Probabilistic Analysis of Mean-Field Games

By René A. Carmona, François Delarue. SIAM Journal of Control and Optimization, 2012

493 citations recorded by the source corpus.

The purpose of this paper is to provide a complete probabilistic analysis of a large class of stochastic differential games with mean field interactions. We implement the Mean-Field Game strategy developed analytically by Lasry and Lions in a purely probabilistic framework, relying on tailor-made forms of the stochastic maximum principle. While we assume that the state dynamics are affine in the states and the controls, and the costs are convex, our assumptions on the nature of the dependence of all the coefficients upon the statistical distribution of the states of the individual players remains of a rather general nature. Our probabilistic approach calls for the solution of systems of forward-backward stochastic differential equations of a McKean--Vlasov type for which no existence result is known, and for which we prove existence and regularity of the corresponding value function. Finally, we prove that a solution of the Mean-Field Game problem as formulated by Lasry and Lions, does indeed provide appr...

Subfields: Economics and Econometrics, Finance

View Paper: https://www.semanticscholar.org/paper/94152a575be0bde96dea6e35f2881b4b96896997

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Economics and Econometrics

In Finance

Citations in the corpus

Numerical method for FBSDEs of McKean–Vlasov type (2017, background): “(73) coming from Pontryagin principle applied to MFG ` X αt atan p E r X αt sq with d X αt “ α t d t ` d W t , see e.g. [14].”
On the convergence of closed-loop Nash equilibria to the mean field game limit (2018, background): “On the other hand, in the open-loop regime, it is by now well known that strong MFE do arise as the limit of n-player equilibria [48, 14], and it was shown in [43] that the same is true for any weak MFE.”
Time-Dependent Mean-Field Games in the Subquadratic Case (2013, background): “Time-Dependent Mean-Field Games 41 [32, 44] and environmental policy [38], finite state problems [19, 23, 24], explicit models [33, 48], obstacle-type problems [25], extended mean-field games [26, 30], probabilistic methods [13, 14], long-time behavior [10, 12], weak solutions to meanfield games [11”
Mean Field Games Models—A Brief Survey (2013, methodology): “This allows us to avoid using backward stochastic differential equations (see, for instance, [80], and [26, 27]) and, therefore, keeping the presentation elementary.”
Linear Quadratic Mean Field Type Control and Mean Field Games with Common Noise, with Application to Production of an Exhaustible Resource (2016, unknown): “A key reference is the work of Carmona and Delarue [22], which characterizes solutions to the mean field type control problem in terms of a stochastic maximum principle for McKean-Vlasov type dynamics.”
A Probabilistic Approach to Classical Solutions of the Master Equation for Large Population Equilibria (2014, methodology): “Another way is to make use of the stochastic Pontryagin principle to represent directly the optimal path in (i) as the forward component of the solution of a forward-backward system, in which case equilibria solving (ii) may be described through a McKean-Vlasov FBSDE along the lines of [7].”
The Master Equation and the Convergence Problem in Mean Field Games (2015, background): “As already noticed in the earlier paper [21], this would ask for a careful (and certainly complicated) discussion on the choice of Ω and then on the behavior of the solution to (78) with respect to the topology put on Ω.”
Convergence, fluctuations and large deviations for finite state mean field games via the Master Equation (2017, background): “Convergence goes in the opposite direction: are Nash equilibria for the N -player game converging to solutions of the Mean Field Game when the number of players goes to infinity? Results in the first direction are much more common and easier to obtain: for the diffusive case without jumps see for in”
Optimal stopping in mean field games, an obstacle problem approach (2017, background): “A very powerful probabilistic point of vue has also been developed [11,16].”
Some remarks on mean field games (2018, background): “We refer to [12, 13] for examples of such a point of view and to [14, 15] for a complete presentation of this probabilistic approach.”
On the connection between symmetric $N$-player games and mean field games (2014, methodology): “This is the same notion as the one used in the probabilistic approach to mean field games; see Carmona and Delarue (2013).”
A general characterization of the mean field limit for stochastic differential games (2014, methodology): “11 we do not expect to obtain a rate of convergence of ǫn, as in [27, 8].”
Mean Field Games with a Dominating Player (2014, background): “[7], one convenient set of assumptions on g1,f1 and h1 which ensures the unique existence of the minimizer, û1(x, x0, ν, q) = argminuL(x, x0, ν, u, q), is the affine and convexity assumption.”
Time dependent mean-field games in the superquadratic case (2013, methodology): “These include numerical methods [2, 3, 36], applications in economics [32, 41] and environmental policy [36], finite state problems [17, 27, 28], explicit models [33, 45], obstacle-type problems [19], congestion [18, 25], extended mean-field games [24, 29], probabilistic methods [13, 14], long-time ”
Mean Field Games and Systemic Risk (2013, methodology): “The Mean Field Game / FBSDE Approach If it was not for the presence of a common noise, we could apply the results of [5] to obtain approximate Nash equilibria from the solution of the Mean Field Game (MFG).”
On non-uniqueness and uniqueness of solutions in finite-horizon Mean Field Games (2017, unknown): “For the probabilistic approach to MFGs, in addition to the above mentioned works [13, 1, 14, 38, 43], we refer to Carmona and Delarue [12], Lacker [37], and Fischer [19].”