# Risk-Sensitive Mean Field Games via the Stochastic Maximum Principle

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## Abstract

In this paper, we consider risk-sensitive mean field games via the risk-sensitive maximum principle. The problem is analyzed through two sequential steps: (i) risk-sensitive optimal control for a fixed probability measure, and (ii) the associated fixed-point problem. For step (i), we use the risk-sensitive maximum principle to obtain the optimal solution, which is characterized in terms of the associated forward–backward stochastic differential equation (FBSDE). In step (ii), we solve for the probability law induced by the state process with the optimal control in step (i). In particular, we show the existence of the fixed point of the probability law of the state process determined by step (i) via Schauder’s fixed-point theorem. After analyzing steps (i) and (ii), we prove that the set of *N* optimal distributed controls obtained from steps (i) and (ii) constitutes an approximate Nash equilibrium or \(\epsilon \)-Nash equilibrium for the *N* player risk-sensitive game, where \(\epsilon \rightarrow 0\) as \(N \rightarrow \infty \) at the rate of \(O(\frac{1}{N^{1/(n+4)}})\). Finally, we discuss extensions to heterogeneous (non-symmetric) risk-sensitive mean field games.

## Keywords

Mean field game theory Risk-sensitive optimal control Forward–backward stochastic differential equations Decentralized control## Notes

### Acknowledgements

The authors would like to thank the Associate Editor and the two anonymous reviewers for careful reading of and helpful suggestions on the earlier version of the manuscript. This research was supported in part by the National Research Foundation of Korea (NRF) Grant funded by the Ministry of Science and ICT, South Korea (NRF-2017R1E1A1A03070936, NRF-2017R1A5A1015311), in part by Institute for Information and Communications Technology Promotion (IITP) Grant funded by the Korea government (MSIT), South Korea (No. 2018-0-00958), and in part by the Office of Naval Research (ONR) MURI Grant N00014-16-1-2710.

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