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Published in:

13-12-2018

Risk-Sensitive Mean Field Games via the Stochastic Maximum Principle

Authors: Jun Moon, Tamer Başar

Published in: Dynamic Games and Applications | Issue 4/2019

Abstract

In this paper, we consider risk-sensitive mean field games via the risk-sensitive maximum principle. The problem is analyzed through two sequential steps: (i) risk-sensitive optimal control for a fixed probability measure, and (ii) the associated fixed-point problem. For step (i), we use the risk-sensitive maximum principle to obtain the optimal solution, which is characterized in terms of the associated forward–backward stochastic differential equation (FBSDE). In step (ii), we solve for the probability law induced by the state process with the optimal control in step (i). In particular, we show the existence of the fixed point of the probability law of the state process determined by step (i) via Schauder’s fixed-point theorem. After analyzing steps (i) and (ii), we prove that the set of N optimal distributed controls obtained from steps (i) and (ii) constitutes an approximate Nash equilibrium or $$\epsilon$$-Nash equilibrium for the N player risk-sensitive game, where $$\epsilon \rightarrow 0$$ as $$N \rightarrow \infty$$ at the rate of $$O(\frac{1}{N^{1/(n+4)}})$$. Finally, we discuss extensions to heterogeneous (non-symmetric) risk-sensitive mean field games.

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Appendix
Available only for authorised users
Footnotes
1
In fact, for step (i), we provide the risk-sensitive maximum principle for the general r-dimensional Brownian motion in “Appendix A” section, which generalizes the result of the one-dimensional Brownian motion in [42] when the diffusion coefficient is independent of control.

2
This assumption can be relaxed to a complete separable metric space [65].

3
A related discussion on this issue is provided in [6, Chapter 6].

4
This existence is dependent on the value of $$\gamma$$, and when $$\gamma$$ is large, the corresponding Riccati equation always admits a unique solution [7, 48].

5
A discussion on Lipschitz continuity of the optimal control in stochastic optimal control theory can be found in [28].

6
In this Appendix, we do not state specific regularity conditions for the corresponding stochastic optimal control problem, since they are quite similar to the assumptions already made in the paper. See [29, 42, 60, 65] for the regularity conditions for the stochastic optimal control problem.

