• David Ambrose, Drexel University. Title and abstract.
  Some nonseparable mean field games with singular measures as initial data

  Much theory of mean field games assumes additive separability of the Hamiltonian, although many systems arising in practice have non-separable Hamiltonians. The existence theory which has been established for non-separable problems assumes a fair bit of regularity on the initial data, so that the initial measure must be absolutely continuous with respect to Lebesgue measure. However, one would like to be able to take the initial measure to be a sum of Dirac masses. In addition to being non-separable, the Hamiltonians we have in mind from applied problems have a further property: they are nonlocal with respect to the measure variable, in that the measure appears inside an integral over the entire spatial domain. For such Hamiltonians, we develop existence theory in function spaces which allows singular measures, such as Dirac masses, as initial data.

• Martino Bardi, University of Padova. Title and abstract.
  Deterministic ergodic MFG with non-separable Hamiltonian

  I consider deterministic Mean Field Games (MFG) where the cost of the velocity of the agents may depend on the distribution $m$ of the agents, leading to Hamiltonians $H(x,p,m)$ where the moment variables $p$ are not separated from $m$. This is motivated by some models of congestion in crowd dynamics proposed by P.L. Lions and studied by various authors under different assumptions. In particular, I study the ergodic MFG system of 1st order PDEs in connection with the static MFG with cost $F(x,m):=-H(x,0,m)$. Under a coercivity condition on the Hamiltonian $H$ with respect to $p$, I show how to build a solution of such system from any solution of the static MFG. This leads to new existence results under rather general assumptions. Next I prove that the measure component of any solution to the ergodic MFG must solve the associated static MFG, under a homogeneity condition in $p$ on the Hamiltonian. Such necessary condition for the solvability of the ergodic MFG implies some new uniqueness results. In conclusion, I describe several cases of well-posedness of the MFG system, by reducing the problem to the much simpler well-posedness of the static MFG.

• Jules Berry, University of Rennes. Title and abstract.
  Approximation of stable solutions to second order mean field game systems

  We expose a general framework for the study of numerical approximations of a certain class of solutions, called stable solutions, of second order mean-field game systems for which uniqueness of solutions is not guaranteed. To illustrate the approach, we focus on a very simple example of stationary second-order MFG system. We then re-express the solutions of the system as zeros of a well chosen nonlinear map and establish the fact that stable solutions are regular points of this map. This fact is then used to study the finite element approximations of solutions and the local convergence of Newton's method.

• Fabio Camilli, Università G.D'Annunzio Chieti-Pescara. Title and abstract.
  Stationary Mean Field Games on networks with sticky transition conditions

  I consider stochastic Mean Field Games on networks with sticky transition conditions. In this setting, the diffusion process governing the agent's dynamics can spend finite time both in the interior of the edges and at the vertices. The corresponding generator is subject to limitations concerning second-order derivatives and the invariant measure breaks down into a combination of an absolutely continuous measure within the edges and a sum of Dirac measures positioned at the vertices. Additionally, the value function, solution to the Hamilton-Jacobi-Bellman equation, satisfies generalized Kirchhoff conditions at the vertices.

• Pierre Cardaliaguet, Paris-Dauphine University. Title and abstract.
  Traffic flow on junctions: mean field limits and optimization

  Traffic flow problems on junctions (or intersections) have been the subject of abundant literature in recent years. The modeling involves scalar conservation laws with discontinuities at the junction points, or, sometimes equivalently, Hamilton-Jacobi equations with discontinuous Hamiltonians. We will present the existence and uniqueness results for these equations, then explain how to derive these continuous models (where traffic is seen as a fluid) from discrete models (describing in detail the individual behavior of vehicles). We will also explain how to optimize the traffic in the case of a simple junction. The talk is based on joint works with N. Forcadel, R. Monneau and P. Souganidis.

• Elisabetta Carlini, Sapienza University of Rome. Title and abstract.
  Algorithm for Deterministic Mean-Field Games

  We propose a numerical scheme for approximating deterministic Mean Field Games (MFGs), based on semi-Lagrangian and Lagrange-Galerkin methods. We discuss a convergence result of the nonlinear scheme to the MFGs system in arbitrary dimensions. Additionally, we introduce an accelerated algorithm to efficiently solve the resulting discrete nonlinear system for which we show a convergence result. Finally, we present numerical results.

