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Comparing the best-reply strategy and mean-field games: The stationary case

Part of: Miscellaneous topics in calculus of variations and optimal control Game theory Elliptic equations and systems Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  20 November 2020

MATT BARKER ,
PIERRE DEGOND Open the ORCID record for PIERRE DEGOND [Opens in a new window]  and
MARIE-THERESE WOLFRAM
MATT BARKER
Affiliation:
Department of Mathematics Imperial College London, London, SW7 2AZ, UK Grantham Institute, Imperial College London, London, SW7 2AZ, UK emails: m.barker17@imperial.ac.uk; p.degond@imperial.ac.uk
PIERRE DEGOND
Affiliation:
Department of Mathematics Imperial College London, London, SW7 2AZ, UK
MARIE-THERESE WOLFRAM
Affiliation:
University of Warwick, Mathematics Institute, Gibbet Hill Road, CV47AL Coventry, UK Radon Institute for Computational and Applied Mathematics, Altenbergerstr. 69, 4040 Linz, Austria email: m.wolfram@warwick.ac.uk
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Abstract

Mean-field games (MFGs) and the best-reply strategy (BRS) are two methods of describing competitive optimisation of systems of interacting agents. The latter can be interpreted as an approximation of the respective MFG system. In this paper, we present an analysis and comparison of the two approaches in the stationary case. We provide novel existence and uniqueness results for the stationary boundary value problems related to the MFG and BRS formulations, and we present an analytical and numerical comparison of the two paradigms in some specific modelling situations.

Keywords

Fokker–Planck equationsPDEs in connection with game theoryeconomicssocial and behavioural sciencesdifferential games and control

MSC classification

Primary: 35Q84: Fokker-Planck equations 35Q91: PDEs in connection with game theory, economics, social and behavioral sciences 49N70: Differential games
Secondary: 35J15: Second-order elliptic equations 35J57: Boundary value problems for second-order elliptic systems 91A23: Differential games

