American Institute of Mathematical Sciences

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October  2016, 3(4): 319-334. doi: 10.3934/jdg.2016017

Competition with high number of agents and a major one

Valeria De Mattei 1,

1. 

Dipartimento di Matematica, Largo Bruno Pontecorvo, Pisa, Italy

Received  June 2015 Revised  February 2016 Published  October 2016

In the framework of mean field game theory, a new optimization problem is presented by adding an additional player, called the principal. After introducing a proper payoff for the principal, continuity and existence of minimum is proved. Some considerations about uniqueness and possible ways of continuing the analysis of this problem are given.
Keywords: principal., optimization, Mean field games, mechanism design, partial differential equations.
Mathematics Subject Classification: Primary: 35Q91, 35Q70; Secondary: 91A1.
Citation: Valeria De Mattei. Competition with high number of agents and a major one. Journal of Dynamics & Games, 2016, 3 (4) : 319-334. doi: 10.3934/jdg.2016017
References:
[1]

Y. Achdou, F. Camilli and I. Capuzzo-Dolcetta, Meand field games: Numerical methods for the planning problem,, SIAM J. Control Optim., 50 (2012), 77.  doi: 10.1137/100790069.  Google Scholar

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O. Guéant, J. M. Lasry and P. L. Lions, Mean field games and applications,, 2009., ().   Google Scholar

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

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J. M. Lasry and P. L. Lions, Cours du College de France,, 2009., ().   Google Scholar

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S. Perkins and D. S. Leslie, Stochastic fictitious play with continuous action sets,, Journal of Economic Theory, 152 (2014), 179.  doi: 10.1016/j.jet.2014.04.008.  Google Scholar

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A. Porretta, Weak solutions to Fokker Planck equations and mean field games,, Archive for Rational Mechanics and Analysis, 216 (2015), 1.  doi: 10.1007/s00205-014-0799-9.  Google Scholar

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D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes,, Springer Verlag, (1979).   Google Scholar

show all references

References:
[1]

Y. Achdou, F. Camilli and I. Capuzzo-Dolcetta, Meand field games: Numerical methods for the planning problem,, SIAM J. Control Optim., 50 (2012), 77.  doi: 10.1137/100790069.  Google Scholar

[2]

M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,, Systems and Control: Foundations and Applications. Birkhauser Boston Inc., (1997).  doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[3]

T. Borgers, An introduction to the Theory of Mechanism Design,, 2015., ().  doi: 10.1093/acprof:oso/9780199734023.001.0001.  Google Scholar

[4]

P. Cardaliaguet, Notes on Mean Field Games,, 2012., ().   Google Scholar

[5]

H. Gintis, Game Theory Evolving,, Princeton University Press, (2009).   Google Scholar

[6]

O. Guéant, J. M. Lasry and P. L. Lions, Mean field games and applications,, 2009., ().   Google Scholar

[7]

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

[8]

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

[9]

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

[10]

J. M. Lasry and P. L. Lions, Cours du College de France,, 2009., ().   Google Scholar

[11]

S. Perkins and D. S. Leslie, Stochastic fictitious play with continuous action sets,, Journal of Economic Theory, 152 (2014), 179.  doi: 10.1016/j.jet.2014.04.008.  Google Scholar

[12]

A. Porretta, Weak solutions to Fokker Planck equations and mean field games,, Archive for Rational Mechanics and Analysis, 216 (2015), 1.  doi: 10.1007/s00205-014-0799-9.  Google Scholar

[13]

D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes,, Springer Verlag, (1979).   Google Scholar

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