Control of McKean–Vlasov dynamics versus mean field games

By René A. Carmona, François Delarue, A. Lachapelle. 2012

303 citations recorded by the source corpus.

We discuss and compare two investigation methods for the asymptotic regime of stochastic differential games with a finite number of players as the number of players tends to the infinity. These two methods differ in the order in which optimization and passage to the limit are performed. When optimizing first, the asymptotic problem is usually referred to as a mean-field game. Otherwise, it reads as an optimization problem over controlled dynamics of McKean–Vlasov type. Both problems lead to the analysis of forward–backward stochastic differential equations, the coefficients of which depend on the marginal distributions of the solutions. We explain the difference between the nature and solutions to the two approaches by investigating the corresponding forward–backward systems. General results are stated and specific examples are treated, especially when cost functionals are of linear-quadratic type.

Subfields: Demography, Finance, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/1b9f9220a9afd9051af3f0f3eb357b1b98189904

How researchers see this paper

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

In Demography

In Finance

In Statistical and Nonlinear Physics

Citations in the corpus

On the connection between symmetric $N$-player games and mean field games (2014, background): “Controlled martingale problems are especially powerful in combination with relaxed controls; see El Karoui et al. [1987], Kushner [1990], and the references therein.”
A Pontryagin Maximum Principle in Wasserstein spaces for constrained optimal control problems (2018, background): “in [21] that in the class of so-called potential mean-field games, the self-organization of an ensemble of agents could be equivalently reformulated as an optimal control problem in Wasserstein spaces involving adequately modified functionals.”
Mean field games via controlled martingale problems: Existence of Markovian equilibria (2014, background): “This stochastic approach leads to a notably clean solution of linear-quadratic mean field games, as in [4, 13], as well as an interesting generalization of mean field games involving one dominant agent competing with a mean field of minor agents [27, 8].”
A general characterization of the mean field limit for stochastic differential games (2014, background): “Existence results for this somewhat delicate boundary case have been obtained in [8, 10, 6, 12] by assuming some additional inequalities between coefficients.”
Mean Field Games Models—A Brief Survey (2013, methodology): “Mean ﬁeld games have also been analyzed using backward-forward stochastic differential equations; see, for instance [80], and [26–29].”
Selection of equilibria in a linear quadratic mean-field game (2018, methodology): “2 FRANÇOIS DELARUE, RINEL FOGUEN TCHUENDOM We also refer to the works of Carmona and Delarue, who studied MFGs with a probabilistic approach, see for instance [7, 10] together with the recent two-volume monograph [8, 9].”
Randomized dynamic programming principle and Feynman-Kac representation for optimal control of McKean-Vlasov dynamics (2016, unknown): “Both of these problems describe equilibriums states of large population of weakly interacting symmetric players and we refer to [14] for a discussion pointing out the differences between the two frameworks: In a nutshell MFGs describe Nash equilibrium in large populations and the optimal control of ”
Mean-field optimal control as Gamma-limit of finite agent controls (2018, methodology): “The most popular methods are based on extending Pontryagin’s maximum principle [2, 6, 9, 14, 17] or deriving a dynamic programming principle, and with it a form of a Hamilton–Jacobi–Bellman equation on a space of probability measures [8, 47, 55].”
Mean field games with common noise (2014, background): “This was observed in [10, 29] in the case σ0 = 0; the authors showed that only certain linear-quadratic mean field games admit (strong) solutions.”
Mean field games with congestion (2017, background): “Delarue [14, 13] and mean field type control by A.”
McKean–Vlasov optimal control: The dynamic programming principle (2019, background): “Though related, these two problems have some subtle differences which were investigated thoroughly by Carmona, Delarue, and Lachapelle [27].”
A mean-field stochastic control problem with partial observations (2017, background): “Therefore, similar to [7] (or [13]) we begin by considering N individual investors, and assume that ith investor’s private state dynamics is of the form: (1.”
On non-uniqueness and uniqueness of solutions in finite-horizon Mean Field Games (2017, unknown): “…based on the FBSDE approach to mean ﬁeld games, examples of MFGs with multiple solutions that fall into the class of problems considered here are given in Section 5.1 of Carmona et al. [13] and Section 4.2 of Tchuendom [43], where the author shows that adding a common noise restores uniqueness.”