ε-nash mean field game theory for nonlinear stochastic dynamical systems with major and minor agents

SIAM Journal on Control and Optimization

Volumn 51, Issue 4, 2013, Pages 3302-3331

  Nourian, Mojtaba ^a,b,c   Caines, Peter E ^b,c  

^a UNIVERSITY OF MELBOURNE   (Australia)

^b MCGILL UNIVERSITY   (Canada)

^c GERAD   (Canada)

Author keywords

Decentralized control; Mean field games; Mixed agents; Nash equilibria; Stochastic dynamic games; Stochastic Hamilton Jacobi Bellman equation; Stochastic McKean Vlasov equation; Stochastic optimal control

Indexed keywords

HAMILTON JACOBI BELLMAN EQUATION; MCKEAN-VLASOV EQUATIONS; MEAN FIELD GAMES; NASH EQUILIBRIA; STOCHASTIC DYNAMICS; STOCHASTIC OPTIMAL CONTROL;

COST FUNCTIONS; DECENTRALIZED CONTROL; GAME THEORY; OPTIMAL CONTROL SYSTEMS; POPULATION STATISTICS; STOCHASTIC CONTROL SYSTEMS; STOCHASTIC MODELS; TELECOMMUNICATION NETWORKS; VLASOV EQUATION;

STOCHASTIC SYSTEMS;

EID: 84888865707     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/120889496     Document Type: Article

References (48)