Information

Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 1 , February 2022 , pp. 79 - 110
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Achdou, Y., Camilli, F. & Capuzzo-Dolcetta, I. (2013) Mean field games: convergence of a finite difference method. SIAM J. Numer. Anal. 51(5), 2585–2612 CrossRefGoogle Scholar
Achdou, Y. & Capuzzo-Dolcetta, I. (2010) Mean field games: numerical methods. SIAM J. Numer. Anal. 48(3), 1136–1162.CrossRefGoogle Scholar
Barker, M. (2019) From mean field games to the best reply strategy in a stochastic framework. J. Dyn. Games 6(4), 291–314.CrossRefGoogle Scholar
Benamou, J. D., Carlier, G. & Santambrogio, F. (2017) Variational mean field games. Model. Simul. Sci. Eng. Technol. 1, 141–171.CrossRefGoogle Scholar
Bertucci, C., Lasry, J.-M. & Lions, P.-L. (2019) Some remarks on mean field games. Commun. Part. Differ. Equ. 44, 205–227.CrossRefGoogle Scholar
Biane, P. & Speicher, R. (2001) Free diffusions, free entropy and free Fisher information. Annales de l’institut Henri Poincare (B) Prob. Stat. 37, 581–606.CrossRefGoogle Scholar
Boccardo, L., Orsina, L. & Porretta, A. (2016) Strongly coupled elliptic equations related to mean-field games systems. J. Differ. Equ. 261, 1796–1834.CrossRefGoogle Scholar
Caines, P. E., Huang, M. & Malhamé, R. P. (2017) Mean field games. In: Basar, T. and Zaccour, G. (editors), Handbook of Dynamic Game Theory, pp. 1–28. Springer, Berlin Heidelberg.Google Scholar
Cardaliaguet, P. & Jameson Graber, P. (2015) Mean field games systems of first order. ESAIM - Cont. Optim. Cal. Variat. 21, 690–722.CrossRefGoogle Scholar
Cardaliaguet, P., Lasry, J.-M., Lions, P.-L. & Porretta, A. (2012) Long time average of mean field games. Netw. Heterogeneous Media 7, 279–301.CrossRefGoogle Scholar
Carmona, R. & Delarue, F. (2018) Probabilistic Theory of Mean Field Games with Applications I, vol. 83. Probability Theory and Stochastic Modelling. Springer International Publishing, Cham.Google Scholar
Carmona, R. & Delarue, F. (2018) Probabilistic Theory of Mean Field Games with Applications II, vol. 84. Probability Theory and Stochastic Modelling. Springer International Publishing, Cham.Google Scholar
Carrillo, J. A., Gvalani, R. S., Pavliotis, G. A. & Schlichting, A.. (2020) Long-time behaviour and phase transitions for the Mckean–Vlasov equation on the torus. Arch. Ration. Mech. Anal. 235, 635–690.CrossRefGoogle Scholar
Carrillo, J. A., McCann, R. J. & Villani, C. (2003) Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev. Mat. Iberoam. 19, 971–1018.CrossRefGoogle Scholar
Carrillo, J. A., McCann, R. J. & Villani, C. (2006) Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Ration. Mech. Anal. 179, 217–263.CrossRefGoogle Scholar
Cesaroni, A. & Cirant, M. (2018) Concentration of ground states in stationary mean-field games systems. Anal. PDE 12, 737–787.CrossRefGoogle Scholar
Chayes, L. & Panferov, V. (2010) The McKean-Vlasov equation in finite volume. J. Stat. Phys. 138, 351–380.CrossRefGoogle Scholar
Cirant, M. (2015) Multi-population Mean Field Games systems with Neumann boundary conditions. J. de Mathématiques Pures et Appliquées 103, 1294–1315.CrossRefGoogle Scholar
Cirant, M. (2016) Stationary focusing mean-field games. Commun. Part. Differ. Equ. 41, 1324–1346.CrossRefGoogle Scholar
Degond, P., Herty, M. & Liu, J. G. (2017) Meanfield games and model predictive control. Commun. Math. Sci. 15, 1403–1422.CrossRefGoogle Scholar
Degond, P., Liu, J. G. & Ringhofer, C. (2014) Large-scale dynamics of mean-field games driven by local nash equilibria. J. Nonlinear Sci. 24, 93–115.CrossRefGoogle Scholar
Evangelista, D., Ferreira, R., Gomes, D. A., Nurbekyan, L. & Voskanyan, V. (2018) First-order, stationary mean-field games with congestion. Nonlinear Anal. Theory Methods Appl. 173, 37–74.CrossRefGoogle Scholar
Evangelista, D. & Gomes, D. A. (2018) On the existence of solutions for stationary mean-field games with congestion. J. Dyn. Differ. Equ. 30, 1365–1388.CrossRefGoogle Scholar
Evans, L. C. (1998) Partial Differential Equations. American Mathematical Society, USA.Google Scholar
Ferreira, R. & Gomes, D. (2018) Existence of weak solutions to stationary mean-field games through variational inequalities. SIAM J. Math. Anal. 50, 5969–6006.CrossRefGoogle Scholar
Ferreira, R., Gomes, D. & Tada, T. (2019) Existence of weak solutions to first-order stationary mean-field games with Dirichlet conditions. Proc. Am. Math. Soc. 147, 4713–4731 (2019).Google Scholar
Gomes, D. & Sánchez Morgado, H. (2013) A stochastic Evans-Aronsson problem. Trans. Am. Math. Soc. 366, 903–929.CrossRefGoogle Scholar
Gomes, D. A., Patrizi, S. & Voskanyan, V. (2014) On the existence of classical solutions for stationary extended mean field games. Nonlinear Anal. Theory Methods Appl. 99, 49–79.CrossRefGoogle Scholar
Gomes, D. A., Pimentel, E. A. & Voskanyan, V. (2016) Regularity Theory for Mean-Field Game Systems. SpringerBriefs in Mathematics. Springer International Publishing, Cham.Google Scholar
Gomes, D. A. & Voskanyan, V. K. (2016) Extended deterministic mean-field games. SIAM J. Cont. Optim. 54, 1030–1055.CrossRefGoogle Scholar
Guéant, O. (2009) A reference case for mean field games models. J. de Mathématiques Pures et Appliquées 92, 276–294.CrossRefGoogle Scholar
Guéant, O. (2012) Mean Field Games Equations with Quadratic Hamiltonian: A Specific Approach, Math. Models Methods Appl. Sci. 22, 1250022.CrossRefGoogle Scholar
Huang, M., Caines, P. E. & Malhamé, R. P. (2006) Distributed multi-agent decision-making with partial observations: asymptotic Nash equilibria. In: Proceedings of the 17th Internat. Symposium on Mathematical Theory of Networks and Systems (MTNS 2006), Kyoto, Japan, pp. 2725–2730.Google Scholar
Huang, M., Caines, P. E. & Malhamé, R. P. (2007) An invariance principle in large population stochastic dynamic games. J. Syst. Sci. Complexity 20, 162–172.CrossRefGoogle Scholar
Huang, M., Malhamé, R. & Caines, P. (2006) Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6, 221–251.Google Scholar
Lasry, J.-M. & Lions, P.-L. (2006) Jeux à champ moyen. I – Le cas stationnaire. Comptes Rendus Mathematique 343, 619–625.CrossRefGoogle Scholar
Lasry, J.-M. & Lions, P.-L. (2006) Jeux à champ moyen. II – Horizon fini et contrôle optimal. Comptes Rendus Mathematique 343, 679–684.CrossRefGoogle Scholar
Lasry, J.-M. & Lions, P.-L. (2007) Mean field games. Japanese J. Math. 2(1), 229–260.CrossRefGoogle Scholar
McCann, R. J. (1997) A convexity principle for interacting gases. Adv. Math. 128, 153–179.CrossRefGoogle Scholar
Schmitt, K. (1978) Boundary value problems for quasilinear second order elliptic equations. Nonlinear Anal. Theory Methods Appl. 2, 263–309.CrossRefGoogle Scholar
Tamura, Y. (1984) On asymptotic behaviours of the solution of a non-linear diffusion process. J. Fac. Sci. Univ. Tokyo Sect. 1A, Math. 31, 195–221.Google Scholar
Tugaut, J. (2014) Phase Transitions of McKean-Vlasov Processes in Double-Wells Landscape. Stochastic 86, 257–284.CrossRefGoogle Scholar