January  2020, 7(1): 1-20. doi: 10.3934/jdg.2020001

Mean-field games and swarms dynamics in Gaussian and non-Gaussian environments

STI/EPFL, Station 17, CH-1015 Lausanne, Switzerland

Corresponding author: Max-Olivier Hongler

Received  November 2018 Revised  August 2019 Published  December 2019

The collective behaviour of stochastic multi-agents swarms driven by Gaussian and non-Gaussian environments is analytically discussed in a mean-field approach. We first exogenously implement long range mutual interactions rules with strengths that are weighted by the real-time distance separating each agent with the swarm barycentre. Depending on the form of this barycentric modulation, a transition between two drastically different collective behaviours can be unveiled. A behavioural bifurcation threshold due to the tradeoff between the desynchronisation effects of the stochastic environment and the synchronising interactions is analytically calculated. For strong enough interactions, the emergence of a soliton propagating wave is established. Alternatively, weaker interactions cannot overcome the environmental noise and evanescent diffusive waves result. In a second and complementary approach, we show that the emergent solitons can alternatively be interpreted as being the optimal equilibrium of mean-field games (MFG) models with ad-hoc running cost functions which are here exactly determined. These MFG's soliton equilibria are therefore endogenously generated. Hence for the classes of models here proposed, an explicit correspondence between exogenous and endogenous interaction rules leading to similar collective effects is explicitly constructed. For non-Gaussian environments our results offer a new class of exactly solvable mean-field games dynamics.

Citation: Max-Olivier Hongler. Mean-field games and swarms dynamics in Gaussian and non-Gaussian environments. Journal of Dynamics & Games, 2020, 7 (1) : 1-20. doi: 10.3934/jdg.2020001
References:
[1]

J. A. AcebrónL. L. BonillaJ. Pérez VicenteF. Riort and R. Spiegler, The kuramoto model. A simple paradigm for synchronization phenomena, Rev. Modern Phys., 77 (2005), 137-185.   Google Scholar

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P. CardaliaguetJ.-M. LasryP.-L. Lions and A. Poretta, Long time average of mean field games, with a nonlocal coupling, SIAM J. Contr. and Optim., 51 (2013), 3558-3591.  doi: 10.1137/120904184.  Google Scholar

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R. Carmona and F. Delarue, Probabilistic Theory Meanfield Games with Applications. II. Mean Field Games with Common Noise and Master Equations, Probability Theory and Stochastic Modelling, 84. Springer, Cham, 2018.  Google Scholar

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E. R. Fernholz, Stochastic Portfolio Theory, Applications of Mathematics (New York), 48. Stochastic Modelling and Applied Probability, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-3699-1.  Google Scholar

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E. CarlenE. Gabetta and E. Regazzini, Probabilistic investigations on the explosion of solutions of the Kac equation with infinite energy initial distribution, J. Appl. Probab., 45 (2008), 95-106.  doi: 10.1017/S0021900200003983.  Google Scholar

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O. Gallay, F Hashemi and M.-O. Hongler, Imitation, proximity and growth, Advances in Complex Syst., 22 (2019), 1950011. Google Scholar

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C. Gardiner, Stochastic Methods. A Handbook for the Natural and Social Ssciences, Fourth edition. Springer Series in Synergetics. Springer-Verlag, Berlin, 2009.  Google Scholar

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D. A. Gomes, E. A. Pimentel and V. Voskanyan, Regularity Theory for Mean-Field Game Systems, SpringerBriefs in Mathematics. Springer, [Cham], 2016. doi: 10.1007/978-3-319-38934-9.  Google Scholar

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S. Gradshteyn and M. Ryzhik, Tables of integrals, series and products, Wiley. Google Scholar

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O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications, Paris-Princeton Lectures on Mathematical Finance 2010, Lecture Notes in Math., Springer, Berlin, 2003 (2011), 205–266. doi: 10.1007/978-3-642-14660-2_3.  Google Scholar

[14]

M.-O. HonglerH. M. Soner and L. Streit, Stochastic control for a class of random evolution models, Appl. Math. Optim., 49 (2014), 113-121.  doi: 10.1007/s00245-003-0786-2.  Google Scholar

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M.-O. Hongler and L. Streit, A probabilistic connection between the Burgers and a discrete Boltzmann equation, Europhys. Lett., 12 (1990), 193-197.  doi: 10.1209/0295-5075/12/3/001.  Google Scholar

