Special Session 184: Mean-Field Games: From Partial Differential Equations to Numerical Methods

Nonlocal mean field games with pseudomeasure or negative Sobolev initial distributions

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

Abstract: To rigorously take the limit of N-player games as N goes to infinity, one important step is to establish existence of solutions for the mean field games PDE system allowing measure-valued data. To do so, we consider two classes of initial data, each of which has probability measures as a subset. The two classes of data are the space of pseudomeasures, and Sobolev spaces with sufficiently negative regularity index. To allow such rough data, we consider a class of non-separable Hamiltonians which are regularizing with respect to the measure, but which arise naturally in a number of applications. In these Hamiltonians, the measure variable appears only in an integral over the whole spatial domain. We present existence theorems for the mean field games PDE system with these Hamiltonians and these classes of initial data. In the negative Sobolev case, this includes the first-order case, i.e. without diffusion.

Non-asymptotic estimates for differential games with large discount

Giovanni Carlon University of Padova Italy Co-Author(s):    Marco Cirant

Abstract: In this talk, I will present recent estimates for solutions to systems of PDEs arising as optimality conditions in a class of differential games with large discount, obtained in collaboration with Marco Cirant. The estimates are pointwise and uniform with respect to both the number of players $N$ and the discount factor $\lambda$, and they do not rely on any monotonicity assumptions on the cost functions of the players. These results, in turn, allow us to investigate several interesting questions about the equilibria of such games, including the size of the gap between different notions of equilibrium and the limiting behavior as $N$ or $\lambda$ tend to infinity.

Uniqueness of solutions for MFG with large discount

Elisa Continelli University of Padova Italy Co-Author(s):    Marco Cirant, Nicol\`{o} De Bernardi

Abstract: We deal with a class of Mean Field Games systems with large discount. We show that solutions to these systems are unique provided that the discount factor is sufficiently large, identifying an asymptotic uniqueness regime that falls outside the usual ones involving monotonicity.

A semi-Lagrangian scheme for First-Order Mean Field Games based on monotone operators

Valentina Coscetti University of Rome "La Sapienza" Italy Co-Author(s):    Elisabetta Carlini

Abstract: We construct a semi-Lagrangian scheme for first-order, time-dependent, and non-local Mean Field Games. The convergence of the scheme to a weak solution of the system is analyzed by exploiting a key monotonicity property. To solve the resulting discrete problem, we implement a Learning Value Algorithm, prove its convergence, and propose an acceleration strategy based on a Policy iteration method. Finally, we present numerical experiments that validate the effectiveness of the proposed schemes and show that the accelerated version significantly improves performance.

On the Local Turnpike Property in Mean Field Control and Games

Nicol\`o De Bernardi Universit\`a degli Studi di Padova Italy Co-Author(s):    Marco Cirant

Abstract: We study the local stability of solutions to ergodic and discounted Mean Field Games systems around stationary equilibria with quadratic Hamiltonians. Inspired by finite-dimensional models, we introduce a new stability assumption which allows for non-monotone local couplings, and show that this weaker condition still ensures a (local) exponential turnpike property for solutions close to the stationary one. We also provide an interpretation of this non-monotonicity assumption in terms of spectral properties of an operator, which might be encoded in the MFG system. This leads to a class of meaningful examples of MFG systems with no monotonicity but still fulfilling our new assumption. This talk is based on joint work with M. Cirant (Padova).

Lipschitz regularity of the trajectories minimizing the total variation in a congested setting

Annette Dumas Universit\`e de Limoges, XLIM France Co-Author(s):    Filippo Santambrogio

Abstract: The problem I will present is motivated by the study of housing dynamics inside a city through a Mean Field Game model. Each agent jumps to move from one place to another and minimizes a cost composed of the number of jumps and an increasing function of the density of the population. A Nash equilibrium is of the form of a probability measure on the set of individual trajectories. This probability measure minimizes a problem which depends on the probability measure itself. By using tools from optimal transport, we will see that the latter problem can be expressed in an Eulerian point of view by minimizing the total variation on the set of curves of measures. The solution to the Eulerian problem exists, is unique and is Lipschitz in time, despite the discontinuous trajectories taken by each agent. With additional hypothesis on the data, boundedness or continuity in space can be obtained with Dirichlet conditions in time. Numerical simulations are carried out on the Eulerian problem by using a splitting method called Fast Dual Proximal Gradient method for which the convergence of the iterations is guaranteed by Beck and Teboulle in 2014. The regularity results allows us to show the existence of a Nash equilibrium.

