The Nobel Laureate Thomas Schelling studied the behaviour of different ethnic groups in american cities in the 60s and used simple simulations to show that a mild preference of each person for not belonging to a too small minority in his/her neighborhood leads to a population distribution where each neighborhood is inhabited mostly by a single group: the segregation phenomenon. We propose some mean field games models of two populations each minimizing a functional that describes the preferences observed by Schelling. We show some existence results for the stationary as well as the evolutive system of PDEs with Neumann boundary conditions. Uniqueness is not expected in these models, as it is easy to exhibit examples with multiple equilibria. We present numerical simulations obtained by various methods. They show some form of segregation if the noise affecting the dynamics is small, but also more complicated and unstable behavior for intermediate noise intensities. This is joint work with Marco Cirant (Università di Milano) and Yves Achdou (Université Paris-Diderot).

Since its introduction by P.L. Lions in his lectures and seminars at the College de France, see also the very helpful notes of Cardialaguet on Lions' lectures, the Master equation has attracted a lot of interest, and various points of view have been expressed, see Carmona-Delarue, Bensoussan-Frehse-Yam , Buckdahn-Li-Peng-Rainer . There are several ways to introduce this type of equation. It involves an argument which is a probability measure, and P.L. Lions has proposed the idea of working with the Hilbert space of random variables which are square integrable. So writing the equation is an issue. Another issue is its origin. We discuss in this paper these various aspects, and for the modeling rely heavily on a seminar at College de France delivered by P.L. Lions on November 14, 2014.

Many problems from mass transport can be reformulated as variational problems under a prescribed divergence constraint (static problems) or subject to a time dependent continuity equation which again can also be formulated as a divergence constraint but in time and space. The variational class of Mean-Field Games introduced by Lasry and Lions may also be interpreted as a generalisation of the time-dependent optimal transport problem. Following Benamou and Brenier, we show that augmented Lagrangian methods are well-suited to treat convex but nonsmooth problems and apply to variational Mean Field Games.

The talk extends Peng's stochastic maximum principle from classical stochastic control problems to those in which the coefficients of the dynamics of the controlled state process do not only depend on the state process and the control themselves but also on the law of the control state process. The characterisation of the optimal control, which is obtained, extends the corresponding result by Shige Peng and contains also extra-terms coming from the mean-field character of the stochastic control problem. Joint work with Juan Li (SDU, Weihai), Jin Ma (USC, Los Angeles).

Dynamic games with a large population of minor agents and one major agent are considered where the minor agents partially observe the state of the major agent. It is shown that the epsilon-Nash equilibrium property holds in this MFG scenario with the best response control actions of each minor agent depending upon the conditional density generated by a non-linear filter for the major agent’s state. Work with Nevroz Sen.

We study a stationary Mean Field Game system defined on a network. We motivate the transition conditions we consider at the vertices and we prove existence and uniqueness of a smooth solution to the system. We also consider a numerical scheme for the problem: in this framework a correct approximation of the transition conditions at the vertices plays a crucial role. We prove existence, uniqueness and convergence of the scheme and we show some numerical experiments (in collaboration with S.Cacace (Roma) and C.Marchi (Padova)).

We recast several models of bank runs which appeared in the finance literature as Mean Field Games of Timing. We propose a mathematical framework for these games, and present some existence and approximation results.

We propose a mean field kinetic model for systems of rational agents interacting in a game theoretical framework. This model is inspired from non-cooperative anonymous games with a continuum of players and Mean-Field Games. The large time behavior of the system is given by a macroscopic closure with a Nash equilibrium serving as the local thermodynamic equilibrium. Applications of the presented theory to social and economical models will be given.

We here discuss the construction of a classical solution to the master equation associated with a mean-field game, both without and with a common noise, under the assumption that the Larry Lions monotonicity condition holds true. Then we investigate the convergence of the Nash equilibria of the N-player-game to the solution of the mean-field game. The key point is to let the solution of the master equation act onto the empirical distribution of the system formed by the N players in equilibrium: this provides an approximated solution to the N Nash system. Taking benefit of the regularity of the solution of the master equation, we manage to estimate the distance with the true solution of the Nash system. Based on two joints works with J.F. Chassagneux and D. Crisan; and P. Cardaliaguet, J.M. Lasry and P.L. Lions.

Mean field games arise as limit models for symmetric N-player games with interaction of mean field type when the number of players N tends to infinity. The limit relation is often understood in the sense that a solution of a mean field game allows to construct approximate Nash equilibria for the corresponding N-player games. The opposite direction is of interest, too: When do sequences of Nash equilibria converge to solutions of an associated mean field game? I will discuss recent rigorous results in this direction for finite-horizon systems in the framework of stochastic open-loop controls.

