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Games 2018, 9(1), 7; doi:10.3390/g9010007

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

1
Department of Mathematics, University of Kansas, Lawrence, KS 66044, USA
2
Learning and Game Theory Laboratory, New York University Abu Dhabi, P.O. Box 129188, Abu Dhabi, UAE
*
Author to whom correspondence should be addressed.
Received: 4 January 2018 / Revised: 29 January 2018 / Accepted: 31 January 2018 / Published: 12 February 2018
(This article belongs to the Special Issue Mean-Field-Type Game Theory)
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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. View Full-Text
Keywords: Nash bargaining solution; mean-field equilibrium; variance; direct method Nash bargaining solution; mean-field equilibrium; variance; direct method
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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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Duncan, T.E.; Tembine, H. Linear–Quadratic Mean-Field-Type Games: A Direct Method. Games 2018, 9, 7.

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