Here, we establish the existence of weak solutions to a wide class of time-dependent monotone mean-field games (MFGs). These MFGs are given as a system of degenerate parabolic equations with initial and terminal conditions. To construct these solutions, we consider a high-order elliptic regularization in space-time. Then, using Schaefer's fixed-point theorem, we obtain the existence and uniqueness for this regularized problem. Using Minty's method, we prove the existence of a weak solution to the original MFG. Finally, the paper ends with a discussion on congestion problems and density constrained MFGs.
We investigate the existence of classical solutions to second-order quadratic Mean-Field Games systems with local and strongly decreasing couplings of the form $-σm^α$, $α\ge 2/N$, where $m$ is the population density and $N$ is the dimension of the state space. We prove the existence of solutions under the assumption that $σ$ is small enough. For large $σ$, we show that existence may fail whenever the time horizon $T$ is large.
This paper deals with the existence of solutions of a class of contact mean field games systems of first order. Cardaliaguet \cite{CAR} found a link between the weak KAM theory for Hamiltonian systems and mean field games systems. We prove that there is still a connection between the weak KAM theory for contact Hamiltonian systems and contact mean field games systems. By the analysis of properties of the Mather set for contact Hamiltonian systems, we prove the main existence result.
In this paper, we investigate the existence and uniqueness of solutions to a stationary mean field game model introduced by J.-M. Lasry and P.-L. Lions. This model features a quadratic Hamiltonian with possibly singular congestion effects. Thanks to a new class of a-priori bounds, combined with the continuation method, we prove the existence of smooth solutions in arbitrary dimensions.
In this paper we consider extended stationary mean field games, that is mean-field games which depend on the velocity field of the players. We prove various a-priori estimates which generalize the results for quasi-variational mean field games in [GPSM12]. In addition we use adjoint method techniques to obtain higher regularity bounds. Then we establish existence of smooth solutions under fairly general conditions by applying the continuity method. When applied to standard stationary mean-field games as in [LL06a], [GSM11] or [GPSM12] this paper yields various new estimates and regularity prop…
Here, we consider stationary monotone mean-field games (MFGs) and study the existence of weak solutions. First, we introduce a regularized problem that preserves the monotonicity. Next, using variational inequalities techniques, we prove the existence of solutions to the regularized problem. Then, using Minty's method, we establish the existence of solutions for the original MFG. Finally, we examine the properties of these weak solutions in several examples. Our methods provide a general framework to construct weak solutions to stationary MFGs with local, nonlocal, or congestion terms.
We study a system of partial differential equations used to describe Bertrand and Cournot competition among a continuum of producers of an exhaustible resource. By deriving new a priori estimates, we prove the existence of classical solutions under general assumptions on the data. Moreover, under an additional hypothesis we prove uniqueness. Keywords: mean field games, Hamilton-Jacobi, Fokker-Planck, coupled systems, optimal control, nonlinear partial differential equations
We study the local in time existence of a regular solution of a nonlinear parabolic backward-forward system arising from the theory of Mean-Field Games (briefly MFG). The proof is based on a contraction argument in a suitable space that takes account of the peculiar structure of the system, which involves also a coupling at the final horizon. We apply the result to obtain existence to very general MFG models, including also congestion problems.
We investigate the existence of solutions to viscous ergodic Mean Field Games systems in bounded domains with Neumann boundary conditions and local, possibly aggregative couplings. In particular we exploit the associated variational structure and search for constrained minimizers of a suitable functional. Depending on the growth of the coupling, we detect the existence of global minimizers in the mass subcritical and critical case, and of local minimizers in the mass supercritical case, notably up to the Sobolev critical case.