• Annalisa Cesaroni, University of Padova. Title and abstract.
  Equilibria in a Kuramoto Mean Field Game

  I will present results obtained in collaboration with Marco Cirant concerning equilibria in a Kuramoto mean-field game model. This model, introduced by R. Carmona, Q. Cormier, and M. Soner, is a game-theoretic version of the classical Kuramoto model, and describes synchronization phenomena in a large population of rational interacting oscillators. Specifically, I will discuss the characterization of equilibria and a local turnpike property.

• Jean-François Chassagneux, ENSAE-CREST & Institut Polytechnique de Paris. Title and abstract.
  Computing the stationary measure of McKean-Vlasov SDEs by ergodic simulation

  We design a fully implementable scheme to compute the invariant distribution of ergodic McKean-Vlasov SDE satisfying a uniform confluence property. Under natural conditions, we prove various convergence results notably we obtain rates for the Wasserstein distance in quadratic mean and almost sure sense.

• Marco Cirant, University of Padova. Title and abstract.
  On the role of monotonicity in the analysis of the Nash system

  Closed-loop Nash equilibria are provided by solutions to the so-called Nash system of PDEs. In this talk, I will explore the derivation of estimates for these solutions, emphasising in particular the role of monotonicity in the stability of finite-population equilibria. I will then discuss how these estimates apply to the convergence problem in MFG.

• Katharina Eichinger, LMO Paris-Saclay and INRIA Paris. Title and abstract.
  The exponential turnpike phenomenon for mean field game systems: weakly monotone drifts and small interactions

  In this talk we prove the exponential turnpike property for a class of mean field games on $\mathbb{R}^d$. The exponential turnpike property states that optimal trajectories stay exponentially close to a stationary state, called turnpike, if they are far from the initial and final time. Our technique is based on coupling by reflection adapted to controlled processes allowing us to treat controlled dynamics governed by an asymptotically convex potential. This enables us to prove existence and uniqueness of mean field game problems and their ergodic counterpart without monotonicity assumptions on the cost but rather a smallness condition on the dependence of the measure variable, and finally the exponential turnpike property. Based on joint work with Alekos Cecchin, Giovanni Conforti and Alain Durmus.

• Rita Ferreira, King Abdullah University of Science and Technology. Title and abstract.
  Uniqueness of solutions to MFGs through monotone methods

  In this talk, we address the question of uniqueness of weak solutions for stationary first-order 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.

• Jameson Graber, Baylor University. Title and abstract.
  Non-uniqueness and selection for a class of first-order mean field games

  There exists a relatively simple class of first-order mean field games for which it is possible to explicitly construct multiple equilibria, both at the level of $N$-player games and for the mean field game. These games are characterized by finding a scalar quantity depending on the final population distribution. Formally, this scalar quantity satisfies a transport equation on the space of probability measures. By adding noise in a certain fashion, it is possible to restore uniqueness for the $N$-player game. Using the theory of entropy solutions for scalar balance laws, we can find an error estimate showing that, as $N$ becomes large and the noise becomes small, the selected equilibrium is close to being a sort of "entropy solution" to the scalar transport equation on probability measures.

• Megan Griffin-Pickering, University of Zürich. Title and abstract.
  Kinetic Theoretic Approaches to Mean Field Games of Acceleration

  Kinetic Theory refers to the study of PDEs originating in statistical physics, which describe large many-particle systems statistically using probability measures on phase space. Kinetic equations share key structural features with Mean Field Games in which players control their acceleration. In such a game, a player’s control over their state vector (in phase space) is restricted: the permitted controls lie in a subspace with dimension half that of the players’ state space. In this talk, I will explain how techniques from kinetic theory can aid in the analysis of Mean Field Games with restricted controls of this kind. I will present recent results on the well-posedness and regularity of kinetic-type Mean Field Games. Based on joint works with David Ambrose and Alpár Mészáros.

