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From mean field games to the best reply strategy in a stochastic framework
Imperial College London, London, SW7 2AZ, UK |
This paper builds on the work of Degond, Herty and Liu in [
References:
[1] |
Y. Achdou, F. Camilli and I. Capuzzo-Dolcetta,
Mean field games: Convergence of a finite difference method, SIAM J. Numer. Anal., 51 (2013), 2585-2612.
doi: 10.1137/120882421. |
[2] |
Y. Achdou, F. Camilli and I. Capuzzo-Dolcetta,
Mean field games: Numerical methods for the planning problem, SIAM J. Numer. Anal., 50 (2012), 77-109.
doi: 10.1137/100790069. |
[3] |
Y. Achdou and A. Porretta,
Convergence of a finite difference scheme to weak solutions of the system of partial differential equations arising in mean field games, SIAM J. Numer. Anal., 54 (2016), 161-186.
doi: 10.1137/15M1015455. |
[4] |
G. Albi, M. Herty and L. Pareschi,
Kinetic description of optimal control problems and applications to opinion consensus, Commun. Math. Sci., 13 (2015), 1407-1429.
doi: 10.4310/CMS.2015.v13.n6.a3. |
[5] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2$^{nd}$ edition, Birkhaüser, Basel, 2008.
doi: 10.1007/978-3-7643-8722-8. |
[6] |
R. J. Aumann,
Markets with a Continuum of Traders, Econometrica, 32 (1964), 39-50.
doi: 10.2307/1913732. |
[7] |
M. Bardi and F. S. Priuli,
Linear-quadratic n-person and mean-field games with ergodic cost, SIAM J. Control Optim., 52 (2014), 3022-3052.
doi: 10.1137/140951795. |
[8] |
A. Blanchet and G. Carlier, From Nash to Cournot-Nash equilibria via the Monge-Kantorovich problem, Philos. T. R. Soc. A, 372 (2014), 20130398, 11 pp.
doi: 10.1098/rsta.2013.0398. |
[9] |
J. P. Bouchaud and M. Mézard,
Wealth condensation in a simple model of economy, Physica A, 282 (2000), 536-545.
doi: 10.1016/S0378-4371(00)00205-3. |
[10] | |
[11] |
P. Cardaliaguet, F. Delarue, J.-M. Lasry and P.-L. Lions, The Master Equation and the Convergence Problem in Mean Field Games, Annals of Mathematics Studies, 201. Princeton University Press, Princeton, NJ, 2019, arXiv: 1509.02505.
doi: 10.2307/j.ctvckq7qf. |
[12] |
P. Cardaliaguet, P. J. Graber, A. Porretta and D. Tonon,
Second order mean field games with degenerate diffusion and local coupling, NODEA-Nonlinear Diff., 22 (2015), 1287-1317.
doi: 10.1007/s00030-015-0323-4. |
[13] |
R. Carmona and F. Delarue, Mean field forward-backward stochastic differential equations, Electron. Commun. Prob., 18 (2013), 15pp.
doi: 10.1214/ECP.v18-2446. |
[14] |
R. Carmona and F. Delarue,
Probabilistic analysis of mean-field games, SIAM J. Control Optim., 51 (2013), 2705-2734.
doi: 10.1137/120883499. |
[15] |
R. Carmona, F. Delarue and D. Lacker,
Mean field games with common noise, Ann. Probab., 44 (2016), 3740-3803.
doi: 10.1214/15-AOP1060. |
[16] |
P. Degond, M. Herty and J.-G. Liu,
Meanfield games and model predictive control, Commun. Math. Sci., 15 (2017), 1403-1422.
doi: 10.4310/CMS.2017.v15.n5.a9. |
[17] |
P. Degond, J.-G. Liu and C. Ringhofer,
Evolution of the distribution of wealth in an economic environment driven by local Nash equilibria, J. Stat. Phys., 154 (2014), 751-780.
doi: 10.1007/s10955-013-0888-4. |
[18] |
P. Degond, J.-G. Liu and C. Ringhofer, Evolution of wealth in a non-conservative economy driven by local Nash equilibria, Philos. T. R. Soc. A, 372 (2014), 20130394, 15 pp.
doi: 10.1098/rsta.2013.0394. |
[19] |
P. Degond, J.-G. Liu and C. Ringhofer,
Large-scale dynamics of mean-field games driven by local Nash equilibria, J. Nonlinear Sci., 24 (2014), 93-115.
doi: 10.1007/s00332-013-9185-2. |
[20] |
F. Delarue and R. Carmona, Probabilistic Theory of Mean Field Games with Applications I, Springer, Cham, 2018.
doi: 10.1007/978-3-319-58920-6. |
[21] |
B. Düring and G. Toscani,
Hydrodynamics from kinetic models of conservative economies, Physica A, 384 (2007), 493-506.
