On the Graphon Mean Field Game equations: Individual agent affine dynamics and mean field dependent performance functions
ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 24

This paper establishes unique solvability of a class of Graphon Mean Field Game equations. The special case of a constant graphon yields the result for the Mean Field Game equations.

DOI : 10.1051/cocv/2022020
Classification : 35M30, 60J60
Keywords: Mean Field Games, Graphon, Hamilton-Jacobi-Bellman equation 
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     author = {Caines, Peter E. and Ho, Daniel and Huang, Minyi and Jian, Jiamin and Song, Qingshuo},
     title = {On the {Graphon} {Mean} {Field} {Game} equations: {Individual} agent affine dynamics and mean field dependent performance functions},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     year = {2022},
     publisher = {EDP-Sciences},
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Caines, Peter E.; Ho, Daniel; Huang, Minyi; Jian, Jiamin; Song, Qingshuo. On the Graphon Mean Field Game equations: Individual agent affine dynamics and mean field dependent performance functions. ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 24. doi: 10.1051/cocv/2022020

[1] P. E. Caines and M. Huang, Graphon mean field games and the gmfg equations. In 2018 IEEE Conference on Decision and Control (CDC). IEEE (2018) 4129–4134. | DOI

[2] P. E. Caines and M. Huang, Graphon mean field games and the gmfg equations: ε-nash equilibria. In 2019 IEEE 58th Conference on Decision and Control (CDC). IEEE (2019) 286–292. | DOI

[3] P. E. Caines and M. Huang, Graphon mean field games and their equations. SIAM J. Control Optim. 59 (2021) 4373–4399. | MR | Zbl | DOI

[4] P. Cardaliaguet, Notes on mean field games. https://www.ceremade.dauphine.fr/~cardaliaguet/MFG20130420.pdf (2013).

[5] L. C. Evans, Partial differential equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (1998). | MR | Zbl

[6] W. H. Fleming and H. M. Soner, Controlled Markov processes and viscosity solutions, vol. 25 of Stochastic Modelling and Applied Probability. second edition, Springer, New York (2006). | MR | Zbl

[7] S. Gao and P. E. Caines, Grapbon linear quadratic regulation of large-scale networks of linear systems. In 2018 IEEE Conference on Decision and Control (CDC). IEEE (2018) 5892–5897. | DOI

[8] D. A. Gomes, E. A. Pimentel and V. Voskanyan, Regularity Theory for Mean-Field Game Systems. SpringerBriefs in Mathematics. Springer International Publishing (2016). | MR | Zbl | DOI

[9] M. Huang, P. E. Caines and R. P. Malhamé, Individual and mass behavior in large population stochastic wireless power control problems: centralized and nash equilibrium solutions. In: Proceedings of the 42nd IEEE CDC (2003) 98–103.

[10] 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) 221–251. | MR | Zbl | DOI

[11] M. Huang, P. E. Caines and R. P. Malhamé, Large-population cost-coupled LQG problems with non-uniform agents: individual-mass behavior and decentralized epsilon-nash equilibria. IEEE Trans. Automat. Control 52 (2007) 1560–1571. | MR | Zbl | DOI

[12] J. L. Kelley, General topology. Courier Dover Publications (2017). | Zbl

[13] N. V. Krylov, Lectures on elliptic and parabolic equations in Hölder spaces, vol. 12 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (1996). | MR | Zbl

[14] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'Ceva, Linear and quasilinear equations of parabolic type. Trans. Math.Monogr., vol. 23. American Mathematical Society, Providence, R.I. (1967). | MR | Zbl

[15] J.-M. Lasry and P.-L. Lions, Mean field games. Jpn. J. Math. 2 (2007) 229–260. | MR | Zbl | DOI

[16] L. Lovász, Large networks and graph limits, Vol. 60. American Mathematical Soc. (2012). | MR | Zbl

[17] M. Nourian and P. E. Caines, ϵ -Nash mean field game theory for nonlinear stochastic dynamical systems with major and minor agents. SIAM J. Control Optim. 51 (2013) 3302–3331. | MR | Zbl | DOI

[18] L. Ryzhik, Notes on mean field games (2018). https://math.stanford.edu/~ryzhik/STANFORD/MEAN-FIELD-GAMES/notes-mean-field.pdf.

Cité par Sources :

The research of P.E. Caines, D. Ho, and Q. Song were supported in part by the RGC of Hong Kong CityU (11201518). The work of P. E. Caines was partially supported by AFOSR grant FA9550-19-1-0138. The research work of M. Huang was supported by NSERC. We acknowledge the valuable comments from anonymous reviewers.