We consider time-dependent Bellman systems arising as necessary conditions for Nash-points of Vlasov-Mc-Kean-Functionals under Neumann-conditions. The pay off functions grow quadratically in the controls and may depend on the field variable ; any power of growth of the pay offs is allowed with respect to the field variable as long domination by sum coerciveness is possible. In this case we obtain global smooth solutions for smooth data . The analytical difficulty is mainly how to obtain a bound for the essential sup of approximations to the bellman system. An unexpected rank condition is needed for the dynamics. (Joint work with A.Bensoussan, Dominique Breit and Jens Frehse)

We construct a small time strong solution to a nonlocal Hamilton{Jacobi equation introduced by Lions, the so-called master equation, originating from the theory of Mean Field Games. We discover a link between metric viscosity solutions to local Hamilton{Jacobi equations studied independently by Ambrosio{Feng and G{Swiech, and the master equation. As a consequence we recover the existence of solutions to the First Order Mean Field Games equations, rst proved by Lions. We make a more rigorous connection between the master equation and the Mean Field Games equations. (This talk is based on a joint work with A. Swiech).

Long term dynamic of mining industries

In the present talk, we discuss monotonicity methods for mean-field games. We suggest a new definition of weak solution, whose existence can be proven under general assumptions. Then, we discuss various properties of these weak solutions. Finally, we present applications to the numerical approximation of mean-field games.

We consider continuous time mean field consumption-accumulation games. The capital stock of each agent evolves according to the Cobb-Douglas production function subject to consumption and stochastic depreciation. The individual HARA-type utility depends on both the own consumption and relative consumption. Under some standard model parameters, we analyze the fixed point problem of the mean field game and specify it by use of a set of ordinary differential equations. The individual strategy is obtained as a linear feedback with the gain reflecting the collective behavior of the population. (Joint work with Son Luu Nguyen of University of Puerto Rico)

Games with common noise are attentively studied now by many authors.We show under certain conditions that a solution to the forward-backward limiting system of the mean-field games with common noise provides an $1/N$-Nash equilibrium for the approximating games with $N$ players. Two additional technical ideas are used as compared to the standard case without noise: interpretation of the common noise as a certain generalized binary interaction and the Kunita theory of stochastic characteristics.

Economics of mining industries : MFG approach

MFG equations in infinite dimension approach

The high levels of variability and unreliability of renewable energy sources such as wind and solar energy, act as one of the primary obstacles to their massive adoption for generation in power systems. In this context, availability of energy storage can help mitigate the increasing mismatches between load and generation that would result from their generalized use, and would help limit reliance on more controllable but environmentally damaging fossil based energy sources for such purpose. We consider the potential organization of the dispersed energy storage naturally present in power systems such as found in the thermal inertia of electrically heated or cooled residential and commercial buildings, but also in electric water heaters and refrigerators for example, into a giant ?leaky battery? resource. The challenges are multiple, not the least of which is the presence of potentially millions of heterogeneous control points where monitoring and actuation would be required in a classical centralized view of the control problem. Instead we show how a linear quadratic Mean Field Game formalism provides a natural tool for decentralization of the controls. The theory is applied to a diffusion model of heating and cooling loads, and both non cooperative and social solutions are presented with a mix of theoretical and numerical results. This is joint work with Arman Kizilkale.

The purpose is to get mathematicians interested in studying a number of partial differential equations (PDEs) that naturally arise in macroeconomics. These PDEs come from models designed to study some of the most important questions in economics. At the same time, they are highly interesting for mathematicians because their structure is often quite difficult. We present a number of examples of such PDEs, discuss what is known about their properties, and list some open questions for future research.

The regularity of solutions to mean field games systems can hardly be established for general local couplings, depending on the pointwise value of the distribution density. In this case, it is often necessary to work with weak solutions. In this talk I will discuss the well-posedness of those systems in the weak setting, explaining how this is related with new results on Fokker-Planck equations through a characterization of weak and renormalized solutions. Applications concern several issues, like the convergence of numerical schemes or the characterization of solutions to the planning problem. I will also discuss the vanishing viscosity limit and the uniqueness for the relaxed weak formulation in the first order case.

The question of how to replace density penalizations with density constraints of the form $\rho\leq 1$ in MFG is a tricky one, in particular if one wants a meaningful notion of equilibrium. In the first MFG meeting in Rome some years ago, I proposed a model, where the movement was affected by the gradient of a pressure, endogenously created by the action of the other player, where they saturate the density constraint. This model is only formal, and no successful study has been possible so far. Moreover, we suspect it to be non-variational. In a joint work with P. Cardaliaguet (Paris-Dauphine) and A. Mészaros (Paris-Sud), we study a different model, which is the one that we obtain as a limit $m\to\infty$ when we add a penalization with the $m$-th power of the density. It gives rise to a Benamou-Brenier-type optimization problem subject to the constraint $\rho\leq 1$, and admits a dual problem where a pressure $p$ appears. The pressure also plays a role in the equilibrium, and $p(t,x)$ is, at least formally, the price that agents have to pay to pass through $x$ at time $t$. It is non-negative, and vanishes where the constraint is not saturated. The main difficulty is that $p$ is a priori only a measure, and it does not make sense to integrate it along a trajectory. A precise treatise requires regularity results very weak but sufficient to provide a meaningful and rigorous notion of trajectorial cost: borrowing techniques developed by Y. Brenier and then L. Ambrosio and A. Figalli for incompressible Euler equations, we can prove that $p$ is $L^2_{loc}$ in time, valued in $BV$ in space.

Vaccination as MFG equilibriums