Ergodic Mean Field Game Research Articles

Ergodic mean field games: existence of local minimizers up to the Sobolev critical case

We investigate the existence of solutions to viscous ergodic Mean Field Games systems in bounded domains with Neumann boundary conditions and local, possibly aggregative couplings. In particular we exploit the associated variational structure and search for constrained minimizers of a suitable functional. Depending on the growth of the coupling, we detect the existence of global minimizers in the mass subcritical and critical case, and of local minimizers in the mass supercritical case, notably up to the Sobolev critical case.

Stationary Discounted and Ergodic Mean Field Games with Singular Controls

We study stationary mean field games with singular controls in which the representative player interacts with a long-time weighted average of the population through a discounted and ergodic performance criteria. This class of games finds natural applications in the context of optimal productivity expansion in dynamic oligopolies. We prove the existence and uniqueness of the mean field equilibria, which are completely characterized through nonlinear equations. Furthermore, we relate the mean field equilibria for the discounted and ergodic games by showing the validity of an Abelian limit. The latter also allows us to approximate Nash equilibria of—so far unexplored—symmetric N-player ergodic singular control games through the mean field equilibrium of the discounted game. Numerical examples finally illustrate in a case study the dependency of the mean field equilibria with respect to the parameters of the games. Funding: The authors acknowledge financial support from the Alan Turing Institute and CMAP, École Polytechnique during this project. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 317210226 – SFB 1283.

Weak KAM Approach to First-Order Mean Field Games with State Constraints

We study the asymptotic behavior of solutions to the constrained MFG system as the time horizon T goes to infinity. For this purpose, we analyze first Hamilton–Jacobi equations with state constraints from the viewpoint of weak KAM theory, constructing a Mather measure for the associated variational problem. Using these results, we show that a solution to the constrained ergodic mean field games system exists and the ergodic constant is unique. Finally, we prove that any solution of the first-order constrained MFG problem on [0, T] converges to the solution of the ergodic system as T goes to infinity.

Ergodic behavior of control and mean field games problems depending on acceleration

The goal of this paper is to study the long time behavior of solutions of the first-order mean field game (MFG) systems with a control on the acceleration. The main issue for this is the lack of small time controllability of the problem, which prevents to define the associated ergodic mean field game problem in the standard way. To overcome this issue, we first study the long-time average of optimal control problems with control on the acceleration: we prove that the time average of the value function converges to an ergodic constant and represent this ergodic constant as a minimum of a Lagrangian over a suitable class of closed probability measure. This characterization leads us to define the ergodic MFG problem as a fixed-point problem on the set of closed probability measures. Then we also show that this MFG ergodic problem has at least one solution, that the associated ergodic constant is unique under the standard monotonicity assumption and that the time-average of the value function of the time-dependent MFG problem with control of acceleration converges to this ergodic constant.

Long-Time Behavior of First-Order Mean Field Games on Euclidean Space

The aim of this paper is to study the long time behavior of solutions to deterministic mean field games systems on Euclidean space. This problem was addressed on the torus ${\mathbb T}^n$ in [P. Cardaliaguet, {\it Long time average of first order mean field games and weak KAM theory}, Dyn. Games Appl. 3 (2013), 473-488], where solutions are shown to converge to the solution of a certain ergodic mean field games system on ${\mathbb T}^n$. By adapting the approach in [A. Fathi, E. Maderna, {\it Weak KAM theorem on non compact manifolds}, NoDEA Nonlinear Differential Equations Appl. 14 (2007), 1-27], we identify structural conditions on the Lagrangian, under which the corresponding ergodic system can be solved in $\mathbb{R}^{n}$. Then we show that time dependent solutions converge to the solution of such a stationary system on all compact subsets of the whole space.

Ergodic mean field games with H\xf6rmander diffusions

We prove existence of solutions for a class of systems of subelliptic PDEs arising from mean field game systems with Hörmander diffusion. These results are motivated by the feedback synthesis mean field game solutions and the nash equilibria of a large class of N-player differential games.

On Quasi-stationary Mean Field Games Models

We explore a mechanism of decision-making in mean field games with myopic players. At each instant, agents set a strategy which optimizes their expected future cost by assuming their environment as immutable. As the system evolves, the players observe the evolution of the system and adapt to their new environment without anticipating. With a specific cost structures, these models give rise to coupled systems of partial differential equations of quasi-stationary nature. We provide sufficient conditions for the existence and uniqueness of classical solutions for these systems, and give a rigorous derivation of these systems from N-players stochastic differential games models. Finally, we show that the population can self-organize and converge exponentially fast to the ergodic mean field games equilibrium, if the initial distribution is sufficiently close to it and the Hamiltonian is quadratic.

Bifurcation and segregation in quadratic two-populations mean field games systems

We search for non-constant normalized solutions to the semilinear elliptic system where ν > 0, Ω ⊂ RN is smooth and bounded, the functions gi are positive and increasing, and both the functions vi and the parameters λi are unknown. This system is obtained, via the Hopf−Cole transformation, from a two-populations ergodic Mean Field Games system, which describes Nash equilibria in differential games with identical players. In these models, each population consists of a very large number of indistinguishable rational agents, aiming at minimizing some long-time average criterion. Firstly, we discuss existence of nontrivial solutions, using variational methods when gi(s) = s, and bifurcation ones in the general case; secondly, for selected families of nontrivial solutions, we address the appearing of segregation in the vanishing viscosity limit, i.e.

On solutions of mean field games with ergodic cost

A general class of mean field games are considered where the governing dynamics are controlled diffusions in Rd. The optimization criterion is the long time average of a running cost function. Under various sets of hypotheses, we establish the existence of mean field game solutions. We also study the long time behavior of the mean field game solutions associated with the finite horizon problem, and under the assumption of geometric ergodicity for the dynamics, we show that these converge to the ergodic mean field game solution as the horizon tends to infinity. Lastly, we study the associated N-player games, show existence of Nash equilibria, and establish the convergence of the solutions associated to Nash equilibria of the game to a mean field game solution as N→∞.

The Derivation of Ergodic Mean Field Game Equations for Several Populations of Players

This note contains a detailed derivation of the equations of the recent mean field games theory (abbr. MFG), developed by M. Huang, P.E. Caines, and R.P. Malhame on one hand and by J.-M. Lasry and P.-L. Lions on the other, associated with a class of stochastic differential games, where the players belong to several populations, each of which consisting of a large number of similar and indistinguishable individuals, in the context of periodic diffusions and long-time-average (or ergodic) costs. After introducing a system of N Hamilton–Jacobi–Bellman (abbr. HJB) and N Kolmogorov–Fokker–Planck (abbr. KFP) equations for an N-player game belonging to such a class of games, the system of MFG equations (consisting of as many HJB equations, and of as many KFP equations as is the number of populations) is derived by letting the number of the members of each population go to infinity. For the sake of clarity and for reader’s convenience, the case of a single population of players, as formulated in the work of J.-M. Lasry and P.-L. Lions, is presented first. The note slightly improves the results in this case too, by dealing with more general dynamics and costs.

Long Time Average of Mean Field Games with a Nonlocal Coupling

We study the long time average, as the time horizon tends to infinity, of the solution of a mean field game system with a nonlocal coupling. We show an exponential convergence to the solution of the associated stationary ergodic mean field game. Proofs rely on semiconcavity estimates and smoothing properties of the linearized system.