January  2015, 2(1): 89-101. doi: 10.3934/jdg.2015.2.89

Discrete time mean field games: The short-stage limit

1. 

Sorbonne Universités, UPMC Univ Paris 06, UMR 7586, IMJ-PRG, case 247, 4 place Jussieu, F-75005, Paris, France

Received  March 2014 Revised  February 2015 Published  June 2015

In this note we provide a model for discrete time mean field games. Our main contributions are an explicit approximation in the discounted case and an approximation result for a mean field game with short-stage duration.
Citation: Juan Pablo Maldonado López. Discrete time mean field games: The short-stage limit. Journal of Dynamics & Games, 2015, 2 (1) : 89-101. doi: 10.3934/jdg.2015.2.89
References:
[1]

Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods,, SIAM J. Numer. Anal., 48 (2010), 1136.  doi: 10.1137/090758477.  Google Scholar

[2]

S. Adlakha, R. Johari and G. Weintraub, Equilibria of dynamic games with many players: Existence, approximation, and market structure,, J. Econom. Theory, 156 (2015), 269.  doi: 10.1016/j.jet.2013.07.002.  Google Scholar

[3]

M. Bardi, Explicit solutions of some linear-quadratic mean field games,, Networks and Heterogeneous Media, 7 (2012), 243.  doi: 10.3934/nhm.2012.7.243.  Google Scholar

[4]

F. Camilli and F. Silva, A semi-discrete approximation for a first order mean field game problem,, Networks and Heterogeneous Media, 7 (2012), 263.  doi: 10.3934/nhm.2012.7.263.  Google Scholar

[5]

E. Carlini and F. Silva, A fully-discrete semi-Lagrangian approximation for a first order mean field game problem,, SIAM J. Numer. Anal., 52 (2014), 45.  doi: 10.1137/120902987.  Google Scholar

[6]

R. Elliot, X. Li and Y.-H. Ni, Discrete time mean-field stochastic linear-quadratic optimal control problems,, Automatica, 49 (2013), 3222.  doi: 10.1016/j.automatica.2013.08.017.  Google Scholar

[7]

D. Fudenberg and J. Tirole, Learning-by-doing and market performance,, The Bell J. of Economics, 14 (1983), 522.  doi: 10.2307/3003653.  Google Scholar

[8]

D. Gomes, J. Mohr and R. Souza, Discrete time, finite state space mean field games,, J. Math. Pures et Ap., 93 (2010), 308.  doi: 10.1016/j.matpur.2009.10.010.  Google Scholar

[9]

O. Guéant, J. Lasry and P. Lions, Mean Field Games and Applications,, in Paris Princeton Lectures on Mathematical Finance 2010, (2010), 205.   Google Scholar

[10]

O. Hernández-Lerma and J. Lasserre, Discrete-Time Markov Control Problems,, Springer-Verlag, (1996).   Google Scholar

[11]

M. Huang, P. Caines and R. Malhamé, Individual and mass behavior in large population stochastic wireless power control problems: Centralized and nash equilibrium solutions,, in Proceedings of the 42nd IEEE Conference on Decision and Control, (2003).   Google Scholar

[12]

M. Huang, P. Caines and R. Malhamé, Large population stochastic dynamic games: Closed-loop Mc Kean-Vlasov systems and the Nash certainty equivalence principle,, Communications in Information and Systems, 6 (2006), 221.  doi: 10.4310/CIS.2006.v6.n3.a5.  Google Scholar

[13]

J. Lasry and P. Lions, Jeux à champ moyen. I. Le cas stationnaire,, C. R. Math. Acad. Sci. Paris, 343 (2006), 619.  doi: 10.1016/j.crma.2006.09.019.  Google Scholar

[14]

J. Lasry and P. Lions, Jeux à champ moyen. II. Horizon fini et controle optimal,, C. R. Math. Acad. Sci. Paris, 343 (2006), 679.  doi: 10.1016/j.crma.2006.09.018.  Google Scholar

[15]

