In this thesis, we consider stationary one-dimensional mean-field games (MFGs) with or without congestion. Our aim is to understand the qualitative features of these games through the analysis of explicit solutions. We are particularly interested in MFGs with a nonmonotonic behavior, which corresponds to situations where agents tend to aggregate. First, we derive the MFG equations from control theory. Then, we compute explicit solutions using the current formulation and examine their behavior. Finally, we represent the solutions and analyze the results. This thesis main contributions are the following: First, we develop the current method to solve MFG explicitly. Second, we analyze in detail non-monotonic MFGs and discover new phenomena: non-uniqueness, discontinuous solutions, empty regions and unhappiness traps. Finally, we address several regularization procedures and examine the stability of MFGs.

In this thesis, we present three projects. First, we investigate the numerical approximation of Hamilton-Jacobi equations with the Caputo time-fractional derivative. We introduce an explicit in time discretization of the Caputo derivative and a finite-difference scheme for the approximation of the Hamiltonian. We show that the approximation scheme so obtained is stable under an appropriate condition on the discretization parameters and converges to the unique viscosity solution of the Hamilton-Jacobi equation. Also, we study the numerical approximation of a system of PDEs which arises from an optimal control problem for the time-fractional Fokker-Planck equation with time-dependent drift. The system is composed of a backward time-fractional Hamilton-Jacobi-Bellman equation and a forward time-fractional Fokker-Planck equation. We approximate Caputo derivatives in the system by means of L1 schemes and the Hamiltonian by finite differences. The scheme for the Fokker-Planck equation is constructed in such a way that the duality structure of the PDE system is preserved on the discrete level. We prove the well-posedness of the scheme and the convergence to the solution of the continuous problem. Finally, we study a particle approximation for one-dimensional first-order Mean-Field-Games with local interactions with planning conditions. Our problem comprises a system of a Hamilton-Jacobi equation coupled with a transport equation. As we are dealing with the planning problem, we prescribe initial and terminal distributions for the transport equation. The particle approximation builds on a semi-discrete variational problem. First, we address the existence and uniqueness of the semi-discrete variational problem. Next, we show that our discretization preserves some conserved quantities. Finally, we prove that the approximation by particle systems preserves displacement convexity. We use this last property to establish uniform estimates for the discrete problem. All results for the discrete problem are illustrated with numerical examples.

The mean-field game theory is the study of strategic decision making in very large populations of weakly interacting individuals. Mean-field games have been an active area of research in the last decade due to its increased significance in many scientific fields. The foundations of mean-field theory go back to the theory of statistical and quantum physics. One may describe mean-field games as a type of stochastic differential game for which the interaction between the players is of mean-field type, i.e the players are coupled via their empirical measure. It was proposed by Larsy and Lions and independently by Huang, Malhame, and Caines. Since then, the mean-field games have become a rapidly growing area of research and has been studied by many researchers. However, most of these studies were dedicated to diffusion-type games. The main purpose of this thesis is to extend the theory of mean-field games to jump case in both discrete and continuous state space. Jump processes are a very important tool in many areas of applications. Specifically, when modeling abrupt events appearing in real life. For instance, financial modeling (option pricing and risk management), networks (electricity and Banks) and statistics (for modeling and analyzing spatial data). The thesis consists of two papers and one technical report which will be submitted soon: In the first publication, we study the mean-field game in a finite state space where the dynamics of the indistinguishable agents is governed by a controlled continuous time Markov chain. We have studied the control problem for a representative agent in the linear quadratic setting. A dynamic programming approach has been used to drive the Hamilton Jacobi Bellman equation, consequently, the optimal strategy has been achieved. The main result is to show that the individual optimal strategies for the mean-field game system represent 1/N-Nash equilibrium for the approximating system of N agents. As a second article, we generalize the previous results to agents driven by a non-linear pure jump Markov processes in Euclidean space. Mathematically, this means working with linear operators in Banach spaces adapted to the integro-differential operators of jump type and with non-linear partial differential equations instead of working with linear transformations in Euclidean spaces as in the first work. As a by-product, a generalization for the Koopman operator has been presented. In this setting, we studied the control problem in a more general sense, i.e. the cost function is not necessarily of linear quadratic form. We showed that the resulting unique optimal control is of Lipschitz type. Furthermore, a fixed point argument is presented in order to construct the approximate Nash Equilibrium. In addition, we show that the rate of convergence will be of special order as a result of utilizing a non-linear pure jump Markov process. In a third paper, we develop our approach to treat a more realistic case from a modelling perspective. In this step, we assume that all players are subject to an additional common noise of Brownian type. We especially study the well-posedness and the regularity for a jump version of the stochastic kinetic equation. Finally, we show that the solution of the master equation, which is a type of second order partial differential equation in the space of probability measures, provides an approximate Nash Equilibrium. This paper, unfortunately, has not been completely finished and it is still in preprint form. Hence, we have decided not to enclose it in the thesis. However, an outlook about the paper will be included.