• Ulrich Horst, Humboldt University Berlin. Title and abstract.
  Extended mean-field games with multi-dimensional singular controls and non-linear jump impact

  We establish a probabilistic framework for analysing extended mean-field games with multi-dimensional singular controls and state-dependent jump dynamics and costs. Two key challenges arise when analysing such games: the state dynamics may not depend continuously on the control and the reward function may not be u.s.c. Both problems can be overcome by restricting the set of admissible singular controls to controls that can be approximated by continuous ones. We prove that the corresponding set of admissible weak controls is given by the weak solutions to a Marcus-type SDE and provide an explicit characterisation of the reward function. The reward function will in general only be u.s.c. To address the lack of continuity we introduce a novel class of MFGs with a broader set of admissible controls, called MFGs of parametrisations. Parametrisations are laws of state/control processes that continuously interpolate jumps. We prove that the reward functional is continuous on the set of parametrisations, establish the existence of equilibria in MFGs of parametrisations, and show that the set of Nash equilibria in MFGs of parametrisations and in the underlying MFG with singular controls coincide. This shows that MFGs of parametrisations provide a canonical framework for analysing MFGs with singular controls and non-linear jump impact. The talk is based on joint work with Robert Denkert.

• Joe Jackson, University of Chicago. Title and abstract.
  A non-asymptotic viewpoint on stochastic differential games with many players

  We consider stochastic differential games with a large number $N$ of players. In mean field game theory, the $N$-player games of interest are distinguished by their symmetry (players are exchangeable) and their scaling (the strength of the interaction between two distinct players is of order $1/N$). In our study, we drop the symmetry requirement, and relax the scaling requirement so that, while the interaction between distinct players is still "weak" in an appropriate sense, some pairs of players may interact much more strongly than others. For example, our results apply in situations when each player interacts directly only with $K$ neighbors in some graph, the interaction between two neighbors is of strength $1/K$, and $N^{1/2} \ll K \ll N$. In this setting, we first show that appropriate monotonicity conditions can be used to derive estimates on the Nash and Pontryagin systems which are independent of $N$. We then apply these estimates to establish that when the interactions between players are weak and the costs are monotone, the closed-loop, open-loop and distributed formulations of the game nearly coincide. Finally, we apply our results to study some questions about the universality of the mean field game limit. This is joint work with Marco Cirant and Davide Redaelli.

• Espen Jakobsen, Norwegian University of Science and Technology. Title and abstract.
  Fully nonlinear parabolic mean field games: Existence and uniqueness results, and some degenerate problems

  We introduce a class of fully-nonlinear parabolic Mean Field Games systems (PDEs) corresponding to Mean Field Games with controlled local and/or nonlocal diffusion. After a heuristic derivation, we will focus on 3 model problems: (i) Nondegenerate local 2nd order problems, (ii) Nondegenerate nonlocal problems (e.g. with fractional Laplacians), and (iii) a degenerate nonlocal problem. In all cases we give existence and uniqueness results and discuss proofs. Some highlights we may be able to touch upon: A new type of control problem, a moment free theory of MFGs, uniqueness of MFGs without strict convexity or strict monotonicity, existence for MFG without uniqueness for Fokker-Planck, uniqueness for a Fokker-Planck equation with degenerate non-Lipschitz coefficient via non-standard viscosity solution doubling of variables. This is based on joint work with Milosz Krupski (Wroclaw) and Indranil Chowdhury (Kanpur) contained in the two papers: https://epubs.siam.org/doi/epdf/10.1137/23M1615528 and https://arxiv.org/abs/2104.06985

• Mattia Martini, Université Côte d’Azur. Title and abstract.
  Fourier Galerkin approximation of mean field control problems

  The goal of this talk is to introduce a finite-dimensional approximation of the solution to a mean field control problem defined on the $d$-dimensional torus. Our approximation is obtained using a Fourier-Galerkin method, whose main principle is to truncate the Fourier expansion of probability measures. We show that our method achieves a polynomial convergence rate directly proportional to the regularity of the data. This convergence rate is faster than that of the usual particle methods found in the literature, making it a more efficient alternative. Additionally, our technique provides an explicit method for constructing an approximate optimal control along with its corresponding trajectory. This talk is based on joint work with François Delarue.

