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

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  • Fabio Priuli

Abstract

We consider stochastic differential games with $$N$$ 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 $$N$$ N Hamilton–Jacobi–Bellman and $$N$$ 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. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Fabio Priuli, 2015. "Linear-Quadratic $$N$$ N -Person and Mean-Field Games: Infinite Horizon Games with Discounted Cost and Singular Limits," Dynamic Games and Applications, Springer, vol. 5(3), pages 397-419, September.
    • Handle: RePEc:spr:dyngam:v:5:y:2015:i:3:p:397-419
    DOI: 10.1007/s13235-014-0129-8
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    References listed on IDEAS

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    1. Olivier Guéant & Pierre Louis Lions & Jean-Michel Lasry, 2011. "Mean Field Games and Applications," Post-Print hal-01393103, HAL.
    2. Vassili Kolokoltsov & Marianna Troeva & Wei Yang, 2014. "On the Rate of Convergence for the Mean-Field Approximation of Controlled Diffusions with Large Number of Players," Dynamic Games and Applications, Springer, vol. 4(2), pages 208-230, June.
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