arXiv subjects

P. Jameson Graber

Publications and source records attributed to P. Jameson Graber.

At least 19 recordsLinked to original sources

arXiv2025

Mean Field Games of Controls with Boundary Conditions & Invariance Constraints

In a mean field game of controls, a large population of identical players seek to minimize a cost that depends on the joint distribution of the states of the players and their controls. We first consider the classes of mean field games of controls in which the value function and the distribution of player states satisfy either Dirichlet or Neumann boundary conditions. We prove that such systems are well-posed either with sufficient smallness conditions or in the case of monotone couplings. Next, we consider mean field games of controls under invariance constraints imposed on the state space. We prove the existence and uniqueness of weak solutions to our mean field game system, and then we prove higher regularity of solutions under some additional assumptions.

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arXiv2025

Mean Field Games of Controls with Fractional Laplacian

We analyze a fractional mean field game of controls system, showing existence of solutions when the order of the fractional Laplacian is $s\in(\frac{1}{2},1)$. Here the running cost depends on the distribution $μ$ of not only the states but also optimal strategies. The coupling is assumed to satisfy the Lasry-Lions monotonicity condition. We derive three types of a priori estimates on solutions. First, we use the monotonicity condition to derive moment estimates on $μ$. Second, we derive abstract estimates on fractional parabolic equations and apply them to the mean field game. Third, we derive new estimates on the time regularity of the distribution $μ$ by analyzing the associated Lévy process. We apply these estimates and the Leray-Schauder fixed point theorem to establish existence of solutions.

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arXiv2025

A "trembling hand perfect" equilibrium for a certain class of mean field games

We study a particular class of mean field games whose solutions can be formally connected to a scalar transport equation on the Wasserstein space of measures. For this class, we construct some interesting explicit examples of non-uniqueness of Nash equilibria. We then address the selection problem of finding rational criteria by which to choose one equilibrium over others. We show that when the theory of entropy solutions is used, we can obtain explicit error estimates for the ``vanishing noise limit,'' where the error is measured in a certain norm that measures the distance between two functions over the set of empirical measures.

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arXiv2025

Control on Hilbert Space and Mean Field Control: the Common Noise Case

The objective of this paper is to provide an equivalent of the theory developed in P.~Cardaliaguet, F.~Delarue, J.M.~Lasry, P.L.~Lions \cite{CDLL}, following the approach of control on Hilbert spaces introduced by the authors in \cite{BGY-2}. We include the common noise in this paper, so the alternative is now complete. Since we consider a control problem, our theory applies only to Mean field control and not to mean field games. The assumptions are adapted to guarantee a unique optimal control, so they insure that the cost functional is strictly convex and coercive.

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arXiv2024

Remarks on potential mean field games

In this expository article, we give an overview of the concept of potential mean field games of first order. We give a new proof that minimizers of the potential are equilibria by using a Lagrangian formulation. We also provide criteria to determine whether or not a game has a potential. Finally, we discuss in some depth the selection problem in mean field games, which consists in choosing one out of multiple Nash equilibria.

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arXiv2024

A Model Problem for First Order Mean Field Games with Discrete Initial Data

In this article, we study a simplified version of a density-dependent first-order mean field game, in which the players face a penalization equal to the population density at their final position. We consider the problem of finding an equilibrium when the initial distribution is a discrete measure. We show that the problem becomes finite-dimensional: the final piecewise smooth density is completely determined by the weights and positions of the initial measure. We establish existence and uniqueness of a solution using classical fixed point theorems. Finally, we show that Newton's method provides an effective way to compute the solution. Our numerical simulations provide an illustration of how density penalization in a mean field game tends to the smoothen the initial distribution.

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arXiv2024

On Some Mean Field Games and Master Equations through the lens of conservation laws

In this manuscript we derive a new nonlinear transport equation written on the space of probability measures that allows to study a class of deterministic mean field games and master equations, where the interaction of the agents happens only at the terminal time. The point of view via this transport equation has two important consequences. First, this equation reveals a new monotonicity condition that is sufficient both for the uniqueness of MFG Nash equilibria and for the global in time well-posedness of master equations. Interestingly, this condition is in general in dichotomy with both the Lasry--Lions and displacement monotonicity conditions, studied so far in the literature. Second, in the absence of monotonicity, the conservative form of the transport equation can be used to define weak entropy solutions to the master equation. We construct several concrete examples to demonstrate that MFG Nash equilibria, whether or not they actually exist, may not be selected by the entropy solutions of the master equation.