We develop a splitting method to prove the well-posedness, in short time, of solutions for two master equations in mean field game (MFG) theory: the second order master equation, describing MFGs with a common noise, and the system of master equations associated with MFGs with a major player. Both problems are infinite dimensional equations stated in the space of probability measures. Our new approach simplifies, shortens and generalizes previous existence results for second order master equations and provides the first existence result for systems associated with MFG problems with a major play…
The mean field games system is a coupled pair of nonlinear partial differential equations arising in differential game theory, as a limit as the number of agents tends to infinity. We prove existence and uniqueness of classical solutions for time-dependent mean field games with Sobolev data. Many works in the literature assume additive separability of the Hamiltonian, as well as further structure such as convexity and monotonicity of the resulting components. Problems arising in practice, however, may not have this separable structure; we therefore consider the non-separable problem. For our e…
We consider time-dependent mean-field games with congestion that are given by a system of a Hamilton-Jacobi equation coupled with a Fokker-Planck equation. The congestion effects make the Hamilton-Jacobi equation singular. These models are motivated by crowd dynamics where agents have difficulty moving in high-density areas. Uniqueness of classical solutions for this problem is well understood. However, existence of classical solutions, was only known in very special cases - stationary problems with quadratic Hamiltonians and some time-dependent explicit examples. Here, we prove short-time exi…
For non-monotone single and two-populations time-dependent Mean-Field Game systems we obtain the existence of an infinite number of branches of non-trivial solutions. These non-trivial solutions are in particular shown to exhibit an oscillatory behaviour when they are close to the trivial (constant) one. The existence of such branches is derived using local and global bifurcation methods, that rely on the analysis of eigenfunction expansions of solutions to the associated linearized problem. Numerical analysis is performed on two different models to observe the oscillatory behaviour of solutio…
In this paper, we study first-order stationary monotone mean-field games (MFGs) with Dirichlet boundary conditions. While for Hamilton--Jacobi equations Dirichlet conditions may not be satisfied, here, we establish the existence of solutions of MFGs that satisfy those conditions. To construct these solutions, we introduce a monotone regularized problem. Applying Schaefer's fixed-point theorem and using the monotonicity of the MFG, we verify that there exists a unique weak solution to the regularized problem. Finally, we take the limit of the solutions of the regularized problem and using Minty…
Existence and multiplicity of solutions to the mean-field games model with mixed interactions
School of Science, Tianjin University of Commerce（天津商业大学理学院）；School of Mathematics, Yunnan Normal University（云南师范大学数学学院）；Yunnan Key Laboratory of Modern Analytical Mathematics and Applications（云南现代分析数学与应用重点实验室）；Department of Mathematics, Chinese University of Hong Kong（香港中文大学数学系）
In this paper we show how to use a Tug-of-War game to obtain existence of a viscosity solution to the infinity laplacian with non-homogeneous mixed boundary conditions. For a Lipschitz and positive function $g$ there exists a viscosity solution of the mixed boundary value problem, $$ \{\begin{array}{ll} \displaystyle -Δ_{\infty}u(x)=0\quad & \text{in} Ω, \displaystyle \frac{\partial u}{\partial n}(x)= g (x)\quad & \text{on} Γ_N, \displaystyle u(x)= 0 \quad & \text{on} Γ_D. \end{array}. $$
Here, we consider a regularized mean-field game model that features a low-order regularization. We prove the existence of solutions with positive density. To do so, we combine a priori estimates with the continuation method. In contrast with high-order regularizations, the low-order regularizations are easier to implement numerically. Moreover, our methods give a theoretical foundation for this approach.
Here, we prove the existence of solutions to first-order mean-field games (MFGs) arising in optimal switching. First, we use the penalization method to construct approximate solutions. Then, we prove uniform estimates for the penalized problem. Finally, by a limiting procedure, we obtain solutions to the MFG problem.
We prove existence theorems for strong solutions of time-dependent mean field games with non-separable Hamiltonian. In a recent announcement, we showed existence of small, strong solutions for mean field games with local coupling. We first generalize that prior work to allow for non-separable Hamiltonians. This proof is inspired by the work of Duchon and Robert on the existence of small-data vortex sheets in incompressible fluid mechanics. Our next existence result is in the case of weak coupling of the system; that is, we allow the data to be of arbitrary size, but instead require that the (s…
In this paper, we establish the existence and uniqueness of weak solutions to first-order discount mean field games and a stability result to give the existence for the ergodic problem. We show an example to illustrate the multiplicity of weak solutions to the ergodic problem. With this motivation, we address a selection condition, which is a necessary condition that any limit of solutions under subsequence satisfies. As an application, we show a nontrivial example to get the convergence of weak solutions.