Mean-field games (MFG) are models of large populations of rational agents who seek to optimize an objective function that takes into account their state variables and the distribution of the state variable of the remaining agents. MFG with congestion model problems where the agents’ motion is hampered in high-density regions. First, we study radial solutions for first- and second-order stationary MFG with congestion on Rd. The radial case, which is one of the simplest non one-dimensional MFG, is relatively tractable. As we observe, the Fokker-Planck equation is integrable with respect to one of the unknowns. Consequently, we obtain a single equation substituting this solution into the Hamilton-Jacobi equation. For the first-order case, we derive explicit formulas; for the elliptic case, we study a variational formulation of the resulting equation. For the first case, we use our approach to compute numerical approximations to the solutions of the corresponding MFG systems. Next, we consider second-order stationary MFG with congestion and prove the existence of stationary solutions. Because moving in congested areas is difficult, agents prefer to move in non-congested areas. As a consequence, the model becomes singular near the zero density. The existence of stationary solutions was previously obtained for MFG with quadratic Hamiltonians thanks to a very particular identity. Here, we develop robust estimates that give the existence of a solution for general subquadratic Hamiltonians. Additionally, we study first-order stationary MFG with congestion with quadratic or power-like Hamiltonians. Using explicit examples, we illustrate two key difficulties: the lack of classical solutions and the existence of areas with vanishing densities. Our main contribution is a new variational formulation for MFG with congestion. With this formulation, we prove the existence and uniqueness of solutions. Finally, we devise a discretization that is combined with optimization algorithms to numerically solve various MFG with congestion.

In this thesis, we consider the planning problem for first-order mean-field games (MFG). These games degenerate into optimal transport when there is no coupling between players. Our aim is to extend the concept of displacement convexity from optimal transport to MFGs. This extension gives new estimates for solutions of MFGs. First, we introduce the Monge-Kantorovich problem and examine related results on rearrangement maps. Next, we present the concept of displacement convexity. Then, we derive first-order MFGs, which are given by a system of a Hamilton-Jacobi equation coupled with a transport equation. Finally, we identify a large class of functions, that depend on solutions of MFGs, which are convex in time. Among these, we find several norms. This convexity gives bounds for the density of solutions of the planning problem.

In this dissertation, we present two research projects, namely finite-state mean-field games and the Hughes model for the motion of crowds. In the first part, we describe finite-state mean-field games and some applications to socio-economic sciences. Examples include paradigm shifts in the scientific community and the consumer choice behavior in a free market. The corresponding finite-state mean-field game models are hyperbolic systems of partial differential equations, for which we propose and validate a new numerical method. Next, we consider the dual formulation to two-state mean-field games, and we discuss numerical methods for these problems. We then depict different computational experiments, exhibiting a variety of behaviors, including shock formation, lack of invertibility, and monotonicity loss. We conclude the first part of this dissertation with an investigation of the shock structure for two-state problems. In the second part, we consider a model for the movement of crowds proposed by R. Hughes in [56] and describe a numerical approach to solve it. This model comprises a Fokker-Planck equation coupled with an Eikonal equation with Dirichlet or Neumann data. We first establish a priori estimates for the solutions. Next, we consider radial solutions, and we identify a shock formation mechanism. Subsequently, we illustrate the existence of congestion, the breakdown of the model, and the trend to the equilibrium. We also propose a new numerical method for the solution of Fokker-Planck equations and then to systems of PDEs composed by a Fokker-Planck equation and a potential type equation. Finally, we illustrate the use of the numerical method both to the Hughes model and mean-field games. We also depict cases such as the evacuation of a room and the movement of persons around Kaaba (Saudi Arabia).

