Fictitious Play for Mixed Strategy Equilibria in Mean Field Games: Numerical Analysis

Table of Links

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

3.4 Numerical Analysis

In this part, our goal is to prove the convergence of algorithm 2 when implicit scheme (3.25) and (3.26) are applied. The convergence analysis mirrors the proof for Theorem 3.1, requiring only adapting the arguments to a discretized version.

Definition 3.4 (implicit discretized system for mixed strategy equilibrium) We define

the complementary condition

will be weaker than the following one

Before stating the main result, we present a property of the implicit scheme for obstacle equations: the discretized solution u continuously depends on the discretized source term f.

Lemma 3.1 Consider the following discrete obstacle problem:

Now we can state the main convergence result in this section.

Proof The spirit of the proof is analog to the one in theorem 3.1. We divide the proof into 3 steps just parallel to the proof of theorem 3.1.

with equality if and only if

3. We conclude that any cluster point (u∗, m∗) is a solution to (3.29). We first verify that u∗ will satisfy the discretized obstacle problem as follows: