A probabilistic weak formulation of mean field games and applications

By René A. Carmona, Daniel Lacker. 2013

216 citations recorded by the source corpus.

Mean field games are studied by means of the weak formulation of stochastic optimal control. This approach allows the mean field interactions to enter through both state and control processes and take a form which is general enough to include rank and nearest-neighbor effects. Moreover, the data may depend discontinuously on the state variable, and more generally its entire history. Existence and uniqueness results are proven, along with a procedure for identifying and constructing distributed strategies which provide approximate Nash equlibria for finite-player games. Our results are applied to a new class of multi-agent price impact models and a class of flocking models for which we prove existence of equilibria.

Subfields: Economics and Econometrics, Finance

View Paper: https://www.semanticscholar.org/paper/206a345ed5d4de6dd1641739feea3f8c302ad8e2

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

On a strong form of propagation of chaos for McKean-Vlasov equations (2018, background): “5 that we know of seems to come from the paper [5], from which we borrow the proof idea.”
A general characterization of the mean field limit for stochastic differential games (2014, methodology): “In a tradition beginning with [21] and continued by the majority of the probabilistic papers on the subject [8, 13, 6, 4, 27], an optimal control from an MFG solution is used to construct approximate equilibria for the finite-player games.”
Applications of Mean Field Games in financial engineering and economic theory (2020, methodology): “A complete solution of this limit MFG in the weak formulation can be found in [27] and in [21, Section 4.”
Some remarks on mean field games (2018, background): “Existence of solutions of such a system was proven for particular Hamiltonians in [22]; we refer to [23] for a more general result of existence for (2) and to [14, 24] for the 206 C.”
Mean field games with common noise (2014, background): “The analysis of [12] and especially [29] are related to ours in that they employ weak formulations of the optimal control problems.”
Mean field games via controlled martingale problems: Existence of Markovian equilibria (2014, background): “A bit of an outlier is the probabilistic weak formulation of [16], which employs a well-known Girsanov transformation of the control problems under the crucial assumption that the volaility is nondegenerate and uninfluenced by the mean field or the control.”
A Probabilistic Approach to Mean Field Games with Major and Minor Players (2014, background): “Recently, an approach based on the weak formulation of stochastic controls was introduced in [11] and models with a common noise studied in [9].”
On the connection between symmetric $N$-player games and mean field games (2014, methodology): “…that a solution of the mean field game allows to construct approximate Nash equilibria for the corresponding N -player games if N is sufficiently large; see, for instance, Huang, Malhamé and Caines (2006), Kolokoltsov, Li and Yang (2011), Carmona and Delarue (2013) and Carmona and Lacker (2015).”
Mean Field Games and Systemic Risk (2013, background): “See, for example, [6, 2, 5, 4] or [7] for a probabilistic approach based on the weak formulation of stochastic control.”
A Probabilistic Approach to Classical Solutions of the Master Equation for Large Population Equilibria (2014, background): “The first one is to represent the value function of the optimization problem (i) as the decoupling field of a forward-backward system, in which case equilibria solving (ii) may be described through a McKean-Vlasov FBSDE along the lines of [10].”
$N$-player games and mean-field games with absorption (2016, methodology): “1) and a fixed point argument, both in the spirit of Carmona and Lacker [7].”
Mean–field moral hazard for optimal energy demand response management (2019, result): “Although this is not standard in the theory of mean–field games in a Markovian setting, this makes sense in our non–Markovian framework, and a similar condition can be found in the work of Carmona and Lacker [12].”