Long time behavior of the master equation in mean field game theory

Cardaliaguet, Pierre; Porretta, Alessio

doi:10.2140/apde.2019.12.1397

Cited by 54 publications

(62 citation statements)

References 29 publications

(106 reference statements)

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“…Numerical methods are being developed, let us cite [1,5] for examples of this growing literature. Let us also mention the questions of long time average [9,8] or learning [6]. In some very particular cases (the so-called "potential case") solutions of the MFG system can be obtain from a PDE optimal control problem [2].…”

Section: Introductionmentioning

confidence: 99%

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Some remarks on mean field games

Bertucci

Lasry

Lions

2019

Communications in Partial Differential Equations

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We study in this paper three aspects of Mean Field Games. The first one is the case when the dynamics of each player depend on the strategies of the other players. The second one concerns the modeling of "noise" in discrete space models and the formulation of the Master Equation in this case. Finally, we show how Mean Field Games reduce to agent based models when the intertemporal preference rate goes to infinity, i.e. when the anticipation of the players vanishes. 2

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Section: Introductionmentioning

confidence: 99%

“…Moreover we insist that in the presence of a common noise, the reduction of the MFG to the system (1) collapses and the study of the master equation is crucial for the understanding of the MFG. We refer the reader to [10,25] for a detailed study of the master equation and to [8,19] for examples of applications.…”

Section: Introductionmentioning

confidence: 99%

Some remarks on mean field games

Bertucci

Lasry

Lions

2019

Communications in Partial Differential Equations

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“…Proof. The proof follows closely the one proposed in [12] and it relies on semiconcavity estimates for the value function u. We recall that if φ ∈ C ∞ (T d ) then…”

Section: Existence Of a Correctormentioning

confidence: 70%

On the long time convergence of potential MFG

Masoero

2019

Nonlinear Differ. Equ. Appl.

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We look at the long time behavior of potential Mean Field Games (briefly MFG) using some standard tools from weak KAM theory. We first show that the time-dependent minimization problem converges to an ergodic constant −λ, then we provide a class of examples where the value of the stationary MFG minimization problem is strictly greater than −λ. This will imply that the trajectories of the time-dependent MFG system do not converge to static equilibria.Potential MFG are those games whose MFG system can be derived as optimality condition of the following minimization problemwhere (m, w) verifies the Fokker Plank equation −∂ t m + m + divw = 0 with m(0) = m 0 and the coupling function F in the MFG system is the derivative with respect to the measure of F. These games have been largely studied (see Lasry and Lions [25] for existence results and, among others [9,4,28] for further properties) but, so far, not much is known regarding their long time behavior outside the assumption of monotonicity, where Cardaliaguet, Lasry, Lions and Porretta [10] proved the convergence to the ergodic system

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“…Let us first recall that, in the open-loop regime, limits of Nash equilibria in the N´player game are MFG equilibria [1,17,24]. On the other hand, in the closed loop Markovian regime, the convergence problem is surprisingly open up to now, although the existence of a solution to the ergodic master equation is known [8]: Indeed in this ergodic set-up, the use of the solution to the master equation is not obvious and the technique of proof of [6] does not seem to apply. Here we concentrate on the limit of equilibria in N´player differential games with generalized Markov strategies.…”

Section: Introductionmentioning

confidence: 99%