In the thesis, we discuss the existence and numerical approximations of solutions of a regularized mean-field game with a low-order regularization. In the first part, we prove a priori estimates and use the continuation method to obtain the existence of a solution with a positive density. Finally, we introduce the monotone flow method and solve the system numerically.

Cette thèse porte sur l’étude de nouveaux modèles de jeux à champ moyen. On étudie dans un premier temps des modèles d’arrêt optimal et de contrôle impulsionnel en l’absence de bruit commun. On construit pour ces modèles une notion de solution adaptée pour laquelle on prouve des résultats d’existence et d’unicité sous des hypothèses naturelles. Ensuite, on s’intéresse à plusieurs propriétés des jeux à champ moyen. On étudie la limite de ces modèles vers des modèles d’évolution pures lorsque l’anticipation des joueurs tend vers 0. On montre l’unicité des équilibres pour des systèmes fortement couples (couples par les stratégies) sous certaines hypothèses. On prouve ensuite certains résultats de régularités sur une ”master equation” qui modélise un jeu à champ moyen avec bruit commun dans un espace d’états discret. Par la suite on présente une généralisation de l’algorithme standard d’Uzawa et on l’applique à la résolution numérique de certains modèles de jeux à champ moyen, notamment d’arrêt optimal ou de contrôle impulsionnel. Enfin on présente un cas concret de jeu à champ moyen qui provient de problèmes faisant intervenir un grand nombre d’appareils connectés dans les télécommunications. / This thesis is concerned with new models of mean field games. First, we study models of optimal stopping and impulse control in the case when there is no common noise. We build an appropriate notion of solutions for those models. We prove the existence and the uniqueness of such solutions under natural assumptions. Then, we are interested with several properties of mean field games. We study the limit of such models when the anticipation of the players vanishes. We show that uniqueness holds for strongly coupled mean field games (coupled via strategies) under certain assumptions. We then prove some regularity results for the master equation in a discrete state space case with common noise. We continue by giving a generalization of Uzawa’s algorithm and we apply it to solve numerically some mean field games, especially optimal stopping and impulse control problems. The last chapter presents an application of mean field games. This application originates from problems in telecommunications which involve a huge number of connected devices.

Mean-field games (MFGs) study the behavior of rational and indistinguishable agents in a large population. Agents seek to minimize their cost based upon statis- tical information on the population’s distribution. In this dissertation, we study the homogenization of a stationary first-order MFG and seek to find a numerical method to solve the homogenized problem. More precisely, we characterize the asymptotic behavior of a first-order stationary MFG with a periodically oscillating potential. Our main tool is the two-scale convergence. Using this convergence, we rigorously derive the two-scale homogenized and the homogenized MFG problems. Moreover, we prove existence and uniqueness of the solution to these limit problems. Next, we notice that the homogenized problem resembles the problem involving effective Hamiltoni- ans and Mather measures, which arise in several problems, including homogenization of Hamilton–Jacobi equations, nonlinear control systems, and Aubry–Mather theory. Thus, we develop algorithms to solve the homogenized problem, the effective Hamil- tonians, and Mather measures. To do that, we construct the Hessian Riemannian flow. We prove the convergence of the Hessian Riemannian flow in the continuous setting. For the discrete case, we give both the existence and the convergence of the Hessian Riemannian flow. In addition, we explore a variant of Newton’s method that greatly improves the performance of the Hessian Riemannian flow. In our numerical experiments, we see that our algorithms preserve the non-negativity of Mather mea- sures and are more stable than related methods in problems that are close to singular. Furthermore, our method also provides a way to approximate stationary MFGs.

This thesis studies mean-field games (MFGs) models of price formation. The thesis focuses explicitly on a MFGs price formation model proposed by Gomes and Saude. The thesis is divided into two parts. The first part examines the deterministic supply case, while the second part extends the model to incorporate a stochastic supply function. We explore different approaches, such as Aubry-Mather theory, to study the properties of the MFGs price formation model and alternative formulations using a convex variational problem with constraints. We propose machine-learning-based numerical methods to approximate the solution of the MFGs price formation model in the deterministic and stochastic setting.