Explicit solutions of some Linear-Quadratic Mean Field Games

Abstract : We consider N-person differential games involving linear systems affected by white noise, running cost quadratic in the control and in the displacement of the state from a reference position, and with long-time-average integral cost functional. We solve an associated system of Hamilton-Jacobi-Bellman and Kolmogorov-Fokker-Plank equations and find explicit Nash equilibria in the form of linear feedbacks. Next we compute the limit as the number N of players goes to infinity, assuming they are almost identical and with suitable scalings of the parameters. This provides a quadratic-Gaussian solution to a system of two differential equations of the kind introduced by Lasry and Lions in the theory of Mean Field Games [19]. Under a natural normalization the uniqueness of this solution depends on the sign of a single parameter. We also discuss some singular limits, such as vanishing noise, cheap control, vanishing discount. Finally, we compare the L-Q model with other Mean Field models of population distribution.
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Martino Bardi. Explicit solutions of some Linear-Quadratic Mean Field Games. Networks and Heterogeneous Media, AIMS-American Institute of Mathematical Sciences, 2012, 7 (2), pp.243 - 261. ⟨10.3934/nhm.2012.7.243⟩. ⟨hal-00664442⟩

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