in the subsequent analysis.





From step 1 we know that



Therefore, we have



We define that




By the learning rate condition (3.3), which we recall here for convenience



we can choose N such that



Hence, for all n ≥ M, we have the following inequality:



This is a contradiction with the continuity of Φ and the assumption that m∗ is a cluster point. Thus the cluster point m∗ should satisfy



for all m ∈ T.


3. We conclude that any cluster point (u ∗, m∗) is a mixed strategy equilibrium in this step. We first verify that u ∗ will satisfy the obstacle problem as following:



It remains to verify



and



Define m∗∗ as in (3.13). By (3.21) and the relation



we know that



From (3.11) and (3.12), this equality implies (3.23) and (3.24).


Hence we have finished the proof.


This paper is available on arxiv under CC 4.0 license.