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Linear–Quadratic Mean-Field-Type Games: A Direct Method

Author

Listed:
  • Tyrone E. Duncan

    (Department of Mathematics, University of Kansas, Lawrence, KS 66044, USA)

  • Hamidou Tembine

    (Learning and Game Theory Laboratory, New York University Abu Dhabi, P.O. Box 129188, Abu Dhabi, UAE)

Abstract

In this work, a multi-person mean-field-type game is formulated and solved that is described by a linear jump-diffusion system of mean-field type and a quadratic cost functional involving the second moments, the square of the expected value of the state, and the control actions of all decision-makers. We propose a direct method to solve the game, team, and bargaining problems. This solution approach does not require solving the Bellman–Kolmogorov equations or backward–forward stochastic differential equations of Pontryagin’s type. The proposed method can be easily implemented by beginners and engineers who are new to the emerging field of mean-field-type game theory. The optimal strategies for decision-makers are shown to be in a state-and-mean-field feedback form. The optimal strategies are given explicitly as a sum of the well-known linear state-feedback strategy for the associated deterministic linear–quadratic game problem and a mean-field feedback term. The equilibrium cost of the decision-makers are explicitly derived using a simple direct method. Moreover, the equilibrium cost is a weighted sum of the initial variance and an integral of a weighted variance of the diffusion and the jump process. Finally, the method is used to compute global optimum strategies as well as saddle point strategies and Nash bargaining solution in state-and-mean-field feedback form.

Suggested Citation

  • Tyrone E. Duncan & Hamidou Tembine, 2018. "Linear–Quadratic Mean-Field-Type Games: A Direct Method," Games, MDPI, vol. 9(1), pages 1-18, February.
  • Handle: RePEc:gam:jgames:v:9:y:2018:i:1:p:7-:d:131464
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    References listed on IDEAS

    as
    1. Engwerda, Jacob C., 1998. "On the open-loop Nash equilibrium in LQ-games," Journal of Economic Dynamics and Control, Elsevier, vol. 22(5), pages 729-762, May.
    2. Harry Markowitz, 1952. "The Utility of Wealth," Journal of Political Economy, University of Chicago Press, vol. 60(2), pages 151-151.
    3. A. Bensoussan, 2006. "Explicit Solutions of Linear Quadratic Differential Games," International Series in Operations Research & Management Science, in: Houmin Yan & George Yin & Qing Zhang (ed.), Stochastic Processes, Optimization, and Control Theory: Applications in Financial Engineering, Queueing Networks, and Manufacturing Systems, chapter 0, pages 19-34, Springer.
    4. Tyrone E. Duncan, 2014. "Linear-Quadratic Stochastic Differential Games with General Noise Processes," International Series in Operations Research & Management Science, in: Fouad El Ouardighi & Konstantin Kogan (ed.), Models and Methods in Economics and Management Science, edition 127, pages 17-25, Springer.
    5. Pradeep Dubey, 1986. "Inefficiency of Nash Equilibria," Mathematics of Operations Research, INFORMS, vol. 11(1), pages 1-8, February.
    6. Nash, John, 1950. "The Bargaining Problem," Econometrica, Econometric Society, vol. 18(2), pages 155-162, April.
    7. Tembine, Hamidou, 2014. "Energy-constrained Mean Field Games in Wireless Networks," Strategic Behavior and the Environment, now publishers, vol. 4(2), pages 187-211, July.
    8. Boualem Djehiche & Hamidou Tembine & Raul Tempone, 2014. "A Stochastic Maximum Principle for Risk-Sensitive Mean-Field Type Control," Papers 1404.1441, arXiv.org.
    9. Dockner,Engelbert J. & Jorgensen,Steffen & Long,Ngo Van & Sorger,Gerhard, 2000. "Differential Games in Economics and Management Science," Cambridge Books, Cambridge University Press, number 9780521637329, October.
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    Cited by:

    1. Alexander Aurell, 2018. "Mean-Field Type Games between Two Players Driven by Backward Stochastic Differential Equations," Games, MDPI, vol. 9(4), pages 1-26, November.

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