Two-days online workshop on Mean Field Games

Les Andelys (France), June 18-19 2020


Titles, abstracts and slides

Y. Averboukh slides

Title: A viability approach to first-order mean field games

Abstract:  The talk is concerned with the study of the dependence of solutions of the deterministic mean field game on initial distribution of players. The main object of the study is the multivalued mapping assigning to initial time and initial distribution of players a set of expected outcomes of the representative player corresponding to solutions of the mean field game. It is natural to call this mapping a value multifunction. Since the mean field game implies that the representative player solves the optimal control problem, the value multifunction acts to the space of continuous function. We introduce the mean field game dynamical system. It is defined on the product of the space of probability measures and the space of continuous functions. We prove that if a multivalued function is viable with respect to the mean field game dynamics, then it is a value multifunction i.e. in this case, given an initial time, initial distribution of players and a continuous function lying in the image of the examining mapping, one can find a solution of the mean field game with the given initial distribution of players such that the expected outcome of the representative player is equal to the given function. Furthermore, the greatest value multifnction is viable with respect to the mean field game dynamical system. Additionally, we obtained the infinitesimal form of the viability condition. It is an analog of famous Nagumo viability theorem for the mean field game dynamical system in the product of space of probability measures and space of continuous function.

F. Bagagiolo slides

Title:  An optimal visiting mean field game.

Abstract:  In this talk I am going to present an n-dimensional deterministic mean field game, where the agents have to minimize a cost which also depends on more than one targets to be “visited”. The targets are points of the state-space and “to visit” can be relaxed in “to pass as close as possible”. The agents can also give up some targets, paying a suitable instantaneous cost. Depending on the targets already visited, the population is then split into several populations, with a transfer between them. The problem is recast in the framework of optimal stopping/optimal switching problems, with the add of a continuum of agents.  Motivations can be found in models of traffic/pedestrian flows and congestion.

The subject is based on an ongoing research project with Adriano Festa and Luciano Marzufero.

A. Barrasso slides

Title: Mean field games with controlled diffusion coefficient, and McKean-Vlasov second order backward SDEs.

Abstract: I will consider a mean field game (MFG) with common noise in which the diffusion coefficients may be controlled, and discuss an existence result, which states that this MFG problem admits a weak relaxed solution under some continuity conditions on the coefficients.

I will then introduce a characterization of the solution of this MFG (when there is no common noise) through a McKean-Vlasov type second order backward SDE.

C. Bertucci slides

Title : Master equation for the planning problem : the finite state space case

Abstract : We show how we can characterize the solution of the master equation associated to the mean field planning problem. This step is fundamental to be able to address common noise for the planning problem. This talk will present some key regularizing estimates on the solution of the master equation, as well as the characterization of the singularity at the terminal time which arise in the planning problem case. This talk will be concerned with the finite state space case. This is a joint work with Jean-Michel Lasry and Pierre-Louis Lions.

A. Cecchin slides

Title: Convergence for finite state mean field control problems

Abstract: We examine mean field control problems (MFCP) on a finite state space and characterize the value function as the unique viscosity solution of a HJB equation in the simplex. In absence of any convexity assumption, we exploit this characterization to prove convergence, as $N$ grows, of the value functions of the centralized $N$-agent optimal control problem to the limit MFCP value function, with a convergence rate of order $1/\sqrt{N}$. Then, assuming convexity, we show that the limit HJB  admits a smooth solution and establish convergence of the $N$-agent optimal trajectories to the unique limiting optimal trajectory, with an explicit rate.

A. Festa slides

Title : A mean field control approach to deep learning

Abstract : Deep learning is one of the most rapidly developing areas in modern artificial intelligence. It refers to a class of machine learning techniques that have dramatically improved the state-of-the-art in speech recognition, visual object recognition, object detection and many other domains such as drug discovery and genomics. In this talk, we deal with the dynamical systems and optimal control perspective to deep neural networks (DNNs) and we consider the supervised learning paradigm, that fits naturally with this viewpoint. In particular, the forward propagation of the input variables in a DNN can be viewed as the discretization of a continuous-time dynamical system, with the parameters of the DNN playing the role of control variables. In this way, deep learning training processes can be recast as optimal control problems and can be solved by using techniques developed for differential equations and optimal control theory. The large amount of data suggest the use of a mean field control approach to the optimal control problems derived: this comes from the fact that the parameters of the DNN control not one, but an entire distribution of inputs and outputs. We tried to solve a binary classification problem by performing simulations based on a discrete version of the DPP. We present some preliminary results for classification problems..

J. Graber slides

Title : Control on a Hilbert space and application to mean field type control

Abstract : We offer a new proof of the existence and uniqueness of classical solutions to the Bellman equation for a mean field type control problem using only techniques from optimal control on a Hilbert space. This is done using a novel "lifting" procedure, which is different from that introduced by P.-L. Lions. We also apply our result to derive solutions to the corresponding master equation, which can also be interpreted as a potential mean field game.

A. Hilbert

Title : A Mean-field Game for an Electricity Grid with Storage Allowing for Jumps

Abstract : We consider a model for an Electricity network with distributed local power generation and storage. Each consumer can buy or sell electricity, where the decision depends on the instantaneous production rate and demand, which in turn has an effect on the electricity spot price. This system is modeled as a network consisting of a large number of nodes, where each node is characterized by its electricity consumption and electricity production (e.g. photovoltaic panels) and manages a local storage device. The dynamics is modelled by a jump-diffusion process. The objective at each node is to minimize energy and storage costs by optimally controlling the storage device. We analyze an N player game where the interaction takes place through a spot price mechanism and we show that it provides a Nash-equilibrium for an N-player game. Existence and uniqueness of an associated general class of RBDSDE and a maximum principle are shown. The work generalizes a paper by Alasseur Matoussi and allows for jump dynamics

I. Chowdhury

Title : Nonlocal mean-field game system associated to controlled jump diffusion

Abstract : In this talk, we address a version of non-local mean field game where the system involves nonlinear jump-diffusion term (associated to controlled SDE driven by pure jump Lévy process). We study the well-posedness results of the system and of the individual equations (Hamilton-Jacobi-Bellman and Fokker-Planck) in two situations: either for degenerate case (complete control) - by considering the singularity of the Lévy operator to be small, or for non-degenerate case (partial control) - by allowing the Lévy operator to be more singular. This is a joint work with Espen Jakobesn and Milosz Krupski.

