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Time-Inconsistent LQ Games for Large-Population Systems and Applications

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Abstract

In this paper, we formulate a time-inconsistent linear-quadratic (LQ) mean-field game problem for large-population systems. By the Nash certainty equivalence methodology, we construct an auxiliary problem consisting of time-inconsistent LQ control problems for every agent, respectively. An equilibrium, instead of optimal solution, is presented explicitly in the form of linear feedback for each agent in this auxiliary problem. Inspired by the framework of tackling time-inconsistency for control problems, we generalize the definition of \(\epsilon \)-Nash equilibrium for large-population games to the time-inconsistent case. Then, the set of linear feedback controls obtained above can be proved to be an \(\epsilon \)-Nash equilibrium for our original problem. Moreover, we solve a resource investment problem and provide a numerical simulation to illustrate the application of our theoretic results.

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Acknowledgements

H. Wang was supported in part by the Natural Science Foundation of China (11901362). R. Xu was supported in part by the Natural Science Foundation of Shandong Province of China (Grant no. ZR2020MA031, ZR2021MA049), National Natural Science Foundation of China (11971266), the Colleges and Universities Twenty Terms Foundation of Jinan City (2021GXRC100).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, China

    Haiyang Wang

  2. School of Mathematics and Statistics, Qilu University of Technology (Shandong Academy of Sciences), Jinan, 250353, China

    Ruimin Xu

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  1. Haiyang Wang
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Correspondence to Ruimin Xu.

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Communicated by Fausto Gozzi.

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Wang, H., Xu, R. Time-Inconsistent LQ Games for Large-Population Systems and Applications. J Optim Theory Appl 197, 1249–1268 (2023). https://doi.org/10.1007/s10957-023-02223-2

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