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

Competition with high number of agents and a major one

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.
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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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

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P. Cardaliaguet, Notes on Mean Field Games,, 2012., (). 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. I. Le cas stationaire,, C. R. Math.Acad. Sci. Paris, 343 (2006), 619. doi: 10.1016/j.crma.2006.09.019. 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, Mean field games,, Japanese Journal of Mathematics, 2 (2007), 229. doi: 10.1007/s11537-007-0657-8. 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

[13]

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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