Special Session 22: Recent advances in mean field games for crowd dynamics

A Stationary First-Order Mean-Field Games with Novel Mixed Boundary Conditions

AbdulRahman M Alharbi Islamic University at Al-Madinah / King AbdUllah University of Science and Technology Saudi Arabia Co-Author(s):    AbdulRahman, Diogo Gomes, and Yuri Ashrafyan

Abstract: In this presentation, I will discuss recent joint work with Diogo Gomes and Yuri Ashrafyan, where we investigate a first-order mean-field game (MFG) model with a novel mixed boundary condition. The new boundary conditions split the boundary into two parts: an entrance with an inflow Neumann boundary condition and an exit with a relaxed Dirichlet condition and a relaxed outflow Neumann condition. We further impose an auxiliary contact-set condition on the exit portion of the boundary that links the other two conditions. This approach offers three advantages. It indicates the exit/entry regions of the boundary based on the general structure without the need to know the exact values of boundary data, addresses the lack of uniqueness issues associated with Neumann boundary conditions, and prevents the artificial inflow of virtual agents through the exit boundary caused by an excessively high Dirichlet condition. The interior behavior of our model adheres to the standard MFG structure, consisting of a coupled system of a first-order separable Hamilton-Jacobi equation and a stationary transport equation. We exploit the separability of the Hamiltonian to establish a corresponding variational formulation for the MFG, which we use to prove the existence of solutions and the uniqueness of the gradient of the value function.

Nonseparable mean field games with pseudomeasure initial distributions

David M. Ambrose Drexel University USA Co-Author(s):

Abstract: In order to use solutions of the mean field games PDE system to control solutions of N-player games as N goes to infinity, we must be able to take initial distributions which are the sum of Dirac masses. Relatedly, the natural space of data is probability measures. In a number of works (including those by the speaker), instead, subsets of the set of probability measures are used, such as probability measures induced by smooth functions. In this work, we instead take a larger space of data than probability measures, considering pseudomeasures as initial distributions. We give a class of nonseparable Hamiltonians for which we can prove existence of solutions of the mean field games PDE system with pseudomeasure initial data. This includes joint work with Milton Lopes Filho, Anna Mazzucato, and Helena Nussenzveig Lopes.

On quasi-stationary Mean Field Games of Controls

Fabio Camilli Sapienza Universita` di Roma Italy Co-Author(s):    Claudio Marchi

Abstract: We consider quasi-stationary Mean Field Games of Controls. In these problems, the strategy-choice mechanism of the agent differs from the classical one: the generic agent cannot predict the evolution of the population and instead chooses its strategy based solely on the information available at the current moment, without anticipating future developments. Furthermore, the dynamics of an individual agent is influenced not only by the distribution of agents but also by the distribution of their optimal strategies. We demonstrate the existence and uniqueness of the solution for the corresponding quasi-stationary Mean Field Games system under various sets of hypotheses and provide examples of models that fall within these parameters.

Algorithm for Deterministic Mean Field Games

Elisabetta Carlini Sapienza University Italy Co-Author(s):

Abstract: We propose a numerical scheme to approximate deterministic Mean Field Games based on semi-Lagrangian and Lagrange-Galerkin methods. We discuss a convergence result of the nonlinear discrete system, obtained by discretization, to the MFGs system in general dimension. We also propose a Policy Algorithm to solve the nonlinear discrete system and present some numerical results.

Degenerate Fully Nonlinear Mean Field Game with Nonlocal Diffusion

Indranil Chowdhury Indian Institute of Technology, Kanpur India Co-Author(s):    Espen R. Jakobsen and Milosz Krupski

Abstract: We consider a strongly degenerate and fully nonlinear MFG system. Our MFG involves a controlled pure jump (nonlocal) Levy diffusion of order less than one, and monotone, smoothing couplings. We study the existence and uniqueness results of such problems. We discuss the key difficulty in obtaining uniqueness for the corresponding Fokker--Planck equation which has degenerate and low regularity diffusion coefficients: since the regularity of the coefficient and the order of the diffusion are interdependent, it holds when the order is sufficiently low.

Weak-strong uniqueness for solutions to MFGs

Rita Ferreira KAUST Saudi Arabia Co-Author(s):    Diogo Gomes and Vardan Voskanyan

Abstract: In this talk, we address the question of uniqueness of weak solutions for stationary first-order Mean-Field Games (MFGs). Despite well-established existence results, establishing uniqueness, particularly for weaker solutions in the sense of monotone operators, remains an open challenge. Building upon the framework of monotonicity methods, we introduce a linearization method that enables us to prove a weak-strong uniqueness result for stationary MFG systems on the $d$-dimensional torus. In particular, we give explicit conditions under which this uniqueness holds.

A mean-field-game approach to overfishing

Ziad Kobeissi Inria Saclay, CentraleSupelec, University Paris-Saclay France Co-Author(s):    Idriss Mazari-Fouquer, Domenec Ruiz-Ballet

Abstract: In this presentation, we propose a novel model for managing fisheries, described by a system of three coupled partial differential equations. The first is a reaction-diffusion equation representing the dynamics of the fish population, which follows standard approaches in the mathematical literature on spatial ecology. The other two equations are derived using a mean-field-game framework to model a large population of fishermen, where the number of fishermen is assumed to be large enough to be treated as infinite. Each fisherman aims to maximize their individual profit, calculated as the revenue from selling fish minus the cost of moving their boat. Under two different structural assumptions about the nonlinearities in the fish dynamics, we prove theoretical results illustrating the tragedy of the commons. Specifically, we show that a lack of coordination among fishermen can significantly harm, or even lead to the extinction of, the fish population. Our findings are supported by several numerical simulations.

