Explicit solutions of some linear-quadratic mean field games

Martino Bardi; Martino Bardi

doi:10.3934/nhm.2012.7.243

Networks and Heterogeneous Media

2012, Volume 7, Issue 2: 243-261. doi: 10.3934/nhm.2012.7.243

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Explicit solutions of some linear-quadratic mean field games

Martino Bardi ^{1
,}

1.
Dipartimento di Matematica, Università di Padova, via Trieste, 63; I-35121 Padova

Received: 01 November 2011 Revised: 01 March 2012
Primary: 91A13, 49N70; Secondary: 93E20, 91A06, 49N10.

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-Planck 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 [22]. 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.
- Mean field games,
- stochastic control,
- linear-quadratic problems,
- differential games,
- models of population distribution
Citation: Martino Bardi. Explicit solutions of some linear-quadratic mean field games[J]. Networks and Heterogeneous Media, 2012, 7(2): 243-261. doi: 10.3934/nhm.2012.7.243

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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-Planck 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 [22]. 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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