American Institute of Mathematical Sciences

Advanced
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 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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