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Published in: Dynamic Games and Applications 4/2019

13-12-2018

Risk-Sensitive Mean Field Games via the Stochastic Maximum Principle

Authors: Jun Moon, Tamer Başar

Published in: Dynamic Games and Applications | Issue 4/2019

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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.

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Appendix
Available only for authorised users
Footnotes
1
In fact, for step (i), we provide the risk-sensitive maximum principle for the general r-dimensional Brownian motion in “Appendix A” section, which generalizes the result of the one-dimensional Brownian motion in [42] when the diffusion coefficient is independent of control.
 
2
This assumption can be relaxed to a complete separable metric space [65].
 
3
A related discussion on this issue is provided in [6, Chapter 6].
 
4
This existence is dependent on the value of \(\gamma \), and when \(\gamma \) is large, the corresponding Riccati equation always admits a unique solution [7, 48].
 
5
A discussion on Lipschitz continuity of the optimal control in stochastic optimal control theory can be found in [28].
 
6
In this Appendix, we do not state specific regularity conditions for the corresponding stochastic optimal control problem, since they are quite similar to the assumptions already made in the paper. See [29, 42, 60, 65] for the regularity conditions for the stochastic optimal control problem.
 
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Metadata
Title
Risk-Sensitive Mean Field Games via the Stochastic Maximum Principle
Authors
Jun Moon
Tamer Başar
Publication date
13-12-2018
Publisher
Springer US
Published in
Dynamic Games and Applications / Issue 4/2019
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-018-00290-z

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