Literature
1.
Achdou Y, Capuzzo-Dolcetta I (2010) Mean field games: numerical methods. SIAM J Numer Anal 48(3):1136–1162
2.
Achdou Y, Camilli F, Capuzzo-Dolcetta I (2012) Mean field games: numerical methods for the planning problem. SIAM J Control Opt 50(1):77–109
3.
Ahuja S (2016) Wellposedness of mean field games with common noise under a weak monotonicity condition. SIAM J Control Opt 54(1):30–48
4.
Andersson D, Djehiche B (2010) A maximum principle for SDEs of mean-field type. Appl Math Opt 63(3):341–356
5.
Başar T (1999) Nash equilibria of risk-sensitive nonlinear stochastic differential games. J Opt Theory Appl 100(3):479–498
6.
Başar T, Olsder GJ (1999) Dynamic noncooperative game theory, 2nd edn. SIAM, Philadelphia MATH
7.
Başar T, Bernhard P (1995) $$\text{ H }^\infty$$ optimal control and related minimax design problems, 2nd edn. Birkhäuser, Boston MATH
8.
Bardi M, Priuli FS (2014) Linear-quadratic N-person and mean-field games with ergodic cost. SIAM J Control Opt 52(5):3022–3052
9.
Bauso D, Tembine H, Başar T (2016) Robust mean field games. Dyn Games Appl 6(3):277–303
10.
Bauso D, Tembine H, Başar T (2016) Opinion dynamics in social networks through mean-field games. SIAM J Control Opt 54(6):3225–3257
11.
Bensoussan A, Frehse J, Yam P (2013) Mean field games and mean field type control theory. Springer, New York MATH
12.
Bensoussan A, Sung KCJ, Yam SCP, Yung SP (2014) Linear–quadratic mean field games. arXiv:​1404.​5741
13.
Bensoussan A, Chau MHM, Yam SCP (2015) Mean field Stackelberg games: aggregation of delayed instructions. SIAM J Control Opt 53(4):2237–2266
14.
Billingsley P (1999) Convergence of probability measures, 2nd edn. Wiley, New York MATH
15.
Bolley F (2008) Separability and completeness for the Wasserstein distance. Seminarire de probabilities XLI, Lecture Notes Math, pp 371–377
16.
Cardaliaguet P (2012) Notes on mean field games. Technical report
17.
Cardaliaguet P, Lehalle CA (2018) Mean field game of controls and an application to trade crowding. Math Financ Econ 12(3):335–363
18.
Carmona R, Delarue F (2013) Mean field forward–backward stochastic differential equations. Electron Commun Probab 18(68):1–15
19.
Carmona R, Delarue F (2013) Probabilistic analysis of mean-field games. SIAM J Control Opt 51(4):2705–2734
20.
Carmona R, Delarue F (2015) Forward–backward stochastic differential equations and controlled McKean–Vlasov dynamics. Ann Probab 43(5):2647–2700
21.
Carmona R, Delarue F, Lachapelle A (2012) Control of McKean–Vlasov dynamics versus mean field games. Math Financ Econ 7(2):131
22.
Conway JB (2000) A course in functional analysis. Springer, Berlin
23.
Cormen TH, Leiserson CE, Rivest RL, Stein C (2009) Introduction to algorithms, 3rd edn. The MIT Press, London MATH
24.
Couillet R, Perlaza SM, Tembine H, Debbah M (2012) Electrical vehicles in the smart grid: a mean field game analysis. IEEE J Sel Areas Commun 30(6):1086–1096
25.
Delarue F (2002) On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case. Stoch Process Appl 99:209–286
26.
Djehiche B, Hamadene S (2016) Optimal control and zero-sum stochastic differential game problems of mean-field type. arXiv:​1603.​06071v3
27.
Duncan TE (2013) Linear–exponential–quadratic Gaussian control. IEEE Trans Autom Control 58(11):2910–2911
28.
Fleming W, Rishel R (1975) Deterministic and stochastic optimal control. Springer, Berlin MATH
29.
Fleming W, Soner HM (2006) Controlled Markov processes and viscosity solutions, 2nd edn. Springer, Berlin MATH
30.
Fleming WH, James MR (1995) The risk-sensitive index and the $$\text{ H }_2$$ and $$\text{ H }_\infty$$ norms for nonlinear systems. Math Control Signals Syst 8:199–221
31.
Horowitz J, Karandikar R (1994) Mean rate of convergence of empirical measures in the Wasserstein metric. J Comput Appl Math 55:261–273
32.
Huang J, Li N (2018) Linear–quadratic mean-field game for stochastic delayed systems. IEEE Trans Autom Control 63(8):2722–2729
33.
Huang M (2010) Large-population LQG games involving a major player: the Nash certainty equivalence principle. SIAM J Control Opt 48(5):3318–3353
34.
Huang M, Caines PE, Malhamé RP (2003) 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, pp 98–103
35.
Huang M, Malhamé Roland P, Caines Peter E (2006) Large population stochastic dynamic games: closed-loop McKean–Vlasov systems and the Nash certainty equivalence principle. Commun Inf Syst 6(3):221–252
36.
Huang M, Caines PE, Malhamé RP (2007) Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized $$\epsilon$$-Nash equilibria. IEEE Trans Autom Control 52(9):1560–1571
37.
Jacobson D (1973) Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games. IEEE Trans Autom Control 18(2):124–131
