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Risk-Sensitive Mean Field Games via the Stochastic Maximum Principle

Author

Listed:
  • Jun Moon

    (Ulsan National Institute of Science and Technology (UNIST))

  • Tamer Başar

    (University of Illinois at Urbana-Champaign)

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 $$\epsilon $$ϵ-Nash equilibrium for the N player risk-sensitive game, where $$\epsilon \rightarrow 0$$ϵ→0 as $$N \rightarrow \infty $$N→∞ at the rate of $$O(\frac{1}{N^{1/(n+4)}})$$O(1N1/(n+4)). Finally, we discuss extensions to heterogeneous (non-symmetric) risk-sensitive mean field games.

Suggested Citation

  • Jun Moon & Tamer Başar, 2019. "Risk-Sensitive Mean Field Games via the Stochastic Maximum Principle," Dynamic Games and Applications, Springer, vol. 9(4), pages 1100-1125, December.
  • Handle: RePEc:spr:dyngam:v:9:y:2019:i:4:d:10.1007_s13235-018-00290-z
    DOI: 10.1007/s13235-018-00290-z
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    References listed on IDEAS

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    1. Gabriel Y. Weintraub & C. Lanier Benkard & Benjamin Van Roy, 2008. "Markov Perfect Industry Dynamics With Many Firms," Econometrica, Econometric Society, vol. 76(6), pages 1375-1411, November.
    2. Dario Bauso & Hamidou Tembine & Tamer Başar, 2016. "Robust Mean Field Games," Dynamic Games and Applications, Springer, vol. 6(3), pages 277-303, September.
    3. Hamidou Tembine & Quanyan Zhu & Tamer Basar, 2011. "Risk-sensitive mean field stochastic differential games," Post-Print hal-00643547, HAL.
    4. Delarue, François, 2002. "On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case," Stochastic Processes and their Applications, Elsevier, vol. 99(2), pages 209-286, June.
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    Cited by:

    1. Régis Chenavaz & Corina Paraschiv & Gabriel Turinici, 2021. "Dynamic Pricing of New Products in Competitive Markets: A Mean-Field Game Approach," Dynamic Games and Applications, Springer, vol. 11(3), pages 463-490, September.
    2. Tamer Başar, 2025. "Nash equilibria of networked games with delayed coupling in the high population regime," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 33(2), pages 415-427, June.
    3. Hanchao Liu & Dena Firoozi & Mich`ele Breton, 2023. "LQG Risk-Sensitive Single-Agent and Major-Minor Mean-Field Game Systems: A Variational Framework," Papers 2305.15364, arXiv.org, revised Mar 2025.

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