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W. Horsthemke and R. Lefever, Theory and Applications in Physics, Chemistry and Biology, Springer Series in Synergetics, 1984. Google Scholar

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M. Y. HuangP. E. Caines and R. P. Malhamé, Large population cost-coupled LQG problems with uniform agents: Individual-mass behaviour and decentralised $\epsilon$-Nash equilibria, IEEE Trans. Automat. Control, 52 (2007), 1560-1571.  doi: 10.1109/TAC.2007.904450.  Google Scholar

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T. IchibaV. PapathanakosA. BannerI. Karatzas and R. Fernholz, Hybrid atlas models, Annals Appl. Probab., 21 (2011), 609-644.  doi: 10.1214/10-AAP706.  Google Scholar

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V. Kolokoltsov, J. Li and W. Yang, Mean field games and nonlinear markov processes, (2011), arXiv: 1112.3744. Google Scholar

[20]

M. D. KönigJ. Lorenz and F. Zilibotti, Innovation vs. imitation and the evolution of productivity distributions, Theor. Economy, 11 (2016), 1053-1102.  doi: 10.3982/TE1437.  Google Scholar

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J.-M. Lasry and P.-L. Lions, Mean field games, Japan J. Math, 2 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8.  Google Scholar

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J.-M. Lasry and P.-L. Lions, Jeux à champs moyen. I. Le cas stationnaire, C. R. Math, Acad. Sci. Paris, 343 (2006), 619-625.  doi: 10.1016/j.crma.2006.09.019.  Google Scholar

[23]

J.-M. Lasry and P.-L. Lions., Jeux à champs moyen. II. Horizon fini et contrôle optimal, C. R. Math, Acad. Sci. Paris, 343 (2006), 679-684.  doi: 10.1016/j.crma.2006.09.018.  Google Scholar

[24]

T. W. Ruijgrok and T. T. Wu, A completely solvable model of the nonlinear boltzmann equation, Physica A, 113 (1982), 401-416.  doi: 10.1016/0378-4371(82)90147-9.  Google Scholar

[25]

I. Swiecicki, T. Gobron and D. Ullmo, Schrödinger approach to mean-field games, Phys. Rev. Lett., 116 (2016), 128701. Google Scholar

[26]

D. UllmoI. Swiecicki and T. Gobron, Quadratic mean field games, Phys. Rep., 799 (2019), 1-35.  doi: 10.1016/j.physrep.2019.01.001.  Google Scholar

[27]

H. Yin. P. G. MehtaS. P. Meyn and U. V. Shanbhag, Synchronisation of coupled oscillators is a game, IEEE Trans. Autom. Control, 57 (2012), 920-935.  doi: 10.1109/TAC.2011.2168082.  Google Scholar

show all references

References:
[1]

J. A. AcebrónL. L. BonillaJ. Pérez VicenteF. Riort and R. Spiegler, The kuramoto model. A simple paradigm for synchronization phenomena, Rev. Modern Phys., 77 (2005), 137-185.   Google Scholar

[2]

Ph. Aghion and P. Howitt, The Economy of Growth, The MIT press, 2009. Google Scholar

[3]

A. D. BannerR. Fernholz and I. Karatzas, Atlas models of equity markets, Annals Appl. Probab., 15 (2005), 2296-2230.  doi: 10.1214/105051605000000449.  Google Scholar

[4]

A. Bensoussan, J. Frehse and P. Yam, Mean Field Games and Mean Field Type Control Theory, SpringerBriefs in Mathematics, Springer, New York, 2013. doi: 10.1007/978-1-4614-8508-7.  Google Scholar

[5]

P. CardaliaguetJ.-M. LasryP.-L. Lions and A. Poretta, Long time average of mean field games, with a nonlocal coupling, SIAM J. Contr. and Optim., 51 (2013), 3558-3591.  doi: 10.1137/120904184.  Google Scholar

[6]

R. Carmona and F. Delarue, Probabilistic Theory Meanfield Games with Applications. II. Mean Field Games with Common Noise and Master Equations, Probability Theory and Stochastic Modelling, 84. Springer, Cham, 2018.  Google Scholar

[7]

E. R. Fernholz, Stochastic Portfolio Theory, Applications of Mathematics (New York), 48. Stochastic Modelling and Applied Probability, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-3699-1.  Google Scholar

[8]

