Mean field games. I: The stationary case. (Jeux à champ moyen. I: Le cas stationnaire.) (French. English summary) Zbl 1153.91009
Summary: We introduce here a general approach to model games with a large number of players. More precisely, we consider \(N\) players Nash equilibria for long term stochastic problems and establish rigorously the ‘mean field’ type equations as \(N\) goes to infinity. We also prove general uniqueness results and determine the deterministic limit. For Part II, see C. R., Math., Acad. Sci. Paris 343, No. 10, 679–684 (2006; Zbl 1153.91010).
MSC:
| 91A15 | Stochastic games, stochastic differential games |
| 35J55 | Systems of elliptic equations, boundary value problems (MSC2000) |
| 35J60 | Nonlinear elliptic equations |
| 91A06 | \(n\)-person games, \(n>2\) |
| 91A10 | Noncooperative games |
| 91A07 | Games with infinitely many players |
Citations:
Zbl 1153.91010References:
| [1] | Aumann, R., Markets with a continuum of trackers, Econometrica, 32, 39-50 (1964) · Zbl 0137.39003 |
| [2] | Bensoussan, A.; Frehse, J., Nonlinear elliptic systems in stochastic game theory, J. Reine Angew. Math., 350, 23-67 (1984) · Zbl 0531.93052 |
| [3] | Bensoussan, A.; Frehse, J., Ergodic Bellman systems for stochastic games in arbitrary dimension, Proc. Roy. Soc. Edin. A, 449, 65-77 (1995) · Zbl 0833.90141 |
| [4] | G. Carmona, Nash equilibria of games with a continuum of players, Reprint, 2004; G. Carmona, Nash equilibria of games with a continuum of players, Reprint, 2004 |
| [5] | Lasry, J.-M.; Lions, P.-L., Towards a self-consistent theory of volatility, J. Math. Pures Appl. (2006) · Zbl 1127.91027 |
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