Continuous Time Finite State Mean Field Games

By Diogo A. Gomes, Joana Mohr, Rafael R. Souza. 2012

182 citations recorded by the source corpus.

In this paper we consider symmetric games where a large number of players can be in any one of d states. We derive a limiting mean field model and characterize its main properties. This mean field limit is a system of coupled ordinary differential 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 stationary 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.

Subfields: Economics and Econometrics, Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/f0fe981f541f6f32c28ece8d327eafb5bf41f429

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Citations in the corpus

On non-uniqueness and uniqueness of solutions in finite-horizon Mean Field Games (2017, unknown): “A similar structure of proof, based on combining backward and forward integral estimates, was also employed for finite state MFGs by [23].”
Time dependent mean-field games in the superquadratic case (2013, methodology): “These include numerical methods [34], [3], [2], applications in economics [39], [30] and environmental policy [34], finite state problems [19], [20], [16], explicit models [31], [43], obstacle-type problems [21], congestion [18], extended mean-field games [22], [27], probabilistic methods [13], [12]”
Convergence, fluctuations and large deviations for finite state mean field games via the Master Equation (2017, background): “The existence of solutions follows from a standard fixed point argument: see Proposition 4 of [16].”
Finite state mean field games with Wright–Fisher common noise (2019, methodology): “MFGs with (a fixed and) finite number of states were introduced by [35, 36, 38]; for a probabilistic approach to finite state MFGs we refer to [11, 14].”
A Probabilistic Approach to Mean Field Games with Major and Minor Players (2014, methodology): “Similar results were proved in [10] for discrete state mean field game without major player.”
Mean Field Games Models—A Brief Survey (2013, background): “The first positive answer to this problem in discrete state and time was given in [46] and then further extended to the continuous time setting in [49].”
On the existence of classical solutions for stationary extended mean field games (2013, background): “Stationary mean-field games have an independent interest but also, as shown in [CLLP] (see also [GMS10], and [GMS11], for discrete state problems) they encode the asymptotic long time behavior of various mean-field games.”
A general characterization of the mean field limit for stochastic differential games (2014, unknown): “[17] studied models with finite state space.”
Time-Dependent Mean-Field Games in the Subquadratic Case (2013, background): “In particular, a number of problems have been worked out in detail by various authors, including numerical methods [LST10], [ACD10], [ACCD12], applications in economics [LLG10a], [Gué09a] and environmental policy [LST10], finite state problems [GMS10], [GMS13], [FG13], explicit models [Gué09b], [NH1”
Applications of Mean Field Games in financial engineering and economic theory (2020, methodology): “See the system of ODEs (12)-(13) in [51] or [21, Section 7.”