This paper concerns value functions of time-dependent tug-of-war games. We first prove the existence and uniqueness of value functions and verify that these game values satisfy a dynamic programming principle. Using the arguments in the proof of existence of game values, we can also deduce asymptotic behavior of game values when $T \to \infty$. Furthermore, we investigate boundary regularity for game values. Thereafter, based on the regularity results for value functions, we deduce that game values converge to viscosity solutions of the normalized parabolic $p$-Laplace equation.
We consider stationary viscous Mean-Field Games systems in the case of local, decreasing and unbounded coupling. These systems arise in ergodic mean-field game theory, and describe Nash equilibria of games with a large number of agents aiming at aggregation. We show how the dimension of the state space, the behavior of the coupling and the Hamiltonian at infinity affect the existence and non-existence of regular solutions. Our approach relies on the study of Sobolev regularity of the invariant measure and a blow-up procedure which is calibrated on the scaling properties of the system. In very …
In this manuscript, we prove existence of viscosity solutions to singular parabolic equations in Carnot groups. We develop the analysis by constructing appropriate deterministic games adapted to the algebraic and differential structures of Carnot groups. We point out that the proof of existence does not require comparison and it is based on an Arzela-Ascoli-type theorem.
In this paper, we explore Bertrand and Cournot Mean Field Games models for market competition with reflection boundary conditions. We prove existence, uniqueness and regularity of solutions to the system of equations, and show that this system can be written as an optimality condition of a convex minimization problem. We also provide a short proof of uniqueness to the system addressed in [Graber, P. and Ben- soussan, A., Existence and uniqueness of solutions for Bertrand and Cournot mean field games, Applied Mathematics & Optimization (2016)], where uniqueness was only proved for small paramet…
We introduce a notion of weak solution of the master equation without idiosyncratic noise in Mean Field Game theory and establish its existence, uniqueness up to a constant and consistency with classical solutions when it is smooth. We work in a monotone setting and rely on Lions' Hilbert space approach. For the first-order master equation without idiosyncratic noise, we also give an equivalent definition in the space of measures and establish the well-posedness.
We analyze a mean-field game model proposed by economists R.E. Lucas and B. Moll (2014) to describe economic systems where production is based on knowledge growth and diffusion. This model reduces to a PDE system where a backward Hamilton-Jacobi-Bellman equation is coupled with a forward KPP-type equation with nonlocal reaction term. We study the existence of traveling waves for this mean-field game system, obtaining the existence of both critical and supercritical waves. In particular we prove a conjecture raised by economists on the existence of a critical balanced growth path for the descri…
Mean-field games (MFGs) are models of large populations of rational agents who seek to optimize an objective function that takes into account their location and the distribution of the remaining agents. Here, we consider stationary MFGs 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 MFGs with quadratic Hamiltonians thanks to a very particular identity.…
We show that the long time average of solutions of first order mean field game systems in finite horizon is governed by an ergodic system of mean field game type. The well-posedness of this later system and the uniqueness of the ergodic constant rely on weak KAM theory.
This paper presents a general existence and uniqueness result for mean field games equations on graphs ($\mathcal{G}$-MFG). In particular, our setting allows to take into account congestion effects of almost any form. These general congestion effects are particularly relevant in graphs in which the cost to move from one node to another may for instance depend on the proportion of players in both the source node and the target node. Existence is proved using a priori estimates and a fixed point argument à la Schauder. We propose a new criterion to ensure uniqueness in the case of Hamiltonian fu…
We consider a Mean Field Games model where the dynamics of the agents is given by a controlled Langevin equation and the cost is quadratic. A change of variables, introduced in [9], transforms the Mean Field Games system into a system of two coupled kinetic Fokker-Planck equations. We prove an existence result for the latter system, obtaining consequently existence of a solution for the Mean Field Games system.
We analyze a zero-sum stochastic differential game between two competing players who can choose unbounded controls. The payoffs of the game are defined through backward stochastic differential equations. We prove that each player's priority value satisfies a weak dynamic programming principle and thus solves the associated fully non-linear partial differential equation in the viscosity sense.
