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Mean-Field Type Games between Two Players Driven by Backward Stochastic Differential Equations

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  • Alexander Aurell

    (Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden)

Abstract

In this paper, mean-field type games between two players with backward stochastic dynamics are defined and studied. They make up a class of non-zero-sum, non-cooperating, differential games where the players’ state dynamics solve backward stochastic differential equations (BSDE) that depend on the marginal distributions of player states. Players try to minimize their individual cost functionals, also depending on the marginal state distributions. Under some regularity conditions, we derive necessary and sufficient conditions for existence of Nash equilibria. Player behavior is illustrated by numerical examples, and is compared to a centrally planned solution where the social cost, the sum of player costs, is minimized. The inefficiency of a Nash equilibrium, compared to socially optimal behavior, is quantified by the so-called price of anarchy. Numerical simulations of the price of anarchy indicate how the improvement in social cost achievable by a central planner depends on problem parameters.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jgames:v:9:y:2018:i:4:p:88-:d:179975
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    References listed on IDEAS

    as
    1. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
    2. Tyrone E. Duncan & Hamidou Tembine, 2018. "Linear–Quadratic Mean-Field-Type Games: A Direct Method," Games, MDPI, vol. 9(1), pages 1-18, February.
    3. Duffie, Darrel & Lions, Pierre-Louis, 1992. "PDE solutions of stochastic differential utility," Journal of Mathematical Economics, Elsevier, vol. 21(6), pages 577-606.
    4. Aurell, Alexander & Djehiche, Boualem, 2019. "Modeling tagged pedestrian motion: A mean-field type game approach," Transportation Research Part B: Methodological, Elsevier, vol. 121(C), pages 168-183.
    5. Duffie, Darrell & Epstein, Larry G, 1992. "Stochastic Differential Utility," Econometrica, Econometric Society, vol. 60(2), pages 353-394, March.
    6. Boualem Djehiche & Hamidou Tembine & Raul Tempone, 2014. "A Stochastic Maximum Principle for Risk-Sensitive Mean-Field Type Control," Papers 1404.1441, arXiv.org.
    7. Buckdahn, Rainer & Li, Juan & Peng, Shige, 2009. "Mean-field backward stochastic differential equations and related partial differential equations," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3133-3154, October.
    8. Pradeep Dubey, 1986. "Inefficiency of Nash Equilibria," Mathematics of Operations Research, INFORMS, vol. 11(1), pages 1-8, February.
    9. Duffie, Darrell & Epstein, Larry G, 1992. "Asset Pricing with Stochastic Differential Utility," The Review of Financial Studies, Society for Financial Studies, vol. 5(3), pages 411-436.
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    Cited by:

    1. Aurell, Alexander & Djehiche, Boualem, 2019. "Modeling tagged pedestrian motion: A mean-field type game approach," Transportation Research Part B: Methodological, Elsevier, vol. 121(C), pages 168-183.

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