38.
Jordain B, Meleard S, Woyczynski W (2008) Nonlinear SDEs driven by Levy processes and related PDEs. Latin Am J Probab 4:1–29 MATH
39.
Karatzas I, Shreve SE (2000) Brownian motion and stochastic calculus. Springer, Berlin MATH
40.
Lasry JM, Lions PL (2007) Mean field games. Jpn J Math 2(1):229–260
41.
Li T, Zhang JF (2008) Asymptotically optimal decentralized control for large population stochastic multiagent systems. IEEE Trans Autom Control 53(7):1643–1660
42.
Lim EBA, Zhou XY (2005) A new risk-sensitive maximum principle. IEEE Trans Autom Control 50(7):958–966
43.
Luenberger DG (1969) Optimization by vector space methods. Wiley, New York MATH
44.
Ma J, Yong J (1999) Forward–backward stochastic differential equations and their applications. Springer, Berlin MATH
45.
Ma J, Callaway DS, Hiskens IA (2013) Decentralized charging control of large populations of plug-in electric vehicles. IEEE Trans Control Syst Technol 21(1):67–78
46.
Moon J, Başar T (2015) Linear-quadratic stochastic differential Stackelberg games with a high population of followers. In: Proceedings of the 54th IEEE conference on decision and control, pp 2270–2275
47.
Moon J, Başar T (2016) Robust mean field games for coupled Markov jump linear systems. Int J Control 89(7):1367–1381
48.
Moon J, Başar T (2017) Linear quadratic risk-sensitive and robust mean field games. IEEE Trans Autom Control 62(3):1062–1077
49.
Moon J, Başar T (2018) Linear quadratic mean field Stackelberg differential games. Automatica 97:200–213
50.
Nourian M, Caines P (2013) $$\epsilon$$-Nash mean field game theory for nonlinear stochastic dynamical systems with major and minor agents. SIAM J Control Opt 51(4):3302–3331
51.
Nourian M, Caines PE, Malhamé RP, Huang Minyi (2013) Nash, social and centralized solutions to consensus problems via mean field control theory. IEEE Trans Autom Control 58(3):639–653
52.
Nourian M, Caines PE, Malhamé RP (2014) A mean field game synthesis of initial mean consensus problems: a continuum approach for non-Gaussian behavior. IEEE Trans Autom Control 59(2):449–455
53.
Pardoux E, Tang S (1999) Forward–backward stochastic differential equations and quasilinear parabolic PDEs. Probab Theory Relat Fields 114:123–150
54.
Parise F, Grammatico S, Colombino M, Lygeros J (2014) Mean field constrained charging policy for large populations of plug-in electric vehicles. In: Proceedings of 53rd IEEE conference on decision and control, pp 5101–5106
55.
Peng S, Wu Z (1999) Fully coupled forward–backward stochastic differential equations and applications to optimal control. SIAM J Control Opt 37(3):825–843
56.
Pham H (2009) Continuous-time stochastic control and optimization with financial applications. Springer, Berlin MATH
57.
Rachev ST, Ruschendorf L (1998) Mass transportation theory: volume I: theory. Springer, Berlin MATH
58.
Smart DR (1974) Fixed point theorems. Cambridge University Press, Cambridge MATH
59.
Tembine H, Zhu Q, Başar T (2014) Risk-sensitive mean field games. IEEE Trans Autom Control 59(4):835–850
60.
Touzi N (2013) Optimal stochastic control, stochastic target problems, and backward SDE. Springer, Berlin MATH
61.
Wang B, Zhang J (2012) Mean field games for large-population multiagent systems with Markov jump parameters. SIAM J Control Opt 50(4):2308–2334
62.
Weintraub GY, Benkard CL, Van Roy B (2008) Markov perfect industry dynamics with many firms. Econometrica 76(6):1375–1411
63.
Whittle P (1990) Risk-sensitive optimal control. Wiley, New York MATH
64.
Yin H, Mehta PG, Meyn SP, Shanbhag UV (2012) Synchronization of coupled oscillators is a game. IEEE Trans Autom Control 57(4):920–935
65.
Yong J, Zhou XY (1999) Stochastic controls: Hamiltonian systems and HJB equations. Springer, Berlin MATH
66.
Zhou XY (1996) Sufficient conditions of optimality for stochastic systems with controllable diffusions. IEEE Trans Autom Control 41(8):1176–1179
67.
Zhu Q, Başar T (2013) Multi-resolution large population stochastic differential games and their application to demand response management in the smart grid. Dyn Games Appl 3(1):66–88
68.
Zhu Q, Tembine H, Başar T (2011) Hybrid risk-sensitive mean-field stochastic differential games with application to molecular biology. In: Proceedings of the 50th IEEE CDC and ECC, pp 4491–4497
Title
Risk-Sensitive Mean Field Games via the Stochastic Maximum Principle
Authors
Jun Moon
Tamer Başar
Publication date
13-12-2018
Publisher
Springer US
Published in
Dynamic Games and Applications / Issue 4/2019
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-018-00290-z

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