Risk-sensitive mean field games with major and minor players^*

¹ Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, School of Mathematical Sciences, East China Normal University, Shanghai 200241, China
² NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, China

^** Corresponding author: tli@math.ecnu.edu.cn

Received: 24 March 2022
Accepted: 1 December 2022

Abstract

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.