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 and 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), 77-109. doi: 10.1137/100790069.

[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., Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1.

[3]

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

[4]

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

[5]

H. Gintis, Game Theory Evolving, Princeton University Press, 2009.

[6]

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

[7]

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

[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), 679-684. doi: 10.1016/j.crma.2006.09.018.

[9]

J. M. Lasry and P. L. Lions, Mean field games, Japanese Journal of Mathematics, 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8.

[10]

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

[11]

S. Perkins and D. S. Leslie, Stochastic fictitious play with continuous action sets, Journal of Economic Theory, 152 (2014), 179-213. doi: 10.1016/j.jet.2014.04.008.

[12]

A. Porretta, Weak solutions to Fokker Planck equations and mean field games, Archive for Rational Mechanics and Analysis, 216 (2015), 1-62. doi: 10.1007/s00205-014-0799-9.

[13]

D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer Verlag, 1979.

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), 77-109. doi: 10.1137/100790069.

[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., Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1.

[3]

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

[4]

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

[5]

H. Gintis, Game Theory Evolving, Princeton University Press, 2009.

[6]

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

[7]

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

[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), 679-684. doi: 10.1016/j.crma.2006.09.018.

[9]

J. M. Lasry and P. L. Lions, Mean field games, Japanese Journal of Mathematics, 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8.

[10]

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

[11]

S. Perkins and D. S. Leslie, Stochastic fictitious play with continuous action sets, Journal of Economic Theory, 152 (2014), 179-213. doi: 10.1016/j.jet.2014.04.008.

[12]

A. Porretta, Weak solutions to Fokker Planck equations and mean field games, Archive for Rational Mechanics and Analysis, 216 (2015), 1-62. doi: 10.1007/s00205-014-0799-9.

[13]

D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer Verlag, 1979.

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