Volume 29, 2023
|Number of page(s)||71|
|Published online||11 January 2023|
Risk-sensitive mean field games with major and minor players*
Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, School of Mathematical Sciences, East China Normal University,
2 NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, China
** Corresponding author: email@example.com
Accepted: 1 December 2022
We investigate a class of mean field games containing a large number of major and minor players. Each player minimizes a quadratic-tracking type risk-sensitive cost functional, where the reference signal is a function of the state average term of the major and minor players. To reduce the complexity for solving the problem, we design a sequence of decentralized strategies by the Nash certainty equivalence principle. Firstly, for the optimal control problems with quadratic type risk-sensitive cost functionals, we propose a new verification theorem. Secondly, we apply the two-layer state aggregation method to construct the fixed-point equations for the estimations of the state average terms and give the conditions for the existence and uniqueness of the fixed points. Then, we design a sequence of decentralized strategies by the estimations of the state average terms based on local information. It is shown that the estimations of the state average terms are consistent with the true values for the closed-loop systems, and the sequence of strategies designed is a decentralized asymptotic Nash equilibrium. Finally, the effectiveness of the theoretical analysis is demonstrated by a numerical example.
Mathematics Subject Classification: 91A10 / 91A25 / 93C05 / 93E03
Key words: Risk-sensitive cost functional / major and minor players / mean field game / decentralized strategy / decentralized asymptotic Nash equilibrium
This work was supported in part by the National Natural Science Foundation of China under Grant 61977024, in part by the Basic Research Project of Shanghai Science and Technology Commission under Grant 20JC1414000 and in part by NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai. The authors are listed in alphabetical order.
© The authors. Published by EDP Sciences, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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