C. Mendico slides

Title : Mean Field Games problems for linear control system and ergodic behavior of Mean Field Games problems depending on acceleration

Abstract : The aim of this talk is to present the recent results obtained in collaboration with P. Cannarsa and P. Cardaliaguet concerning the study of MFG problems in which the dynamics of each agents is given by a controlled linear differential equation and the analysis of the long time behavior of solutions for MFG problems depending on the acceleration of each players, respectively. In particular, in the first part of this talk I will concentrate on the existence and uniqueness of mild solutions of MFG problems for linear control systems and their relations with the classical weak solutions of the PDE systems. In the second part, I will describe the results we are obtaining regarding the ergodic behavior of MFG problems depending on acceleration with particular attention to the main differences we encountered with respect to the literature known by now on this subject.

S. Munoz slides

Title : Classical and weak solutions to local first order mean field games through elliptic regularity

Abstract : I will present new results about the regularity and well posedness of the local, first-order forward-backward Mean Field Games system, assuming a polynomially growing cost function and a Hamiltonian of quadratic growth. I consider systems and terminal data which are strictly coercive in the density and study two different regimes depending on whether there exists a lower bound for the running cost function. The work relies on a transformation due to P.-L. Lions which gives rise to an elliptic partial differential equation with oblique boundary conditions, that is strictly elliptic when the coupling is unbounded below. In this case, I prove the existence of a classical solution. When the problem is degenerate elliptic, I obtain existence and uniqueness of weak solutions analogous to those obtained by P. Cardaliaguet and P.J. Graber for the problem with a terminal condition that is independent of the density.

L. Pfeiffer slides

Title : An existence result for a class of mean-field games of controls

Abstract : The talk will focus on a mean field game model involving an endogenous price variable, impacting the cost functional of each agent and related to the distribution of the controls of the population of agents. The model can be seen as a dynamic version of the classical Cournot competition model, with infinitely many agents. I will present an existence result, based on a fixed-point approach, and a duality result.

Reference: J.F. Bonnans, S. Hadikhanloo, L. Pfeiffer, Schauder estimates for a class of mean-field games of controls, Appl. Math. Optim, online first.

M. Ricciardi slides

Title : The Master Equation in a Bounded Domain with Neumann Conditions

Abstract : In this talk we study the existence and uniqueness of solutions for the first-order Master Equation with Neumann conditions, and we use it to prove, under suitable assumptions, the convergence of the Nash system for the N-players game towards the solution of the Master Equation. The talk is clearly inspired to the ideas of Cardaliaguet, Delarue, Lasry, Lions, who studied the periodic case, but many technicalities have to be handled in order to take care of the boundary of the state space ?. This leads to a deep study of the estimations up to the boundary, and a completely new Neumann condition with respect to the measure appears in the formulation of the Master Equation.

D. Tonon slides

Title : Mean Field Schrödinger problem

Abstract : In this talk we introduce the Mean Field Schrödinger problem (MFSP), that is the problem of finding the most likely evolution of a cloud of interacting Brownian particles conditionally on observations. Using classical results on the representation of entropy functionals, one can in many cases recast the MFSP as a mean field control problem whose optimality conditions are expressed by a PDE system consisting in a Fokker-Planck equation coupled with a Hamilton Jacobi Bellman equation with boundary marginal constraints. In this sense MFSP shows some analogies with Mean Field games problems. However, the coupling terms are often of different nature from those covered in the MFG literature, thus posing new challenges. In addiction to this mean field control interpretation, the optimal value of MFSP can be formally seen in duality with the solution to an Hamilton Jacobi Bellman equation on the space of probability measures; we will present the first steps in the study of the well-posedness of such equation.

M.-T. Wolfram slides

Title : Mean field games vs. best reply strategy

Abstract : Mean field games (MFG) and the best reply strategy describe the competitive dynamics of large interacting agents systems. The latter can be interpreted as an approximation of the respective MFG system. In this talk we highlight the differences of the two approaches and discuss their analysis in the stationary case. Furthermore we illustrate and compare solutions in specific situations and with computational experiments.

X. Yang slides

Title: The Hessian Riemannian Flow and Newton's Method for Effective Hamiltonians and Mather Measures

Abstract: Effective Hamiltonians arise in several problems, including homogenization of Hamilton--Jacobi equations, nonlinear control systems, Hamiltonian dynamics, and Aubry--Mather theory. In Aubry--Mather theory, related objects, Mather measures, are also of great importance. Here, we combine ideas from mean-field games with the Hessian Riemannian flow to compute effective Hamiltonians and Mather measures simultaneously. We prove the convergence of the Hessian Riemannian flow in the continuous setting. For the discrete case, we give both the existence and the convergence of the Hessian Riemannian flow. In addition, we explore a variant of Newton's method that greatly improves the performance of the Hessian Riemannian flow. In our numerical experiments, we see that our algorithms preserve the non-negativity of Mather measures and are more stable than related methods in problems that are close to singular. Furthermore, our method also provides a way to approximate stationary MFGs.

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