Tag - Mean-field game theory

In this talk I will describe how to construct global in time classical solutions to the master equation arising in mean field games. Our method works for a general class of non-separable Hamiltonians and final data that satisfy a suitable monotonicity condition. This stems from the so-called displacement convexity condition introduced and used successfully in the theory of optimal mass transportation. Our results hold true independently of the intensity of the idiosyncratic noise.

This talk is devoted to the convergence problem of the Nash Equilibria in an N-player differential game towards the optimal strategies in the Mean-Field Games framework. The novelty here is that the dynamic of the generic player includes a reflection process which guarantees the invariance of the state space Ω. This implies that the MFG system presents Neumann boundary conditions for the value function u and the density of the population m. The first part of the talk is devoted to the study of the well-posedness of the Master Equation, essential tool in order to study the convergence problem. The reflection process in the N-players game leads to two Neumann conditions in the Master Equation formulation. In the second part we analyse the convergence problem, borrowing and readapting the ideas from the periodic case, studied by Cardaliaguet, Delarue, Lasry, Lions. The results can also be generalized in the case of Dirichlet boundary conditions.

In an extended mean field game the vector field governing the flow of the population can be different from that of the individual player at some mean field equilibrium. This new class strictly includes the standard mean field games. It is well known that, without any monotonicity conditions, mean field games typically contain multiple mean field equilibria and the wellposedness of their corresponding master equations fails. In this paper, a partial order for the set of probability measure flows is proposed to compare different mean field equilibria. The minimal and maximal mean field equilibria under this partial order are constructed and satisfy the flow property. The corresponding value functions, however, are in general discontinuous. We thus introduce a notion of weak-viscosity solutions for the master equation and verify that the value functions are indeed weak-viscosity solutions. Moreover, a comparison principle for weak-viscosity semi-solutions is established and thus these two value functions serve as the minimal and maximal weak-viscosity solutions in appropriate sense. In particular, when these two value functions coincide, the value function becomes the unique weak-viscosity solution to the master equation. The novelties of the work persist even when restricted to the standard mean field games.

In this talk, we will present a work in collaboration with Z. Kobeissi and D. Ruiz-Balet where we analyse an optimal harvesting problem from the point of view of Mean Field Games. Our goal is to show that, one the one hand, a purely selfish harvesting strategies, whereby each single fisherman acts in his best interest, can drive the population to extinction while a coordinated plan of action, where fishermen would coordinate, would actually lead to a survival of the fishes’ population, and to a higher harvested yield for every single fisherman. Mathematically, we will use a travelling wave approach, focusing on a bistable model for which, when no fisherman is present, travelling wave solutions are invading. We will model the behaviour of fisherman using the MFG formalism.

I will discuss the issue of existence of solutions to viscous Mean Field Games systems in the so-called anti-monotone regime, that describe Nash equilibria in differential games involving a large population of identical players aiming at aggregating. The problem can be recast into the optimal control of a system whose state is driven by a Fokker-Planck equation. I will show the role of the aggregation strength in the existence of equilibria, which may correspond to global or local minima of a suitable functional, or their nonexistence. The stationary and the evolutive case, which correspond to long-time and fixed time horizon optimization respectively, will be discussed and compared.

First-order kinetic mean-field games formally describe the Nash equilibria of deterministic differential games where agents control their acceleration, asymptotically in the limit as the number of agents tends to infinity. The known results for the well-posedness theory of mean field games with control on the acceleration assume either that the running and final costs are regularizing functionals of the density variable, or the presence of noise, i.e. a second-order system. In this talk we describe how to construct global in time weak solutions to a first order mean-field games system involving kinetic transport operators, where the costs are local (hence non-regularizing) functions of the density variable with polynomial growth. We show the uniqueness of these solutions on the support of the agent density. The heart of the analysis is to characterize solutions through two convex optimization problems in duality. We will introduce a notion of 'reachable set', built from the initial agent distribution, that allows us to work with initial measures with or without compact support. In this way we are able to obtain crucial estimates on minimizing sequences for merely bounded and continuous initial measures. These are then carefully combined with L¹-type averaging lemmas from kinetic theory to obtain pre-compactness for the minimizing sequence. Finally, under stronger convexity and monotonicity assumptions on the data, we can prove higher-order Sobolev estimates of the solutions.

This talk is devoted to the formulation of ensemble optimal control problems governed by kinetic models. The starting point for this presentation is the Liouville equation, which is the fundamental building block of models that govern the evolution of probability and material density functions like the Fokker-Planck equation and the Boltzmann equation. In this framework, optimal control problems arise in a multitude of application fields ranging from plasma physics to pedestrians' motion, where it is required to design control mechanisms that are able to drive the underlying stochastic process or microscopic system in order to perform given tasks. It is shown that many of these tasks can be formulated in terms of ensemble (expected value) cost functionals. It is also illustrated how ensemble cost functionals allow to draw a connection between open-loop Fokker-Planck control problems and Hamilton-Jacobi-Bellman problems arising in the computation of closed-loop controls. The talk is concluded with a brief discussion on Fokker-Planck Nash games for modelling the avoidance problem in, e.g., pedestrian motion.