• Nikiforos Mimikos-Stamatopoulos, Université Côte d’Azur. Title and abstract.
  Sharp convergence rates for mean field control in the region of strong regularity

  This is joint work with P. Cardaliaguet, J. Jackson, and P. E. Souganidis. We study the convergence problem for mean field control, also known as optimal control of McKean-Vlasov dynamics. We assume that the data is smooth but not convex, and thus the limiting value function $U : [0, T]\times\mathcal{P}_2(\mathbb{R}^d)\to\mathbb{R}$ may not be differentiable. In this setting, the first and last named authors recently identified an open and dense set $\mathcal{O}$ on which the limiting value function is $C^1$, and solves the relevant infinite-dimensional Hamilton-Jacobi equation in a classical sense. In the present paper, we use these regularity results (and some non-trivial extensions of them) to derive sharp rates of convergence. In particular, we show that the value functions for the $N$-particle control problems converge towards $U$ with a rate of $1/N$, uniformly on subsets of $\mathcal{O}$ which are compact in the $p$-Wasserstein space for some $p > 2$. A similar result is also established at the level of the optimal feedback controls. Importantly, the rate $1/N$ is the optimal rate in this setting even if $U$ is smooth, and the optimal global rate of convergence is known to be slower than $1/N$. Thus our results show that the optimal rate of convergence is faster inside of $\mathcal{O}$ than it is outside. As a consequence of the convergence of optimal feedbacks, we obtain a concentration inequality for optimal trajectories of the $N$-particle problem started from i.i.d. initial conditions.

• Sebastian Munoz, University of California, Los Angeles. Title and abstract.
  Self-similar intermediate asymptotics for first-order mean field games

  We present recent results regarding the long time behavior of solutions, in the whole space, to first-order mean field games system with a local coupling. Addressing a question left open in previous work of Cardaliaguet, Porretta, and the author, we show a sharp convergence result of the solutions to the unique self-similar profile. We proceed by analyzing a continuous rescaling of the solution, and identifying an appropriate Lyapunov functional.

• Alessio Porretta, Tor Vergata University of Rome. Title and abstract.
  Some results on the mean-field Schrödinger problem from an MFG perspective

  The mean-field Schrödinger problem consists in finding the most likely evolution of a system of interacting particles given initial and final observations of the density configuration. Translated into optimal transport formulation, this is an entropic optimal transport with additional interacting potential, whose optimality system is just one sample of a MFG planning problem. Relying on some ideas from displacement convexity, I will present a few results concerning the problem of uniqueness of the optimal trajectory, relying on convexity properties of the data and log-concavity estimates of the density.

• Davide Redaelli, Tor Vergata University of Rome. Title and abstract.
  Differential games with sparse interactions and infinite-dimensional Nash systems

  In the last years, we have witnessed an increasing interest in the understanding of large population limits of Nash equilibria — or particle systems, in general — under more general assumptions than those of exchangeability and negligibility that characterise classic MFGs. On the one hand, if the interactions are governed by dense graphs, then Graphon MF(G) theory helps to describe effectively such limits; on the other, as of today, sparse interactions have been studied to a lesse extent. I will discuss some recent results on this last topic, manly obtained in collaboration with Marco Cirant, showing how Nash equilibria of games with sparse interactions can be described — in the large population limit — by infinite-dimensional systems of Hamilton-Jacobi-Bellman equations, and giving insights on how such systems can be dealt with.

• Benjamin Seeger, University of North Carolina at Chapel Hill. Title and abstract.
  Hamilton-Jacobi equations on spaces of probability measures

  Hamilton-Jacobi equations posed on spaces of measures arise in a number of settings dealing with mean field descriptions of interacting agent systems. In this talk, recent methods for analyzing the well-posedness of such equations will be discussed, as well as applications of these results to studying such mean field problems.