doi: 10.1016/j.physa.2007.05.062. |
[22] |
E. Feleqi,
The derivation of ergodic mean field game equations for several populations of players, Dyn. Games Appl., 3 (2013), 523-536.
doi: 10.1007/s13235-013-0088-5. |
[23] |
A. Friedman,
Stochastic differential games, J. Differ. Equations, 11 (1972), 79-108.
doi: 10.1016/0022-0396(72)90082-4. |
[24] |
M. Herty and M. Zanella,
Performance bounds for the mean-field limit of constrained dynamics, Discrete Cont. Dyn.–A, 37 (2017), 2023-2043.
doi: 10.3934/dcds.2017086. |
[25] |
M. Huang, P. E. Caines and R. P. Malhamé,
An invariance principle in large population stochastic dynamic games, J. Syst. Sci. Complex., 20 (2007), 162-172.
doi: 10.1007/s11424-007-9015-4. |
[26] |
M. Huang, P. E. Caines and R. P. Malhamé, Distributed multi-agent decision-making with partial observations: Asymptotic Nash equilibria, in Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems, (2006), 2725–2730. |
[27] |
M. Huang, P. E. Caines and R. P. Malhamé,
Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized $ \varepsilon$-nash equilibria, IEEE T. Automat. Contr., 52 (2007), 1560-1571.
doi: 10.1109/TAC.2007.904450. |
[28] |
M. Huang, P. E. Caines and R. P. Malhamé,
Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Communications in Information and Systems, 6 (2006), 221-251.
doi: 10.4310/CIS.2006.v6.n3.a5. |
[29] |
M. Huang, P. E. Caines and R. P. Malhamé, Nash certainty equivalence in large population stochastic dynamic games: Connections with the physics of interacting particle systems, in Proceedings of the 45th IEEE Conference on Decision and Control, (2006), 4921–4926. |
[30] |
M. Huang and S. L. Nguyen,
Mean field games for stochastic growth with relative consumption, Appl. Math. Opt., 74 (2016), 643-668.
doi: 10.1007/s00245-016-9395-8. |
[31] |
B. Jourdain, S. Méléard and W. Woyczynski,
Nonlinear SDEs driven by Lévy processes and related PDEs, ALEA–Lat. Am. J. Probab., 4 (2008), 1-29.
|
[32] |
A. C. Kizilkale and R. P. Malhamé, Collective target tracking mean field control for markovian jump-driven models of electric water heating loads,, in Control of Complex Systems: Theory and Applications (eds. K. G. Vamvoudakis and S. Jagannathan) Elsevier, (2016), 559–584.
doi: 10.1016/B978-0-12-805246-4.00020-3. |
[33] |
P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, 2$^{nd}$ edition Springer-Verlag, Berlin Heidelberg, 1992.
doi: 10.1007/978-3-662-12616-5. |
[34] |
A. Lachapelle and M. -T. Wolfram,
On a mean field game approach modeling congestion and aversion in pedestrian crowds, Transport. Res. B–Meth., 45 (2011), 1572-1589.
doi: 10.1016/j.trb.2011.07.011. |
[35] |
J. M. Lasry and P. L. Lions,
Jeux à champ moyen. Ⅰ — Le cas stationnaire, C. r. math., 343 (2006), 619-625.
doi: 10.1016/j.crma.2006.09.019. |
[36] |
J. M. Lasry and P. L. Lions,
Jeux à champ moyen. Ⅱ — Horizon fini et contrôle optimal, C. R. Math., 343 (2006), 679-684.
doi: 10.1016/j.crma.2006.09.018. |
[37] |
J. M. Lasry and P. L. Lions,
Mean field games, Jpn. J. Math., 2 (2007), 229-260.
doi: 10.1007/s11537-007-0657-8. |
[38] |
A. Mas-Colell,
On a theorem of Schmeidler, J. Math. Econ., 13 (1984), 201-206.
doi: 10.1016/0304-4068(84)90029-6. |
[39] |
D. Q. Mayne and H. Michalska,
Receding horizon control of nonlinear systems, IEEE T. Automat. Contr., 35 (1990), 814-824.
doi: 10.1109/9.57020. |
[40] |
B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, Springer-Verlag, Berlin Heidelberg, 2003.
doi: 10.1007/978-3-642-14394-6. |
[41] |
D. Schmeidler,
Equilibrium points of nonatomic games, J. Stat. Phys., 7 (1973), 295-300.
doi: 10.1007/BF01014905. |
[42] |
A.-S. Sznitman, Topics in propagation of chaos, in Ecole d'Eté de Probabilités de Saint-Flour XIX–1989 (Ed. P.-L. Hennequin), Springer, Berlin Heidelberg, 1464 (1991), 165–251.
doi: 10.1007/BFb0085169. |
show all references
References:
[1] |
Y. Achdou, F. Camilli and I. Capuzzo-Dolcetta,
Mean field games: Convergence of a finite difference method, SIAM J. Numer. Anal., 51 (2013), 2585-2612.
doi: 10.1137/120882421. |
[2] |
Y. Achdou, F. Camilli and I. Capuzzo-Dolcetta,
Mean field games: Numerical methods for the planning problem, SIAM J. Numer. Anal., 50 (2012), 77-109.
doi: 10.1137/100790069. |
[3] |
Y. Achdou and A. Porretta,
Convergence of a finite difference scheme to weak solutions of the system of partial differential equations arising in mean field games, SIAM J. Numer. Anal., 54 (2016), 161-186.
doi: 10.1137/15M1015455. |
[4] |
G. Albi, M. Herty and L. Pareschi,
Kinetic description of optimal control problems and applications to opinion consensus, Commun. Math. Sci., 13 (2015), 1407-1429.
doi: 10.4310/CMS.2015.v13.n6.a3. |
[5] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2$^{nd}$ edition, Birkhaüser, Basel, 2008.
doi: 10.1007/978-3-7643-8722-8. |
[6] |
R. J. Aumann,
Markets with a Continuum of Traders, Econometrica, 32 (1964), 39-50.
doi: 10.2307/1913732. |
[7] |
M. Bardi and F. S. Priuli,
Linear-quadratic n-person and mean-field games with ergodic cost, SIAM J. Control Optim., 52 (2014), 3022-3052.
doi: 10.1137/140951795. |
[8] |
A. Blanchet and G. Carlier, From Nash to Cournot-Nash equilibria via the Monge-Kantorovich problem, Philos. T. R. Soc. A, 372 (2014), 20130398, 11 pp.
doi: 10.1098/rsta.2013.0398. |
[9] |
J. P. Bouchaud and M. Mézard,
Wealth condensation in a simple model of economy, Physica A, 282 (2000), 536-545.
doi: 10.1016/S0378-4371(00)00205-3. |
[10] | |
[11] |
P. Cardaliaguet, F. Delarue, J.-M. Lasry and P.-L. Lions, The Master Equation and the Convergence Problem in Mean Field Games, Annals of Mathematics Studies, 201. Princeton University Press, Princeton, NJ, 2019, arXiv: 1509.02505.
doi: 10.2307/j.ctvckq7qf. |
[12] |
P. Cardaliaguet, P. J. Graber, A. Porretta and D. Tonon,
Second order mean field games with degenerate diffusion and local coupling, NODEA-Nonlinear Diff., 22 (2015), 1287-1317.
doi: 10.1007/s00030-015-0323-4. |
[13] |
R. Carmona and F. Delarue, Mean field forward-backward stochastic differential equations, Electron. Commun. Prob., 18 (2013), 15pp.
doi: 10.1214/ECP.v18-2446. |
[14] |
R. Carmona and F. Delarue,
Probabilistic analysis of mean-field games, SIAM J. Control Optim., 51 (2013), 2705-2734.
doi: 10.1137/120883499. |
[15] |
R. Carmona, F. Delarue and D. Lacker,
Mean field games with common noise, Ann. Probab., 44 (2016), 3740-3803.
doi: 10.1214/15-AOP1060. |
[16] |
P. Degond, M. Herty and J.-G. Liu,
Meanfield games and model predictive control, Commun. Math. Sci., 15 (2017), 1403-1422.
doi: 10.4310/CMS.2017.v15.n5.a9. |
[17] |
P. Degond, J.-G. Liu and C. Ringhofer,
Evolution of the distribution of wealth in an economic environment driven by local Nash equilibria, J. Stat. Phys., 154 (2014), 751-780.
doi: 10.1007/s10955-013-0888-4. |
[18] |
P. Degond, J.-G. Liu and C. Ringhofer, Evolution of wealth in a non-conservative economy driven by local Nash equilibria, Philos. T. R. Soc. A, 372 (2014), 20130394, 15 pp.
doi: 10.1098/rsta.2013.0394. |
[19] |
P. Degond, J.-G. Liu and C. Ringhofer,
Large-scale dynamics of mean-field games driven by local Nash equilibria, J. Nonlinear Sci., 24 (2014), 93-115.
doi: 10.1007/s00332-013-9185-2. |
[20] |
F. Delarue and R. Carmona, Probabilistic Theory of Mean Field Games with Applications I, Springer, Cham, 2018.
doi: 10.1007/978-3-319-58920-6. |
[21] |
B. Düring and G. Toscani,
Hydrodynamics from kinetic models of conservative economies, Physica A, 384 (2007), 493-506.