One`s experience vs. population`s knowledge in mean field games and control

Hicham Kouhkouh University of Graz Austria Co-Author(s):

Abstract: I shall present a novel viewpoint on ergodic Mean-Field Games and ergodic Mean-Field Control (or control of McKean-Vlasov processes), showing that an equilibrium and an optimal strategy of a large number of interacting agents can be obtained by one ``experienced`` agent.

Machine Learning For Master Equations in Mean Field Games

Mathieu Lauriere NYU Shanghai Peoples Rep of China Co-Author(s):    Asaf Cohen, Ethan Zell

Abstract: Mean field games have been introduced to study games with many players. Since their introduction, they have found numerous potential applications and the theory has been extensively developed. While forward-backward systems of partial or stochastic differential equations can be used to characterize Nash equilibria with a fixed initial distribution, the Master equation introduced by P.-L. Lions provides a tool to solve the problem globally, for any initial condition. However this equation is a partial differential equation posed on the space of measures, which raises significant challenges to solve it numerically. In this talk, we will present several computational methods that have been proposed to tackle Master equations. Theoretical convergence results and numerical experiments will be presented. Mostly based on joint work with Asaf Cohen and Ethan Zell.

Minimal solutions of master equations for extended mean field games

Chenchen Mou City University of Hong Kong Peoples Rep of China Co-Author(s):    Jianfeng Zhang

Abstract: 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. This is based on a joint work with Jianfeng Zhang.

Monotone inclusion methods for a class of second-order non-potential mean-field games

Levon Nurbekyan Emory University USA Co-Author(s):    Siting Liu (UCR), Yat Tin Chow (UCR)

Abstract: We propose a monotone splitting algorithm for solving a class of second-order non-potential mean-field games. Following [Achdou, Capuzzo-Dolcetta, Mean Field Games: Numerical Methods, SINUM (2010)], we introduce a finite-difference scheme and observe that the scheme represents first-order optimality conditions for a primal-dual pair of monotone inclusions. Based on this observation, we prove that the finite-difference system obtains a solution that can be provably recovered by an extension of the celebrated primal-dual hybrid gradient (PDHG) algorithm.

Analysis and Numerical Approximation of Mean Field Game Partial Differential Inclusions

Yohance Osborne Durham University England Co-Author(s):    Iain Smears

Abstract: The Mean Field Game (MFG) system of Partial Differential Equations (PDE), introduced by Lasry \& Lions in 2006, models Nash equilibria of large population stochastic differential games of optimal control where the players of the game have unique optimal controls, and the convex Hamiltonian of the underlying optimal control problem is differentiable. In this talk, we introduce a new class of model problems called Mean Field Game Partial Differential Inclusions (MFG PDI), which extend the MFG system of Lasry and Lions to situations where players may have possibly nonunique optimal controls, and the resulting Hamiltonian of the underlying optimal control problem is not required to be differentiable. We prove the existence of unique weak solutions to MFG PDI for a broad class of Hamiltonians that are convex, Lipschitz, but possibly nondifferentiable, under a monotonicity condition similar to one considered previously by Lasry \& Lions. Moreover, we introduce a class of monotone finite element discretizations of the weak formulation of MFG PDI and present theorems on the strong convergence of the approximations to the value function in the $L^2(H_0^1)$-norm and the strong convergence of the approximations to the density function in $L^p(L^2)$-norms. We conclude the talk with discussion of numerical experiments involving non-smooth solutions.

Stability Analysis of a Non-Separable Mean-Field Games for Pedestrian Flow in Large Corridors

Eliot Pacherie CNRS & Cergy University France Co-Author(s):    Mohamed Ghattassi and Nader Masmoudi

Abstract: We investigate the existence and stability of small perturbations of constant states of the generalized Hughes model for pedestrian flow in an infinitely large corridor. We show that constant flows are stable under a condition on the density. Our findings indicates that when the density is less than half of the maximum density, which is the Lasry-Lions monotonicity condition, we can control the perturbation and prove postive stability results for the nonlinear Generalized Hughes model.

Existence of Solutions to MFG Problems via Monotone Operators

Melih Ucer King Abdullah University of Science and Technology Saudi Arabia Co-Author(s):    Rita Ferreira, Diogo Gomes

Abstract: The theory of monotone operators is the basis of a standard method of proving existence of solutions to Dirichlet problems involving a second-order elliptic PDE of the divergence form. In this context, it generalizes the direct method in the calculus of variations as well as the Lax-Milgram theorem. On the other hand, presence of a monotonicity structure in mean field games has been known since the early papers of Lions et al. Moreover, this monotonicity property was exploited by Ferreira et al for proving existence of solutions in a weak sense, to various MFG problems. More recently, in ongoing joint work with Ferreira and Gomes, we discovered how to systematize the aforementioned existence proofs using the abstract machinery of the monotone operator theory. Consequently, we achieve simpler and unified proofs for a wider range of problems. Furthermore, with new a priori bounds based on a novel idea, we obtain solutions in a stronger sense.