• Iain Smears, University College London. Title and abstract.
  Analysis and numerical approximation of mean field game partial differential inclusions

  We consider the coupled system of the Hamilton-Jacobi-Bellman equation and Fokker-Planck equation from mean field games. A widespread assumption in the literature is that the Hamiltonian must be differentiable with respect to variable for the gradient of the value function. However, it is well-known from optimal control theory that in many applications, the resulting Hamiltonian is not differentiable. This leads to the crucial question of how to make sense of the PDE system in the nondifferentiable setting. From a modelling perspective, this corresponds to the issue of nonuniqueness of the optimal controls, and the question of how the players can choose controls in a way that maintains a Nash equilibrium.

  In this talk, we show that a suitable generalization of the problem is provided by relaxing the Fokker-Planck equation to a partial differential inclusion (PDI) involving the subdifferential of the Hamiltonian, which expresses mathematically the possibility of more complex structures in the Nash equilibria in the nondifferentiable case, where players in the same state may be required to make distinct choices among the various optimal controls.

  Our analytical contributions include theorems on the existence of solutions of the resulting MFG PDI system under very general conditions on the problem data, allowing for both local/nonlocal and nonsmoothing nonlinear couplings, for both the steady-state and the time-dependent cases in the stochastic setting. We also show that the MFG PDI system conserves uniqueness of the solution for monotone couplings, as a generalization of the result of Lasry and Lions. We also consider the regularization of the Hamiltonian, and show the convergence of the solutions of the regularized MFG to solutions of the PDI. In the second part of the talk, we consider the numerical solution of the MFG PDI, where we propose and analyse a stabilized finite element method for the PDI system. We prove the convergence of the method in various Bochner-Sobolev spaces for the fully coupled time-dependent problem, and we study the rates of convergence of the method in the steady-state setting. Numerical experiments for both steady-state and time-dependent problems illustrate the quantitative performance of the method.

• Mete Soner, Princeton University. Title and abstract.
  Learning algorithms for mean field optimal control

  We construct an algorithm for the numerical approximation of the optimal feedback control of mean-field optimal control problems using empirical risk minimization and neural networks with problem specific architectures. We approximate the model by an N-particle system and leverage the exchangeability of the particles to obtain substantial computational efficiency. We discuss several numerical examples and the convergence analysis is provided.

• Melih Ucer, King Abdullah University of Science and Technology. Title and abstract.
  Banach space monotone operator methods for existence of solutions in mean-field games

  In joint work with Ferreira and Gomes, we proved existence of solutions in the distributional sense to some first-order mean-field games with local coupling using the monotone operator theory. In this work, we mainly consider stationary problems with periodic boundary conditions and we have mild assumptions on the Hamiltonian; specifically, we only impose monotonicity and power-growth conditions mimicking $ H(p,m) = (|p|^\alpha/m^\tau) - m^\beta$, while we do not impose any specific form for the Hamiltonian such as additive separability. The essential idea is to cast the problem as an operator equation on a suitable Sobolev space, however the challenge with this approach is that one typically has only $L^p$ estimates on $m$ and $W^{1,q}$ estimates on $u$, while one cannot always trivially cast the MFG operator on a space $L^p\times W^{1,q}$ (in other words, we do not have strong assumptions on the Hamiltonian to ensure the necessary integrability of all the terms). Nevertheless, we found appropriate approximations to the MFG operator, then passed to the limit with Minty’s method to establish a weak variational inequality, and proved that a weak variational inequality implies the solution in the desired sense. We had to choose the approximations carefully to ensure that the weak variational inequality provided by Minty’s method applies to a large class of test functions, in order to be able to complete the final step.

• Xin Zhang, New York University. Title and abstract.
  Exciting games and Monge-Ampèe equations

  In this talk, we consider a competition between $d+1$ players, and aim to identify the “most exciting game” of this kind. This is translated, mathematically, into a stochastic optimization problem over martingales that live on the $d$-dimensional subprobability simplex and terminate on the vertices of the simplex, with a cost function related to a scaling limit of Shannon entropies. We uncover a surprising connection between this problem and the seemingly unrelated field of Monge-Ampère equations, and identify the optimal martingale via a detailed analysis of boundary asymptotics of a Monge-Ampère equation.

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