Convergence, fluctuations and large deviations for finite state mean field games via the Master Equation

Stochastic Processes and their Applications

Volumn 129, Issue 11, 2019, Pages 4510-4555

  Cecchin, Alekos ^a   Pelino, Guglielmo ^a  

^a UNIVERSITY OF PADOVA   (Italy)

Author keywords

Central Limit Theorem; Finite state space; Jump Markov processes; Large Deviation Principle; Master Equation; Mean field games; N person games; Nash equilibrium; Propagation of chaos

Indexed keywords

MARKOV PROCESSES;

CENTRAL LIMIT THEOREM; FINITE STATE SPACES; JUMP MARKOV PROCESS; LARGE DEVIATION PRINCIPLE; MASTER EQUATIONS; MEAN FIELD GAMES; N-PERSON GAMES; NASH EQUILIBRIA;

GAME THEORY;

EID: 85058804507     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.spa.2018.12.002     Document Type: Article

References (36)

1. Basna, R., Hilbert, A., Kolokoltsov, V.N., An epsilon-nash equilibrium for nonlinear markov games of mean-feld-type on finite spaces. Commun. Stoch. Anal. 8:4 (2014), 449–468.
2. E. Bayraktar, A. Cohen, Analysis of a Finite State Many Player Game Using its Master Equation, preprint 2017, arXiv:1707.02648.
3. Bensoussan, A., Frehse, J., Yam, P., Mean Field Games and Mean Field Type Control Theory Springer Briefs in Mathematics, 2013, Springer, New York.
4. Bensoussan, A., Sung, K., Yam, S., Yung, S., Linear-quadratic mean field games. J. Optim. Theory Appl. 169:2 (2016), 496–529.
5. P. Cardaliaguet, Notes on mean field games, Technical report, Université de Paris Dauphine, 2013.
6. P. Cardaliaguet, F. Delarue, J.M. Lasry, P.L. Lions, The master equation and the convergence problem in mean field games, preprint 2015, arXiv:1509.02505.
7. Carmona, R., Delarue, F., Probabilistic analysis of mean-field games. SIAM J. Control Optim. 51:4 (2013), 2705–2734.
8. Carmona, R., Lacker, D., A probabilistic weak formulation of mean field games and applications. Ann. Appl. Probab. 25:3 (2015), 1189–1231.
9. R. Carmona, P. Wang, Finite State Mean Field Games with Major and Minor Players, preprint 2016, arXiv:1610.05408.
10. Cecchin, A., Fischer, M., Probabilistic approach to finite state mean field games. Appl. Math. Optim., 2018 pre-published on-line, http://dx.doi.org/10.1007/s00245-018-9488-7.
11. Comets, F., Eisele, Th., Asymptotic dynamics, non-critical and critical fluctuations for a geometric long-range interacting model. Comm. Math. Phys. 118 (1988), 531–567.
12. Dai Pra, P., Runggaldier, W.J., Sartori, E., Tolotti, M., Large portfolio losses: a dynamic contagion model. Ann. Appl. Probab. 19:1 (2009), 347–394.
13. F. Delarue, D. Lacker, K. Ramanan, From the master equation to mean field game limit theory: a central limit theorem, preprint 2018, arXiv:1804.08542.
14. F. Delarue, D. Lacker, K. Ramanan, From the master equation to mean field game limit theory: large deviations and concentration of measure, preprint 2018, arXiv:1804.08550.
15. Dupuis, P., Ellis, R., Weiss, A., Large deviations for Markov processes with discontinuous statistics, I: General upper bounds. Ann. Probab. 19:3 (1991), 1280–1297.
16. P. Dupuis, K. Ramanan, W. Wu, Large deviation principle for finite-state mean field interacting particle systems, preprint 2016, arXiv:1601.06219v1.
17. Ferreira, R., Gomes, D., On the convergence of finite state mean-field games through the Γ-convergence. J. Math. Anal. Appl. 418:1 (2014), 211–230.
18. Fischer, M., On the connection between symmetric N-player games and mean field games. Ann. Appl. Probab. 27:2 (2017), 757–810.
19. Fournier, N., Guillin, A., On the rate of convergence in Wasserstein distance of the empirical measure. Probab. Theory Related Fields 162 (2015), 707–738.
20. Gomes, D., Mohr, J., Souza, R.R., Discrete time, finite state space mean field games. J. Math. Pures Appl. (9) 93:2 (2010), 308–328.
21. Gomes, D., Mohr, J., Souza, R.R., Continuous time finite state mean field games. Appl. Math. Optim. 68:1 (2013), 99–143.
22. Gomes, D., Velho, R.M., Wolfram, M.-T., Dual two-state mean-field games. Proceedings CDC, 2014.
23. Gomes, D., Velho, R.M., Wolfram, M.-T., Socio-economic applications of finite state mean field games. Proc. Roy. Soc. A, 2014 Bd. 372-2028(S.).
24. Graham, C., McKean-Vlasov Itô-Skorokhod equations, and nonlinear diffusions with discrete jump sets. Stochastic Process. Appl. 40:1 (1992), 69–82.
25. Guéant, O., Existence and uniqueness result for mean field games with congestion effect on graphs. Appl. Math. Optim. 72:2 (2015), 291–303.
26. Huang, M., Mallamé, R.P., Caines, P.E., Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6:3 (2006), 221–252.
27. Ikeda, N., Watanabe, S., Stochastic Differential Equations and Diffusion Processes, second ed. North-Holland Mathematical Library, vol. 24, 1989, North Holland/ Kodansha, Amsterdam.
28. Lacker, D., A general characterization of the mean field limit for stochastic differential games. Probab. Theory Related Fields 165:3 (2016), 581–648.
29. Lasry, J.-M., Lions, P.-L., Mean field games. Jpn. J. Math. 2:1 (2007), 229–260.
30. Léonard, C., Large deviations for long range interacting particle systems with jumps. Ann. Inst. Henri Poincaré Probab. Stat. 31:2 (1995), 289–323.
31. P.L. Lions, Lectures at Collège de France, www.college-de-france.fr.
32. Rachev, S.T., Rüschendorf, L., Mass Transportation Problems. 1998, Springer-Verlag.
33. Shwartz, A., Weiss, A., Large Deviations for Performance Analysis. 1995, Chapman and Hall.
34. Shwartz, A., Weiss, A., Large deviations with diminishing rates. Math. Oper. Res. 30:2 (2005), 281–310.
35. Sznitman, A.-S., Topics in propagation of chaos. Hennequin, P.-L., (eds.) Ecole d'Eté de Probabilités de Saint-Flour XIX - 1989 Lecture Notes in Math., vol. 1464, 1991, Springer-Verlag, Berlin, 165–251.
36. Wu, W., Large Deviations and Quasipotential for Finite-State Mean Field Interacting Particle Systems. (Ph.D. thesis), 2014, Brown University, Providence, RI.