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.
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
4 Numerical Experiments and 4.1 A Non-local OSMFG Example
5 Conclusion, Acknowledgement, and References
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: