October  2019, 6(4): 291-314. doi: 10.3934/jdg.2019020

From mean field games to the best reply strategy in a stochastic framework

Imperial College London, London, SW7 2AZ, UK

* Corresponding author: Matt Barker

Received  November 2018 Revised  July 2019 Published  October 2019

Fund Project: Matt Barker is funded by NERC through the SSCP DTP at the Grantham Institute.

This paper builds on the work of Degond, Herty and Liu in [16] by considering $ N $-player stochastic differential games. The control corresponding to a Nash equilibrium of such a game is approximated through model predictive control (MPC) techniques. In the case of a linear quadratic running-cost, considered here, the MPC method is shown to approximate the solution to the control problem by the best reply strategy (BRS) for the running cost. We then compare the MPC approach when taking the mean field limit with the popular mean field game (MFG) strategy. We find that our MPC approach reduces the two coupled PDEs to a single PDE, greatly increasing the simplicity and tractability of the original problem. We give two examples of applications of this approach to previous literature and conclude with future perspectives for this research.

Citation: Matt Barker. From mean field games to the best reply strategy in a stochastic framework. Journal of Dynamics & Games, 2019, 6 (4) : 291-314. doi: 10.3934/jdg.2019020
References:
[1]

Y. AchdouF. 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.  Google Scholar

[2]

Y. AchdouF. 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.  Google Scholar

[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.  Google Scholar

[4]

G. AlbiM. 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.  Google Scholar

[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.  Google Scholar

[6]

R. J. Aumann, Markets with a Continuum of Traders, Econometrica, 32 (1964), 39-50.  doi: 10.2307/1913732.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

P. Cardaliaguet, Notes on mean field games, Preprint. Google Scholar

[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.  Google Scholar

[12]

P. CardaliaguetP. J. GraberA. 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.  Google Scholar

[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.  Google Scholar

[14]

R. Carmona and F. Delarue, Probabilistic analysis of mean-field games, SIAM J. Control Optim., 51 (2013), 2705-2734.  doi: 10.1137/120883499.  Google Scholar

[15]

R. CarmonaF. Delarue and D. Lacker, Mean field games with common noise, Ann. Probab., 44 (2016), 3740-3803.  doi: 10.1214/15-AOP1060.  Google Scholar

[16]

P. DegondM. 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.  Google Scholar

[17]

P. DegondJ.-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.  Google Scholar

[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.  Google Scholar

[19]

P. DegondJ.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

A. Friedman, Stochastic differential games, J. Differ. Equations, 11 (1972), 79-108.  doi: 10.1016/0022-0396(72)90082-4.  Google Scholar

[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.  Google Scholar

[25]

M. HuangP. 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.  Google Scholar

[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. Google Scholar

[27]

M. HuangP. 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.  Google Scholar

[28]

M. HuangP. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[31]

B. JourdainS. Méléard and W. Woyczynski, Nonlinear SDEs driven by Lévy processes and related PDEs, ALEA–Lat. Am. J. Probab., 4 (2008), 1-29.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

A. Mas-Colell, On a theorem of Schmeidler, J. Math. Econ., 13 (1984), 201-206.  doi: 10.1016/0304-4068(84)90029-6.  Google Scholar

[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.  Google Scholar

[40]

B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, Springer-Verlag, Berlin Heidelberg, 2003. doi: 10.1007/978-3-642-14394-6.  Google Scholar

[41]

D. Schmeidler, Equilibrium points of nonatomic games, J. Stat. Phys., 7 (1973), 295-300.  doi: 10.1007/BF01014905.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

Y. AchdouF. 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.  Google Scholar

[2]

Y. AchdouF. 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.  Google Scholar

[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.  Google Scholar

[4]

G. AlbiM. 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.  Google Scholar

[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.  Google Scholar

[6]

R. J. Aumann, Markets with a Continuum of Traders, Econometrica, 32 (1964), 39-50.  doi: 10.2307/1913732.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

P. Cardaliaguet, Notes on mean field games, Preprint. Google Scholar

[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.  Google Scholar

[12]

P. CardaliaguetP. J. GraberA. 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.  Google Scholar

[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.  Google Scholar

[14]

R. Carmona and F. Delarue, Probabilistic analysis of mean-field games, SIAM J. Control Optim., 51 (2013), 2705-2734.  doi: 10.1137/120883499.  Google Scholar

[15]

R. CarmonaF. Delarue and D. Lacker, Mean field games with common noise, Ann. Probab., 44 (2016), 3740-3803.  doi: 10.1214/15-AOP1060.  Google Scholar

[16]

P. DegondM. 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.  Google Scholar

[17]

P. DegondJ.-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.  Google Scholar

[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.  Google Scholar

[19]

P. DegondJ.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

A. Friedman, Stochastic differential games, J. Differ. Equations, 11 (1972), 79-108.  doi: 10.1016/0022-0396(72)90082-4.  Google Scholar

[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.  Google Scholar

[25]

M. HuangP. 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.  Google Scholar

[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. Google Scholar

[27]

M. HuangP. 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.  Google Scholar

[28]

M. HuangP. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[31]

B. JourdainS. Méléard and W. Woyczynski, Nonlinear SDEs driven by Lévy processes and related PDEs, ALEA–Lat. Am. J. Probab., 4 (2008), 1-29.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

A. Mas-Colell, On a theorem of Schmeidler, J. Math. Econ., 13 (1984), 201-206.  doi: 10.1016/0304-4068(84)90029-6.  Google Scholar

[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.  Google Scholar

[40]

B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, Springer-Verlag, Berlin Heidelberg, 2003. doi: 10.1007/978-3-642-14394-6.  Google Scholar

[41]

D. Schmeidler, Equilibrium points of nonatomic games, J. Stat. Phys., 7 (1973), 295-300.  doi: 10.1007/BF01014905.  Google Scholar

[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.  Google Scholar

Figure 1.  Schematic diagram describing the links between $ N $-player games, mean field games and model predictive control
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