**Fictitious Play for Mixed Strategy Equilibria in Mean Field Games: Abstract and Introduction**

Fictitious Play for Mixed Strategy Equilibria in Mean Field Games: Conclusion and References by@gamifications

by Gamifications FTW PublicationsSeptember 25th, 2024

**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 conclusion, this paper proposes a novel generalized fictitious play algorithm for computing mixed strategy equilibria in OSMFGs. The key innovations include leveraging an iterative process of solving pure strategy systems to approximate mixed equilibria, as well as expanding the design flexibility for the learning rate parameter. Rigorous convergence results are provided, and finite difference schemes are constructed to efficiently solve the obstacle and Fokker-Planck equations during each iteration.

Future work includes extensions to problems with common noise, where the equilibria consist of randomized stopping times that depend on the realized common noise path. The generalized fictitious play framework could also be applied to other competitive games involving optimal stopping decisions. Additionally, further analysis on quantifying the convergence rate and computational complexity could provide deeper theoretical insights. Overall, this paper introduces a novel algorithm and analysis to overcome the limitations of current OSMFG methods, opening the door for handling broader classes of large-scale dynamic games with optimal stopping.

ZZ is supported by the National Key R&D Program of China, Project Number 2021YFA1001200, and the NSFC, grant Number 12031013, 12171013. YL is supported by the NSFC, grant Number 12090022. We thank Xu’an Dou, Jian-Guo Liu and Jiajun Tong for helpful discussions.

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This paper is available on arxiv under CC 4.0 license.

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