1. D. Andersson and B. Djehiche, A maximum principle for SDEs of mean-field type, Appl. Math. Optim., 63 (2011), pp. 341-356.
2. J. M. Bismut, Linear quadratic optimal stochastic control with random coefficients, SIAM J. Control Optim., 14 (1976), pp. 419-444.
3. R. Boel and P. Varaiya, Optimal control of jump processes, SIAM J. Control Optim., 15 (1977), pp. 92-119.
4. R. Buckdahn, P. Cardaliaguet, and M. Quincampoix, Some recent aspects of differential game theory, Dynam. Games Appl., 1 (2011), pp. 74-114.
5. R. Buckdahn, B. Djehiche, J. Li, and S. Peng, Mean-field backward stochastic differential equations: A limit approach, Ann. Probab., 37 (2009), pp. 1524-1565.
6. R. Buckdahn, J. Li, and S. Peng, Mean-field backward stochastic differential equations and related partial differential equations, Stochastic Process Appl., 119 (2009), pp. 3133-3154.
7. P. E. Caines, Bode lecture: Mean field stochastic control, in Proceedings of the 48th IEEE CDC, Shanghai, China, 2009; also available online from http://www.ieeecssoll. org/lectures/2009/mean-field-stochastic-control.
8. C. Dogbé, Modeling crowd dynamics by the mean-field limit approach, Math. Comput. Modelling, 52 (2010), pp. 1506-1520.
9. B. Düring, P. Markowich, J. F. Pietschmann, and M. T. Wolfram, Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders, in Proc. Roy. Soc. A, 465 (2009), pp. 3687-3707.
10. N. Englezos and I. Karatzas, Utility maximization with habit formation: Dynamic programming and stochastic PDEs, SIAM J. Control Optim., 48 (2009), pp. 481-520.
11. O. Guéant, J.-M. Lasry, and P.-L. Lions, Mean field games and applications, in Paris-Princeton Lectures on Mathematical Finance, Springer-Verlag, Heidelberg, 2011, pp. 205-266.
12. O. Haimanko, Nonsymmetric values of nonatomic and mixed games, Math. Oper. Res., 25 (2000), pp. 591-605. (Pubitemid 32135795)
13. S. Hart, Values of mixed games, Internat. J. Game Theory, 2 (1973), pp. 69-85.
14. M. Huang, P. E. Caines, and R. P. Malhamé, Individual and mass behaviour in large population stochastic wireless power control problems: Centralized and Nash equilibrium solutions, in Proceedings of the 42nd IEEE CDC, Maui, HI, 2003, pp. 98-103.
15. M. Huang, P. E. Caines, and R. P. Malhamé, An invariance principle in large population stochastic dynamic games, J. Syst. Sci. Complex., 20 (2007), pp. 162-172. (Pubitemid 46800807)
16. M. Huang, P. E. Caines, and R. P. Malhamé, Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized -Nash equilibria, IEEE Trans. Automat. Control, 52 (2007), pp. 1560-1571. (Pubitemid 47456068)
17. M. Huang, R. P. Malhamé, and P. E Caines, Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Commun. Inf. Syst., 6 (2006), pp. 221-252.
18. M. Huang, Large-population LQG games involving a major player: The Nash certainty equivalence principle, SIAM J. Control Optim., 48 (2010), pp. 3318-3353.
19. A. C. Kizilkale and P. E. Caines, Mean field (NCE) stochastic control: Populations of major and egoist-altruist agents, in Proceedings of the 50th IEEE CDC-ECC, Orlando, FL, 2011, pp. 5547-5552.
20. A. C. Kizilkale and P. E. Caines, Mean field stochastic adaptive control, IEEE Trans. Automat. Control, 58 (2013), pp. 905-920.
21. A. C. Kizilkale, S. Mannor, and P. E. Caines, Large scale real-time bidding in the smart grid: A mean field framework, in Proceedings of the 51st IEEE CDC, Maui, HI, 2012, pp. 3680-3687.
22. V. N. Kolokoltsov, J. Li, and W. Yang, Mean Field Games and Nonlinear Markov Processes, arXiv:1112.3744, 2011.
23. L. B. Koralov and Y. G. Sinai, Theory of Probability and Random Processes, Springer-Verlag, Heidelberg, 2007.
24. J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. II Horizon fini et contrôle optimal, C. R. Math., 343 (2006), pp. 679-684. (Pubitemid 44762838)
25. J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. I, Le cas stationnaire, C. R. Math., 343 (2006), pp. 619-625. (Pubitemid 44689282)
26. J.-M. Lasry and P.-L. Lions, Mean field games, Japan. J. Math., 2 (2007), pp. 229-260.
27. T. Li and J.-F. Zhang, Asymptotically optimal decentralized control for large population stochastic multiagent systems, IEEE Trans. Automat. Control, 53 (2008), pp. 1643-1660.
28. Z. Ma, D. S. Callaway, and I. A. Hiskens, Decentralized charging control of large populations of plug-in electric vehicles, IEEE Trans. Control Syst. Technol., 21 (2013), pp. 67-78.
29. A. Neyman, Values of games with infinitely many players, Handbook Game Theory Economic Appl., 3 (2002), pp. 2121-2167.
30. S. L. Nguyen and M. Huang, Linear-quadratic-Gaussian mixed games with continuumparametrized minor players, SIAM J. Control Optim., 50 (2012), pp. 2907-293.
31. S. L. Nguyen and M. Huang, Mean field LQG games with mass behavior responsive to a major player, in Proceedings of the 51st IEEE CDC, Maui, HI, 2012, pp. 5972-5797.
32. M. Nourian, P. E. Caines, R. P. Malhamé, and M. Huang, Mean field LQG control in leader-follower stochastic multi-agent systems: Likelihood ratio based adaptation, IEEE Trans. Automat. Control, 57 (2012), pp. 2801-2816.
33. M. Nourian, P. E. Caines, R. P. Malhamé, and M. Huang, Nash, social and centralized solutions to consensus problems via mean field control theory, IEEE Trans. Automat. Control, 58 (2013), pp. 639-653.
34. M. Nourian, P. E. Caines, and R. P. Malhamé, Mean field analysis of controlled Cucker- Smale type flocking: Linear analysis and perturbation equations, in Proceedings of the 18th IFAC WC, Milan, Italy, 2011, pp. 4471-4476.
35. M. Nourian and P. E. Caines, -Nash mean field game theory for nonlinear stochastic dynamical systems with mixed agents, in Proceedings of the 51st IEEE CDC, Maui, HI, 2012, pp. 2090-2095.
36. M. Nourian and P. E. Caines, Appendices: -Nash Mean Field Game Theory for Nonlinear Stochastic Dynamical Systems with Major and Minor Agents. McGill University Tech. report, arXiv:1209.5684, 2013.
37. M. Nourian and P. E. Caines, -Nash mean field theory for nonlinear stochastic dynamic games with major-minor agents, in Proceedings of the 20th MTNS, Melbourne, Australia, 2012.
38. E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett., 14 (1990), pp. 55-61. (Pubitemid 20649275)
39. S. Peng, Stochastic Hamilton-Jacobi-Bellman equations, SIAM J. Control Optim., 30 (1992), pp. 284-304.
40. S. Peng, Backward stochastic differential equations and applications to optimal control, Appl. Math. Optim., 27 (1993), pp. 125-144.
41. A. S. Sznitman, Topics in propagation of chaos, Ecole d'Eté de Probabilités de Saint-Flour 1989, Lecture Notes in Math. 1464, Springer, New York, 1991, pp. 164-251.
42. B. C. Wang and J.-F. Zhang, Distributed control of multi-agent systems with random parameters and a major agent, Automatica, 48 (2012), pp. 2093-2106.
43. G. Y. Weintraub, C. L. Benkard, and B. Van Roy, Oblivious equilibrium: A mean field approximation for large-scale dynamic games, in Adv. Neural Inf. Process. Syst., MIT Press, 2006.
44. G. Y. Weintraub, C. L. Benkard, and B. Van Roy, Markov perfect industry dynamics with many firms, Econometrica, 76 (2008), pp. 1375-1411.
45. H. Yin, P. G. Mehta, S. P. Meyn, and U. V. Shanbhag, Synchronization of coupled oscillators is a game, IEEE Trans. Automat. Control, 57 (2012), pp. 920-935.
46. X. Yin, Z. Ma, and L. Dong, Decentralized loading coordinations for large-population plug-in electric vehicles and a few controllable bulk loads, in Proceedings of the 50th IEEE CDC, Orlando, FL, 2011, pp. 3092-3097.
47. J. Yong and X. Y. Zhou, Stochastic Control: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999.
48. J. Yong, A leader-follower stochastic linear quadratic differential game, SIAM J. Control Optim., 41 (2002), pp. 1015-1041.