Large-population LQG games involving a major player: The Nash certainty equivalence principle

SIAM Journal on Control and Optimization

Volumn 48, Issue 5, 2009, Pages 3318-3353

  Huang, Minyi ^a  

^a CARLETON UNIVERSITY   (Canada)

Author keywords

Decentralized control; LQG control; Mean field models; Nash equilibria; Stochastic differential games

Indexed keywords

AGGREGATES; DECENTRALIZED CONTROL; MEAN FIELD THEORY; STOCHASTIC MODELS; STOCHASTIC SYSTEMS;

CERTAINTY EQUIVALENCE; CONSISTENCY CONDITIONS; LINEAR QUADRATIC GAUSSIAN; LQG CONTROL; MEAN FIELD APPROXIMATION; MEAN FIELD MODELS; NASH EQUILIBRIA; STOCHASTIC DIFFERENTIAL GAME;

GAME THEORY;

EID: 79952140932     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/080735370     Document Type: Article

References (38)

1. R. J. AUMANN, Game theory, in The New Palgrave: A Dictionary of Economics, J. Eatwell, M. Milgate, and P. Newman, eds., Vol. 2, Macmillan, London, 1987, pp. 460-482.
2. H. ABOU-KANDIL and P. BERTRAND, Analytic solution for a class of linear quadratic open-loop Nash games, Int. J. Control, 43(1986), pp. 997-1002.
3. T. BASAR and G. J. OLSDER, Dynamic Noncooperative Game Theory, 2nd ed., Academic Press, London, UK, 1995.
4. T. BASAR and S. LI, Distributed computation of Nash equilibria in linear-quadratic stochastic differential games, SIAM J. Control Optim., 27(1989), pp. 563-578.
5. C. T. BAUCH and D. J. D. EARN, Vaccination and the theory of games, Proc. Natl. Acad. Sci. U. S. A., 101(2004), pp. 13391-13394.
6. R. BREBAN, R. VARDAVAS, and S. BLOWER, Mean-field analysis of an inductive reasoning game: Application to influenza vaccination, Phys. Rev. E, 76(2007), 031127.
7. A. BENSOUSSAN, Points de Nash dans le cas de fonctionelles quadratiques et jeux differentials lineaires a N personnes, SIAM J. Control, 12(1974), pp. 460-499.
8. A. BENSOUSSAN, Stochastic Control of Partially Observable Systems, Cambridge University Press, Cambridge, UK, 1992.
9. M. C. DELFOUR, Linear quadratic differential games: Saddle point and Riccati differential equation, SIAM J. Control Optim., 46(2007), pp. 750-774.
10. G. M. ERICKSON, Differential game models of advertising competition, European J. Oper. Res., 83(1995), pp. 431-438.
11. F. FOGELMAN and M. QUINZII, Asymptotic values of mixed games, Math. Oper. Res., 5(1980), pp. 86-93.
12. G. FREILING, G. JANK, and S. R. LEE, Existence and uniqueness of open-loop Stackelberg equilibria in linear-quadratic differential games, J. Optim. Theory Appl., 110(2001), pp. 515-544.
13. D. FUDENBERG and D. K. LEVINE, The Theory of Learning in Games, MIT Press, Cambridge, MA, 1998.
14. D. FUDENBERG, D. LEVINE, and W. PESENDORFER, When are nonanonymous players negligible? J. Econ. Theory, 79(1998), pp. 46-71.
15. Z. GALIL, The nucleolus in games with major and minor players, Internat. J. Game Theory, 3(1974), pp. 129-140.
16. A. GRAHAM, Kronecker Products and Matrix Calculus: with Applications, Ellis Horwood, Chichester, UK, 1981.
17. O. HAIMANKO, Nonsymmetric values of nonatomic and mixed games, Math. Oper. Res., 25(2000), pp. 591-605.
18. S. HART, Values of mixed games, Internat. J. Game Theory, 2(1973), pp. 69-86.
19. 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 Conference on Decision and Control, Maui, HI, 2003, pp. 98-103
20. M. HUANG, P. E. CAINES, and R. P. MALHAME, Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized e-Nash equilibria, IEEE Trans. Automat. Control, 52(2007), pp. 1560-1571.
21. M. HUANG, R. P. MALHAME, and P. E. CAINES, Nash equilibria for large-population linear stochastic systems of weakly coupled agents, in Ana lysis, Control and Optimization of Complex Dynamic Systems, E. K. Boukas and R. P. Malhame, eds., Springer, New York, 2005, pp. 215-252.
22. M. HUANG, R. P. MALHAME, and P. E. CAINES, Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Comm. Inform. Systems, 6(2006), pp. 221-252.
23. T. KAILATH, Linear Systems, Prentice-Hall, Englewood Cliffs, NJ, 1980.
24. M. A. KHAN and Y. SUN, Non-cooperative games with many players, in Handbook of Game Theory with Economic Applications, Vol. 3, R. J. Aumann and S. Hart, eds., Elsevier Science, North-Holland, Amsterdam, 2002, pp. 1761-1808.
25. V. E. LAMBSON, Self-enforcing collusion in large dynamic markets, J. Econ. Theory, 34(1984), pp. 282-291.
26. J.-M. LASRY and P.-L. LIONS, Jeux à champ moyen. I. Le cas stationnaire, C. R. Math. Acad. Sci. Paris, 343(2006), pp. 619-625.
27. J.-M. LASRY and P.-L. LIONS, Jeux à champ moyen. II. Horizon fini et contrôle optimal, C. R. Math. Acad. Sci. Paris, 343(2006), pp. 679-684.
28. J.-M. LASRY and P.-L. LIONS, Mean field games, Japan. J. Math., 2(2007), pp. 229-260.
29. D. L. LUKES, Equilibrium feedback control in linear games with quadratic costs, SIAM J. Control, 9(1971), pp. 234-252.
30. J. MAYNARD SMITH, Evolution and the Theory of Games, Cambridge University Press, Cambridge, UK, 1982.
31. J. W. MILNOR and L. S. SHAPLEY, Values oflarge games I I: Oceanic games, Math. Oper. Res., 3(1978), pp. 290-307.
32. A. NEYMAN, Weighted majority games have an asymptotic value, Math. Oper. Res., 13(1988), pp. 556-580.
33. G. P. PAPAVASSILOPOULOS, J. V. MEDANIC, and J. B. CRUZ, JR., On the existence of Nash strategies and solutions to coupled Riccati equations in linear-quadratic Nash games, J. Optim. Theory Appl., 28(1979), pp. 49-76.
34. J. VON NEUMANN and O. MORGENSTERN, Theory of Games and Economic Behavior, 3rd ed., Princeton University Press, Princeton, NJ, 1953.
35. G. Y. WEINTRAUB, C. L. BENKARD, and B. VAN ROY, Oblivious equilibrium: A mean field approximation for large-scale dynamic games, in Advances in Neural Information Processing Systems, MIT Press, Cambridge, MA, 2005. Available online at http://books.nips.cc.
36. G. Y. WEINTRAUB, C. L. BENKARD, and B. VAN ROY, Markov perfect industry dynamics with many firms, Econometrica, 76(2008), pp. 1375-1411.
37. W. M. WONHAM, Linear Multivariable Control: A Geometric Approach, 2nd ed., Springer-Verlag, New York, 1979.
38. J. YONG, A leader-follower stochastic linear quadratic differential game, SIAM J. Control Optim., 41(2002), pp. 1015-1041.