Mean field optimization problems: stability results and Lagrangian discretization

Kang LIU Institut de Mathematiques de Bourgogne France Co-Author(s):    Laurent Pfeiffer

Abstract: We formulate and investigate a convex optimization problem defined on a set of probability measures $\mu$ with prescribed marginal $m$, which we call Mean Field Optimization (MFO) problem. The cost function depends on an aggregate term, defined as the expectation of $\mu$ with respect to a contribution function. This problem is of particular interest in the context of Lagrangian potential mean field games and their discretization. We provide a first-order optimality condition and prove strong duality. We investigate stability properties of the MFO problem with respect to the prescribed marginal, from both primal and dual perspectives. In our stability analysis, we propose a method for recovering an approximate solution to an MFO problem with the help of an approximate solution to an MFO with a different marginal $m$, typically an empirical distribution. We combine this method with the stochastic Frank-Wolfe algorithm of our previous work to derive a complete resolution method.

Weak-Strong Uniqueness for Second-Order Mean-Field Games

Bashayer Majrashi KAUST Saudi Arabia Co-Author(s):    Rita Ferreira, Diogo A. Gomes, Bashayer H. Majrashi

Abstract: We extend the weak--strong uniqueness principle for mean-field game (MFG) systems to a broad class of second-order stationary and time-dependent problems. Under standard monotonicity, growth, and coercivity assumptions on the Hamiltonian and relying strictly on the integrability exponents guaranteed by the existing theory for monotone MFG systems, we show that any weak solution must coincide with a given strong solution. Our analysis covers models with spatially dependent scalar diffusion coefficients, using monotonicity arguments and a coefficient-adapted mollification strategy to manage the variable diffusion terms. We extend this strategy to establish weak--strong uniqueness in the corresponding second-order, initial--terminal, time-dependent setting. Finally, to address the critical quadratic growth regime, we derive a new a priori second-order estimate for a stationary MFG system with logarithmic coupling, quantifying the control of second spatial derivatives of $u$ weighted by $m$ and of $Dm$ in terms of the data and thereby establishing weak--strong uniqueness in this setting. Our results provide a unified framework for uniqueness and regularity in a wide array of MFG models.

Numerical approximation of first-order time-dependent, non-separable mean field games under displacement monotonicity

Yohance Osborne Durham University England Co-Author(s):    Alpar R Meszaros

Abstract: In this talk, we introduce a numerical method to approximate first-order time-dependent mean field game (MFG) systems posed in $R^d$ with non-separable, displacement monotone Hamiltonians and terminal costs, for arbitrary finite time-horizons and (possibly singular) initial player distributions with finite second moment. The numerical method is based on an implicit Euler discretization in time, together with sampling in space, of the characteristic Hamiltonian system associated with the continuous MFG system. We establish convergence of the scheme by proving an asymptotic error bound that implies optimal rates of convergence in the $L^{\infty}(L^2)$-norm for the approximations as the number of spatial samples tends to infinity jointly with the temporal time-step vanishing. We conclude the talk with numerical experiments that illustrate the performance of the scheme for a range of time horizons.

A non-asymptotic approach to games with many players and the universality of the mean field game limit

Davide Francesco Redaelli University of Rome Tor Vergata Italy Co-Author(s):    Cirant M., Jackson J.

Abstract: I will present a non-asymptotic approach to quantifying the gap between different notions of equilibrium in stochastic differential games with many players. Under suitable semi-monotonicity conditions, one can derive estimates for the Nash and Pontryagin systems that are uniform in the number of players. These estimates, in turn, yield quantitative convergence results for open-loop and closed-loop equilibria when interactions between players are weak. The results apply to games with interactions that are not necessarily symmetric and possibly much sparser than in classic mean field game theory, and they confirm the universality of the mean field game limit for games governed by sufficiently dense networks. Based on a joint work with Marco Cirant (Padua) and Joe Jackson (Chicago).

