and the distribution of digital products.
Fictitious Play for Mixed Strategy Equilibria in Mean Field Games: Numerical Analysis
:::info 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.
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Table of Links2 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
3.4 Numerical AnalysisIn 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.
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\ Definition 3.4 (implicit discretized system for mixed strategy equilibrium) We define
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\ the complementary condition
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\ will be weaker than the following one
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\ 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:
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\ Now we can state the main convergence result in this section.
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\ 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.
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\ with equality if and only if
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\ 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:
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:::info This paper is available on arxiv under CC 4.0 license.
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