doi: 10.1016/j.physa.2007.05.062. |
[22] |
E. Feleqi,
The derivation of ergodic mean field game equations for several populations of players, Dyn. Games Appl., 3 (2013), 523-536.
doi: 10.1007/s13235-013-0088-5. |
[23] |
A. Friedman,
Stochastic differential games, J. Differ. Equations, 11 (1972), 79-108.
doi: 10.1016/0022-0396(72)90082-4. |
[24] |
M. Herty and M. Zanella,
Performance bounds for the mean-field limit of constrained dynamics, Discrete Cont. Dyn.–A, 37 (2017), 2023-2043.
doi: 10.3934/dcds.2017086. |
[25] |
M. Huang, P. E. Caines and R. P. Malhamé,
An invariance principle in large population stochastic dynamic games, J. Syst. Sci. Complex., 20 (2007), 162-172.
doi: 10.1007/s11424-007-9015-4. |
[26] |
M. Huang, P. E. Caines and R. P. Malhamé, Distributed multi-agent decision-making with partial observations: Asymptotic Nash equilibria, in Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems, (2006), 2725–2730. |
[27] |
M. Huang, P. E. Caines and R. P. Malhamé,
Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized $ \varepsilon$-nash equilibria, IEEE T. Automat. Contr., 52 (2007), 1560-1571.
doi: 10.1109/TAC.2007.904450. |
[28] |
M. Huang, P. E. Caines and R. P. Malhamé,
Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Communications in Information and Systems, 6 (2006), 221-251.
doi: 10.4310/CIS.2006.v6.n3.a5. |
[29] |
M. Huang, P. E. Caines and R. P. Malhamé, Nash certainty equivalence in large population stochastic dynamic games: Connections with the physics of interacting particle systems, in Proceedings of the 45th IEEE Conference on Decision and Control, (2006), 4921–4926. |
[30] |
M. Huang and S. L. Nguyen,
Mean field games for stochastic growth with relative consumption, Appl. Math. Opt., 74 (2016), 643-668.
doi: 10.1007/s00245-016-9395-8. |
[31] |
B. Jourdain, S. Méléard and W. Woyczynski,
Nonlinear SDEs driven by Lévy processes and related PDEs, ALEA–Lat. Am. J. Probab., 4 (2008), 1-29.
|
[32] |
A. C. Kizilkale and R. P. Malhamé, Collective target tracking mean field control for markovian jump-driven models of electric water heating loads,, in Control of Complex Systems: Theory and Applications (eds. K. G. Vamvoudakis and S. Jagannathan) Elsevier, (2016), 559–584.
doi: 10.1016/B978-0-12-805246-4.00020-3. |
[33] |
P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, 2$^{nd}$ edition Springer-Verlag, Berlin Heidelberg, 1992.
doi: 10.1007/978-3-662-12616-5. |
[34] |
A. Lachapelle and M. -T. Wolfram,
On a mean field game approach modeling congestion and aversion in pedestrian crowds, Transport. Res. B–Meth., 45 (2011), 1572-1589.
doi: 10.1016/j.trb.2011.07.011. |
[35] |
J. M. Lasry and P. L. Lions,
Jeux à champ moyen. Ⅰ — Le cas stationnaire, C. r. math., 343 (2006), 619-625.
doi: 10.1016/j.crma.2006.09.019. |
[36] |
J. M. Lasry and P. L. Lions,
Jeux à champ moyen. Ⅱ — Horizon fini et contrôle optimal, C. R. Math., 343 (2006), 679-684.
doi: 10.1016/j.crma.2006.09.018. |
[37] |
J. M. Lasry and P. L. Lions,
Mean field games, Jpn. J. Math., 2 (2007), 229-260.
doi: 10.1007/s11537-007-0657-8. |
[38] |
A. Mas-Colell,
On a theorem of Schmeidler, J. Math. Econ., 13 (1984), 201-206.
doi: 10.1016/0304-4068(84)90029-6. |
[39] |
D. Q. Mayne and H. Michalska,
Receding horizon control of nonlinear systems, IEEE T. Automat. Contr., 35 (1990), 814-824.
doi: 10.1109/9.57020. |
[40] |
B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, Springer-Verlag, Berlin Heidelberg, 2003.
doi: 10.1007/978-3-642-14394-6. |
[41] |
D. Schmeidler,
Equilibrium points of nonatomic games, J. Stat. Phys., 7 (1973), 295-300.
doi: 10.1007/BF01014905. |
[42] |
A.-S. Sznitman, Topics in propagation of chaos, in Ecole d'Eté de Probabilités de Saint-Flour XIX–1989 (Ed. P.-L. Hennequin), Springer, Berlin Heidelberg, 1464 (1991), 165–251.
doi: 10.1007/BFb0085169. |

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