E. CarlenE. Gabetta and E. Regazzini, Probabilistic investigations on the explosion of solutions of the Kac equation with infinite energy initial distribution, J. Appl. Probab., 45 (2008), 95-106.  doi: 10.1017/S0021900200003983.  Google Scholar

[9]

O. Gallay, F Hashemi and M.-O. Hongler, Imitation, proximity and growth, Advances in Complex Syst., 22 (2019), 1950011. Google Scholar

[10]

C. Gardiner, Stochastic Methods. A Handbook for the Natural and Social Ssciences, Fourth edition. Springer Series in Synergetics. Springer-Verlag, Berlin, 2009.  Google Scholar

[11]

D. A. Gomes, E. A. Pimentel and V. Voskanyan, Regularity Theory for Mean-Field Game Systems, SpringerBriefs in Mathematics. Springer, [Cham], 2016. doi: 10.1007/978-3-319-38934-9.  Google Scholar

[12]

S. Gradshteyn and M. Ryzhik, Tables of integrals, series and products, Wiley. Google Scholar

[13]

O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications, Paris-Princeton Lectures on Mathematical Finance 2010, Lecture Notes in Math., Springer, Berlin, 2003 (2011), 205–266. doi: 10.1007/978-3-642-14660-2_3.  Google Scholar

[14]

M.-O. HonglerH. M. Soner and L. Streit, Stochastic control for a class of random evolution models, Appl. Math. Optim., 49 (2014), 113-121.  doi: 10.1007/s00245-003-0786-2.  Google Scholar

[15]

M.-O. Hongler and L. Streit, A probabilistic connection between the Burgers and a discrete Boltzmann equation, Europhys. Lett., 12 (1990), 193-197.  doi: 10.1209/0295-5075/12/3/001.  Google Scholar

[16]

W. Horsthemke and R. Lefever, Theory and Applications in Physics, Chemistry and Biology, Springer Series in Synergetics, 1984. Google Scholar

[17]

M. Y. HuangP. E. Caines and R. P. Malhamé, Large population cost-coupled LQG problems with uniform agents: Individual-mass behaviour and decentralised $\epsilon$-Nash equilibria, IEEE Trans. Automat. Control, 52 (2007), 1560-1571.  doi: 10.1109/TAC.2007.904450.  Google Scholar

[18]

T. IchibaV. PapathanakosA. BannerI. Karatzas and R. Fernholz, Hybrid atlas models, Annals Appl. Probab., 21 (2011), 609-644.  doi: 10.1214/10-AAP706.  Google Scholar

[19]

V. Kolokoltsov, J. Li and W. Yang, Mean field games and nonlinear markov processes, (2011), arXiv: 1112.3744. Google Scholar

[20]

M. D. KönigJ. Lorenz and F. Zilibotti, Innovation vs. imitation and the evolution of productivity distributions, Theor. Economy, 11 (2016), 1053-1102.  doi: 10.3982/TE1437.  Google Scholar

[21]

J.-M. Lasry and P.-L. Lions, Mean field games, Japan J. Math, 2 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8.  Google Scholar

[22]

J.-M. Lasry and P.-L. Lions, Jeux à champs moyen. I. Le cas stationnaire, C. R. Math, Acad. Sci. Paris, 343 (2006), 619-625.  doi: 10.1016/j.crma.2006.09.019.  Google Scholar

[23]

J.-M. Lasry and P.-L. Lions., Jeux à champs moyen. II. Horizon fini et contrôle optimal, C. R. Math, Acad. Sci. Paris, 343 (2006), 679-684.  doi: 10.1016/j.crma.2006.09.018.  Google Scholar

[24]

T. W. Ruijgrok and T. T. Wu, A completely solvable model of the nonlinear boltzmann equation, Physica A, 113 (1982), 401-416.  doi: 10.1016/0378-4371(82)90147-9.  Google Scholar

[25]

I. Swiecicki, T. Gobron and D. Ullmo, Schrödinger approach to mean-field games, Phys. Rev. Lett., 116 (2016), 128701. Google Scholar

[26]

D. UllmoI. Swiecicki and T. Gobron, Quadratic mean field games, Phys. Rep., 799 (2019), 1-35.  doi: 10.1016/j.physrep.2019.01.001.  Google Scholar

[27]

H. Yin. P. G. MehtaS. P. Meyn and U. V. Shanbhag, Synchronisation of coupled oscillators is a game, IEEE Trans. Autom. Control, 57 (2012), 920-935.  doi: 10.1109/TAC.2011.2168082.  Google Scholar

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