Ergodic Mean Field Games of Controls with State Constraints

Kyle Rosengartner Baylor University USA Co-Author(s):    Jameson Graber

Abstract: In a mean field game of controls, players seek to minimize a cost that depends on the joint distribution of players` states and controls. We consider an ergodic problem for second-order mean field games of controls with state constraints, in which equilibria are characterized by solutions to a second-order MFGC system where the value function blows up at the boundary, the density of players vanishes at a commensurate rate, and the joint distribution of states and controls satisfies the appropriate fixed-point relation. We prove that such systems are well-posed in the case of monotone coupling and Hamiltonians with at most quadratic growth.

Fully nonlinear mean field games with nondifferentiable Hamiltonians

Iain Smears University College London England Co-Author(s):    Thomas Sales

Abstract: We analyse fully nonlinear second-order mean field games (MFG) with nondifferentiable Hamiltonians, which take the form of a coupled system of a fully nonlinear Hamilton--Jacobi--Bellman equation and a Kolmogorov--Fokker--Planck partial differential inclusion (PDI) featuring the set-valued subdifferential of the Hamiltonian. We show the existence of solutions of some stationary MFG systems with quite general coupling operators and nonnegative distributional source terms, on general bounded convex domains, under the primary assumptions of uniform ellipticity and the Cordes condition on the diffusion coefficient. The existence proof is founded on an original, and equivalent, reformulation of the PDI as a nonstandard variational inequality (VI), that offers significant flexibility in passages to the limit. Furthermore, the uniqueness of the solution of the PDI/VI system is proved in the case of strictly monotone couplings. We then show how the MFG PDI/VI system in the fully nonlinear setting can be obtained as the limit of a sequence of PDE systems with differentiable Hamiltonians.

Solving First-Order Time-Dependent Mean-Field Games via Monotone Operator Theory

Melih Ucer KAUST Turkey Co-Author(s):    Diogo Gomes

Abstract: We study first-order, local, time-dependent mean-field games on the torus with a monotone Hamiltonian, using a variational inequality principle developed by Ferreira, Gomes, et al. (2018, 2019, 2021). We prove two main results. First, we establish the existence of a solution to the variational inequality via monotone operator theory in Banach spaces. Second, we show that any such solution satisfying natural integrability conditions is a weak solution in the sense introduced by Cardaliaguet (2015) for separable mean-field games. Together, these results establish the variational inequality principle as a general framework for mean-field games with non-separable Hamiltonians, well beyond the scope of minimization-based methods, yet recovering the same solution concept that those methods yield in the separable case.

A quantitative limit theory for quadratic mean-field controls

Songbo S Wang Universit\`{e} C\^{o}te d`Azur France Co-Author(s):    Songbo Wang

Abstract: Using an entropic formulation of the mean-field optimal control problem, we recast its convergence problem as the classical limit of mean-field Gibbs measures. This framework yields optimal convergence rates under path-dependent interactions and establishes sharp propagation of chaos estimates.

Mean-Field Games in Hilbert Spaces with Degenerate Noise: A Viscosity Approach

Lukas Wessels Universite Cote d`Azur France Co-Author(s):    Andrzej Swiech

Abstract: We investigate a Mean-Field Game (MFG) posed in an infinite-dimensional Hilbert space and driven by degenerate noise. The associated MFG system consists of a Hamilton--Jacobi--Bellman (HJB) equation for the value function coupled with a nonlinear Fokker--Planck (FP) equation for the distribution of the particles, both governed by a degenerate Kolmogorov operator. The degeneracy of the noise introduces significant analytical challenges. In particular, the HJB equation is treated in the viscosity sense, while the FP equation is interpreted in a suitable weak formulation. A major difficulty stems from the degeneracy of the Kolmogorov operator, which makes the uniqueness of solutions to the FP equation particularly delicate. Under appropriate structural assumptions, we establish well-posedness of the MFG system. As an application, we consider Mean-Field Games arising from stochastic delay differential equations, highlighting how delay effects naturally lead to infinite-dimensional and degenerate dynamics. This talk is based on joint work with Andrzej \`{S}wi\k{e}ch.