Appl Math Optim (2013) 68:99–143
Continuous Time Finite State Mean Field Games
Diogo A. Gomes · Joana Mohr · Rafael Rigão Souza
Published online: 23 April 2013
© Springer Science+Business Media New York 2013
Abstract In this paper we consider symmetric games where a large number of play-
ers can be in any one of d states. We derive a limiting mean ﬁeld model and charac-
terize its main properties. This mean ﬁeld limit is a system of coupled ordinary dif-
ferential equations with initial-terminal data. For this mean ﬁeld problem we prove a
trend to equilibrium theorem, that is convergence, in an appropriate limit, to station-
ary solutions. Then we study an N + 1-player problem, which the mean ﬁeld model
attempts to approximate. Our main result is the convergence as N →∞of the mean
ﬁeld model and an estimate of the rate of convergence. We end the paper with some
further examples for potential mean ﬁeld games.
D. Gomes was partially supported by CAMGSD-LARSys through FCT-Portugal and by grants
PTDC/MAT-CAL/0749/2012, UTA-CMU/MAT/0007/2009 PTDC/MAT/114397/2009,
UTAustin-MAT/0057/2008, and by the bilateral agreement Brazil-Portugal (CAPES-FCT) 248/09.
R.R.S. was partially supported by the bilateral agreement Brazil-Portugal (CAPES-FCT) 248/09.
J.M. was partially supported by the bilateral agreement Brazil-Portugal (CAPES-FCT) 248/09.
D.A. Gomes (
Center for Mathematical Analysis, Geometry, and Dynamical Systems, Departamento de
Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal
CSMSE Division, King Abdullah University of Science and Technology (KAUST), Thuwal
23955-6900, Saudi Arabia
J. Mohr · R.R. Souza
Instituto de Matemática, UFRGS, 91509-900 Porto Alegre, Brazil