J. Lasry and P. Lions, Mean field games,, Japanese Journal of Mathematics, 2 (2007), 229.  doi: 10.1007/s11537-007-0657-8.  Google Scholar

[16]

A. Neyman, Stochastic games with short-stage duration,, Dyn. Games Appl., 3 (2013), 236.  doi: 10.1007/s13235-013-0083-x.  Google Scholar

show all references

References:
[1]

Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods,, SIAM J. Numer. Anal., 48 (2010), 1136.  doi: 10.1137/090758477.  Google Scholar

[2]

S. Adlakha, R. Johari and G. Weintraub, Equilibria of dynamic games with many players: Existence, approximation, and market structure,, J. Econom. Theory, 156 (2015), 269.  doi: 10.1016/j.jet.2013.07.002.  Google Scholar

[3]

M. Bardi, Explicit solutions of some linear-quadratic mean field games,, Networks and Heterogeneous Media, 7 (2012), 243.  doi: 10.3934/nhm.2012.7.243.  Google Scholar

[4]

F. Camilli and F. Silva, A semi-discrete approximation for a first order mean field game problem,, Networks and Heterogeneous Media, 7 (2012), 263.  doi: 10.3934/nhm.2012.7.263.  Google Scholar

[5]

E. Carlini and F. Silva, A fully-discrete semi-Lagrangian approximation for a first order mean field game problem,, SIAM J. Numer. Anal., 52 (2014), 45.  doi: 10.1137/120902987.  Google Scholar

[6]

R. Elliot, X. Li and Y.-H. Ni, Discrete time mean-field stochastic linear-quadratic optimal control problems,, Automatica, 49 (2013), 3222.  doi: 10.1016/j.automatica.2013.08.017.  Google Scholar

[7]

D. Fudenberg and J. Tirole, Learning-by-doing and market performance,, The Bell J. of Economics, 14 (1983), 522.  doi: 10.2307/3003653.  Google Scholar

[8]

D. Gomes, J. Mohr and R. Souza, Discrete time, finite state space mean field games,, J. Math. Pures et Ap., 93 (2010), 308.  doi: 10.1016/j.matpur.2009.10.010.  Google Scholar

[9]

O. Guéant, J. Lasry and P. Lions, Mean Field Games and Applications,, in Paris Princeton Lectures on Mathematical Finance 2010, (2010), 205.   Google Scholar

[10]

O. Hernández-Lerma and J. Lasserre, Discrete-Time Markov Control Problems,, Springer-Verlag, (1996).   Google Scholar

[11]

M. Huang, P. Caines and R. Malhamé, Individual and mass behavior in large population stochastic wireless power control problems: Centralized and nash equilibrium solutions,, in Proceedings of the 42nd IEEE Conference on Decision and Control, (2003).   Google Scholar

[12]

M. Huang, P. Caines and R. Malhamé, Large population stochastic dynamic games: Closed-loop Mc Kean-Vlasov systems and the Nash certainty equivalence principle,, Communications in Information and Systems, 6 (2006), 221.  doi: 10.4310/CIS.2006.v6.n3.a5.  Google Scholar

[13]

J. Lasry and P. Lions, Jeux à champ moyen. I. Le cas stationnaire,, C. R. Math. Acad. Sci. Paris, 343 (2006), 619.  doi: 10.1016/j.crma.2006.09.019.  Google Scholar

[14]

J. Lasry and P. Lions, Jeux à champ moyen. II. Horizon fini et controle optimal,, C. R. Math. Acad. Sci. Paris, 343 (2006), 679.  doi: 10.1016/j.crma.2006.09.018.  Google Scholar

[15]

J. Lasry and P. Lions, Mean field games,, Japanese Journal of Mathematics, 2 (2007), 229.  doi: 10.1007/s11537-007-0657-8.  Google Scholar

[16]

A. Neyman, Stochastic games with short-stage duration,, Dyn. Games Appl., 3 (2013), 236.  doi: 10.1007/s13235-013-0083-x.  Google Scholar

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