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Games 2018, 9(4), 88; https://doi.org/10.3390/g9040088

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

Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden
Received: 3 September 2018 / Revised: 16 October 2018 / Accepted: 23 October 2018 / Published: 1 November 2018
(This article belongs to the Special Issue Mean-Field-Type Game Theory)
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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. View Full-Text
Keywords: mean-field type game; non-zero-sum differential game; cooperative game; backward stochastic differential equations; linear-quadratic stochastic control; social cost; price of anarchy mean-field type game; non-zero-sum differential game; cooperative game; backward stochastic differential equations; linear-quadratic stochastic control; social cost; price of anarchy
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Aurell, A. Mean-Field Type Games between Two Players Driven by Backward Stochastic Differential Equations. Games 2018, 9, 88.

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