ϵ-Nash mean-field games for stochastic linear-quadratic systems with delay and applications

Abstract

In this paper, we focus on mean-field linear-quadratic games for stochastic large-population systems with time delays. The $\epsilon$-Nash equilibrium for decentralized strategies in linear-quadratic games is derived via the consistency condition. By means of variational analysis, the system of consistency conditions can be expressed by forward-backward stochastic differential equations. Numerical examples illustrate the sensitivity of solutions of advanced backward stochastic differential equations to time delays, the effect of the the population’s collective behaviors, and the consistency of mean-field estimates.

Keywords

Mean-field game / Linear-quadratic problem / Time delay / Large-population / ϵ-Nash equilibrium

Acknowledgements

Thanks the anonymous reviewers and editors for their valuable comments and suggestions which led to the improvement of the original manuscript. H. P. Ma was supported by the National Natural Science Foundation of China (Grant No. 11801154), R. J. Li was supported by the Guangzhou Science and Technology Program Project Project (Grant No. 202201011057), and W. F. Wang was supported by the Natural Science Foundation of Hubei Province (Grant No. 2023AFC006).

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