On the Asymptotic Nature of First Order Mean Field Games

Fischer, Markus; Silva, Francisco J.

doi:10.1007/s00245-020-09711-1

Cited by 16 publications

(10 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, let us emphasize that the well-posedness question of systems of Hamilton-Jacobi equations in the deterministic setting is not a settled issue in the literature. It's worth mentioning the recent work [28], which studies this convergence question in the deterministic setting in a suitable weak sense, without relying on the well-posedness of either the Nash system or the master equation.…”

Section: Summary Of Our Main Resultsmentioning

confidence: 99%

Global Well‐Posedness of Master Equations for Deterministic Displacement Convex Potential Mean Field Games

Gangbo¹,

Mészáros

2022

Comm Pure Appl Math

View full text Add to dashboard Cite

This manuscript constructs global in time solutions to master equations for potential mean field games. The study concerns a class of Lagrangians and initial data functions that are displacement convex, and so this property may be in dichotomy with the so-called Lasry-Lions monotonicity, widely considered in the literature. We construct solutions to both the scalar and vectorial master equations in potential mean field games, when the underlying space is the whole space R d , and so it is not compact.

show abstract

Section: Summary Of Our Main Resultsmentioning

confidence: 99%

Global Well‐Posedness of Master Equations for Deterministic Displacement Convex Potential Mean Field Games

Gangbo¹,

Mészáros

2022

Comm Pure Appl Math

View full text Add to dashboard Cite

show abstract

“…Note that, given a measure Q ∈ P(C( Ω)), one can obtain the associated time-dependent measure m by setting m t = e t# Q for t ≥ 0. The Lagrangian approach is a classical approach in optimal transport problems (see, e.g., [2,56]) which has been used to define equilibria of first-order mean field games in some recent works, such as [8,11,16,19,21,31,32,50,55].…”

Section: The Minimal-time Mean Field Game and Its Equilibriamentioning

confidence: 99%

Nonsmooth mean field games with state constraints

Arjmand¹,

Mazanti²

2022

ESAIM: COCV

View full text Add to dashboard Cite

In this paper, we consider a mean field game model inspired by crowd motion where agents aim to reach a closed set, called target set, in minimal time. Congestion phenomena are modeled through a constraint on the velocity of an agent that depends on the average density of agents around their position. The model is considered in the presence of state constraints: roughly speaking, these constraints may model walls, columns, fences, hedges, or other kinds of obstacles at the boundary of the domain which agents cannot cross. After providing a more detailed description of the model, the paper recalls some previous results on the existence of equilibria for such games and presents the main difficulties that arise due to the presence of state constraints. Our main contribution is to show that equilibria of the game satisfy a system of coupled partial differential equations, known mean field game system, thanks to recent techniques to characterize optimal controls in the presence of state constraints. These techniques not only allow to deal with state constraints but also require very few regularity assumptions on the dynamics of the agents.

show abstract

“…More recent progress appeared in [32], treating mean field games of controls with common noise. Similar techniques have resolved the open-loop convergence problem for other kinds of MFG models, such as discrete-time models [1], first-order (noiseless) models [36], and correlated equilibria [14], as well as for cooperative (mean field control) models in great generality [49,33,31]. A new approach was developed recently in [57,58] based on propagation of chaos for mean field (F)BSDEs.…”

Section: Introductionmentioning

confidence: 99%

Closed-loop convergence for mean field games with common noise

Lacker¹,

Flem²

2021

Preprint

View full text Add to dashboard Cite

This paper studies the convergence problem for mean field games with common noise. We define a suitable notion of weak mean field equilibria, which we prove captures all subsequential limit points, as n → ∞, of closed-loop approximate equilibria from the corresponding n-player games. This extends to the common noise setting a recent result of the first author, while also simplifying a key step in the proof and allowing unbounded coefficients and non-i.i.d. initial conditions. Conversely, we show that every weak mean field equilibrium arises as the limit of some sequence of approximate equilibria for the n-player games, as long as the latter are formulated over a broader class of closed-loop strategies which may depend on an additional common signal.

show abstract

On the Asymptotic Nature of First Order Mean Field Games

Cited by 16 publications

References 34 publications

Global Well‐Posedness of Master Equations for Deterministic Displacement Convex Potential Mean Field Games

Global Well‐Posedness of Master Equations for Deterministic Displacement Convex Potential Mean Field Games

Nonsmooth mean field games with state constraints

Closed-loop convergence for mean field games with common noise

Contact Info

Product

Resources

About