Linear-quadratic N-person and mean-field games with ergodic cost

SIAM Journal on Control and Optimization

Volumn 52, Issue 5, 2014, Pages 3022-3052

  Bardi, Martino ^a   Priuli, Fabio S ^b  

^a UNIVERSITY OF PADOVA   (Italy)

^b CNR   (Italy)

Author keywords

Consensus problems; Feedback nash equilibria; Large population limit; Linear quadratic problems; Mean field games; Multiagent control; N person differential games; Stochastic control

Indexed keywords

DIFFERENTIAL EQUATIONS; FEEDBACK; GAME THEORY; GAUSSIAN DISTRIBUTION; MULTI AGENT SYSTEMS; RICCATI EQUATIONS; STOCHASTIC SYSTEMS;

CONSENSUS PROBLEMS; DIFFERENTIAL GAMES; LARGE POPULATION; MEAN-FIELD GAMES; MULTIAGENT CONTROL; NASH EQUILIBRIA; QUADRATIC PROBLEM; STOCHASTIC CONTROL;

COSTS;

EID: 84911092308     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/140951795     Document Type: Article

References (39)

1. Y. ACHDOU AND I. CAPUZZO-DOLCETTA, Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), pp. 1136-1162.
2. M. BARDI, Explicit solutions of some linear-quadratic mean field games, Netw. Heterog. Media, 7 (2012), pp. 243-261.
3. M. BARDI, A. CESARONI, AND L. MANCA, Convergence by viscosity methods in multiscale financial models with stochastic volatility, SIAM J. Financial Math., 1 (2010), pp. 230-265.
4. M. BARDI AND E. FELEQI, Nonlinear Elliptic Systems and Mean Field Games, preprint.
5. M. BARDI AND F. S. PRIULI, LQG mean-field games with ergodic cost, in Procedings of the 52nd IEEE-CDC, Florence, 2013, pp. 2493-2498.
6. T. BASAR AND G. J. OLSDER, Dynamic Noncooperative Game Theory, 2nd ed., Academic Press, London, 1995.
7. D. BAUSO, H. TEMBINE, AND T. BASAR, Robust Mean Field Games with Application to Production of Exhaustible Resource, in Proceedings of the 7th IFAC Symposium on Robust Control Design, Aalborg, Denmark, 2012.
8. A. BENSOUSSAN AND J. FREHSE, Ergodic Bellman systems for stochastic games in arbitrary dimension, Proc. R. Soc. Lond. Ser. A, 449 (1995), pp. 65-77.
9. A. BENSOUSSAN AND J. FREHSE, Regularity results for nonlinear elliptic systems and applications, Appl. Math. Sci. 151, Springer, Berlin, 2002.
10. A. BENSOUSSAN, J. FREHSE, AND S. C. P. YAM, Mean Field Games and Mean Field Type Control Theory, Springer Briefs Math., Springer, Berlin, 2013.
11. A. BENSOUSSAN, K. C. J. SUNG, S. C. P. YAM, AND S. P. YUNG, Linear-Quadratic Mean Field Games, preprint.
12. A. BHASKAR, Taussky's theorem, symmetrizability and modal analysis revisited, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), pp. 2455-2480.
13. D. A. BINI, B. IANNAZZO, AND B. MEINI, Numerical Solution of Algebraic Riccati Equations, Fundamentals of Algorithms 9, SIAM, Philadelphia, 2012.
14. A. BRESSAN AND B. PICCOLI, Introduction to the Mathematical Theory of Control, AIMS Ser. Appl. Math., American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2007.
15. P. CARDALIAGUET, Notes on Mean Field Games, (from P.-L. Lions' lectures at Collége de France), 2013.
16. P. CARDALIAGUET, J.-M. LASRY, P.-L. LIONS, AND A. PORRETTA, Long time average of mean field games, Netw. Heterog. Media, 7 (2012), pp. 279-301.
17. M. CIRANT, Nonlinear PDEs in Ergodic Control, Mean Field Games and Prescribed Curvature Problems, Ph.D. thesis, Università di Padova, Padova, Italy, 2013.
18. G. DA PRATO, An Introduction to Infinite-Dimensional Analysis, Springer, Berlin, 2006.
19. J. C. ENGWERDA, LQ Dynamic Optimization and Differential Games, John Wiley & Sons, Hoboken, NJ, 2005.
20. E. FELEQI, The derivation of ergodic mean field game equations for several populations of players, Dyn. Games Appl., 3 (2013), pp. 523-536.
21. D. A. GOMES, J. MOHR, AND R. R. SOUZA, Discrete time, finite state space mean field games, J. Math. Pures Appl. (9), 93 (2010), pp. 308-328.
22. O. GUÉANT, A reference case for mean field games models, J. Math. Pures Appl. (9), 92 (2009), pp. 276-294.
23. O. GUÉANT, J.-M. LASRY, AND P.-L. LIONS, Mean field games and applications, in Paris-Princeton Lectures on Mathematical Finance 2010, Lecture Notes in Math. 2003, R. A. Carmona et al., eds., Springer, Berlin, 2011, pp. 205-266,
24. 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, Hawaii, 2003, pp. 98-103.
25. 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, Commun. Inf. Syst., 6 (2006), pp. 221-251.
26. 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.
27. J. KARATZAS AND S. SHREVE, Brownian Motion and Stochastic Calculus, Grad. Texts in Math. 113, Springer, Berlin, 1988.
28. R. Z. KHASMINSKII, Stochastic Stability of Differential Equations, Sijthoff & Noordhoff, Alphen aan den Rijn-Germantown, 1980.
29. V. N. KOLOKOLTSOV, M. TROEVA, AND W. YANG, On the rate of convergence for the mean-field approximation of controlled diffusions with large number of players, Dyn. Games Appl., 4 (2014), pp. 208-230.
30. J.-M. LASRY AND P.-L. LIONS, Jeuxà champ moyen. I. Le cas stationnaire, C. R. Acad. Sci. Paris, 343 (2006), pp. 619-625.
31. J.-M. LASRY AND P.-L. LIONS, Jeuxà champ moyen. II. Horizon fini et contrôle optimal, C. R. Acad. Sci. Paris, 343 (2006), pp. 679-684.
32. J.-M. LASRY AND P.-L. LIONS, Mean field games, Jpn. J. Math., 2 (2007), pp. 229-260.
33. P. LANCASTER AND L. RODMAN, Algebraic Riccati Equations, Oxford University Press, New York, 1995.
34. 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.
35. M. NOURIAN AND P. E. CAINES, ∈-Nash Mean Field Game Theory for Nonlinear Stochastic Dynamical Systems with Major and Minor Agents, SIAM J. Control Optim., 51 (2013), pp. 3302-3331.
36. 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.
37. F. S. PRIULI, Linear-Quadratic N-person and Mean-Field Games: Infinite Horizon with Discounted Cost and Singular Limits, submitted; also available online from http://arxiv.org/abs/1403.4090.
38. D. SERRE, Matrices: Theory and Applications, Springer, Berlin, 2002.
39. O. TAUSSKY AND H. ZASSENHAUS, On the similarity transformation between a matrix and its transpose, Pacific J. Math., 9 (1959), pp. 893-896.