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Linear-Quadratic \(N\)-Person and Mean-Field Games: Infinite Horizon Games with Discounted Cost and Singular Limits

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Abstract

We consider stochastic differential games with \(N\) nearly identical players, linear-Gaussian dynamics, and infinite horizon discounted quadratic cost. Admissible controls are feedbacks for which the system is ergodic. We first study the existence of affine Nash equilibria by means of an associated system of \(N\) Hamilton–Jacobi–Bellman and \(N\) Kolmogorov–Fokker–Planck partial differential equations, proving that for small discount factors quadratic-Gaussian solutions exist and are unique. Then, we prove the convergence of such solutions to the unique quadratic-Gaussian solution of the pair of Mean Field equations. We also discuss some singular limits, such as vanishing discount, vanishing noise, and cheap control.

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Notes

  1. Observe that the value \(\bar{\ell }\) depends on the fixed \(N\ge \bar{N}\), so that it is in fact \(\bar{\ell }=\bar{\ell }_N\). However, the number of players is being kept fixed throughout the rest of this step of the proof, so we can drop the the dependency in the notation without risk of confusion.

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Acknowledgments

The author was partially supported by the Fondazione CaRiPaRo Project “Nonlinear Partial Differential Equations: models, analysis, and control-theoretic problems”.

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  1. Istituto per le Applicazioni del Calcolo “M. Picone”, C.N.R, Rome, Italy

    Fabio S. Priuli

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  1. Fabio S. Priuli
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Correspondence to Fabio S. Priuli.

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Priuli, F.S. Linear-Quadratic \(N\)-Person and Mean-Field Games: Infinite Horizon Games with Discounted Cost and Singular Limits. Dyn Games Appl 5, 397–419 (2015). https://doi.org/10.1007/s13235-014-0129-8

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