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arXiv2023

Cluster formation in iterated Mean Field Games

We study a simple first-order mean field game in which the coupling with the mean field is only in the final time and gives an incentive for players to congregate. For a short enough time horizon, the equilibrium is unique. We consider the process of \emph{iterating} the game, taking the final population distribution as the initial distribution in the next iteration. Restricting to one dimension, we take this to be a model of coalition building for a population distributed over some spectrum of opinions. Our main result states that, given a final coupling of the form $G(x,m) = \int φ(x-z)\dif m(z)$ where $φ$ is a smooth, even, non-positive function of compact support, then as the number of iterations goes to infinity the population tends to cluster into discrete groups, which are spread out as a function of the size of the support of $φ$. We discuss the potential implications of this result for real-world opinion dynamics and political systems.

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arXiv2023

On monotonicity conditions for Mean Field Games

In this paper we propose two new monotonicity conditions that could serve as sufficient conditions for uniqueness of Nash equilibria in mean field games. In this study we aim for $unconditional\ uniqueness$ that is independent of the length of the time horizon, the regularity of the starting distribution of the agents, or the regularization effect of a non-degenerate idiosyncratic noise. Through a rich class of simple examples we show that these new conditions are not only in dichotomy with each other, but also with the two widely studied monotonicity conditions in the literature, the Lasry-Lions monotonicity and displacement monotonicity conditions.

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arXiv2023

Control on Hilbert Spaces and Application to Some Mean Field Type Control Problems

We propose a new approach to studying classical solutions of the Bellman equation and Master equation for mean field type control problems, using a novel form of the "lifting" idea introduced by P.-L. Lions. Rather than studying the usual system of Hamilton-Jacobi/Fokker-Planck PDEs using analytic techniques, we instead study a stochastic control problem on a specially constructed Hilbert space, which is reminiscent of a tangent space on the Wasserstein space in optimal transport. On this Hilbert space we can use classical control theory techniques, despite the fact that it is infinite dimensional. A consequence of our construction is that the mean field type control problem appears as a special case. Thus we preserve the advantages of the lifiting procedure, while removing some of the difficulties. Our approach extends previous work by two of the coauthors, which dealt with a deterministic control problem for which the Hilbert space could be generic.

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arXiv2022

Master Equation for Cournot Mean Field Games of Control with Absorption

We establish the existence and uniqueness of a solution to the master equation for a mean field game of controls with absorption. The mean field game arises as a continuum limit of a dynamic game of exhaustible resources modeling Cournot competition between producers. The proof relies on an analysis of a forward-backward system of nonlocal Hamilton-Jacobi/Fokker-Planck equations with Dirichlet boundary conditions. In particular, we establish new a priori estimates to prove that solutions are differentiable with respect to the initial measure.

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arXiv2022

Parameter sensitivity analysis for mean field games of production

We study a mean field game system introduced by Chan and Sircar (AMO, 2015) to model production of an exhaustible resource. In particular, we study the sensitivity of the solution with respect to a parameter $\varepsilon$, which measures the degree to which producers are interchangeable. We prove that on some interval $[0,\varepsilon_0]$, where $\varepsilon_0 > 0$, the solution is infinitely differentiable with respect to $\varepsilon$. The result is based on a set of new a priori estimates for forward-backward systems of linear partial differential equations.

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arXiv2020

Nonlocal Bertrand and Cournot Mean Field Games with General Nonlinear Demand Schedule

In this article we prove the existence of classical solutions to a system of mean field games arising in the study of exhaustible resource production under market competition. Individual trajectories are modeled by a controlled diffusion process with jumps, which adds a nonlocal term to the PDE system. The assumptions on the Hamiltonian are sufficiently general to cover a large class of examples proposed in the literature on Bertrand and Cournot mean field games. Uniqueness also holds under a sufficient restriction on the structure of the Hamiltonian, which in practice amounts to a small upper bound on the substitutability of goods.

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arXiv2020

Local well-posedness in Sobolev spaces for first-order barotropic causal relativistic viscous hydrodynamics

We study the theory of relativistic viscous hydrodynamics introduced in arXiv:1109.0985 and arXiv:1907.12695, which provided a causal and stable first-order theory of relativistic fluids with viscosity in the case of barotropic fluids. The local well-posedness of its equations of motion has been previously established in Gevrey spaces. Here, we improve this result by proving local well-posedness in Sobolev spaces.

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arXiv2019

The planning problem in Mean Field Games as regularized mass transport

In this paper, using variational approaches, we investigate the first order planning problem arising in the theory of mean field games. We show the existence and uniqueness of weak solutions of the problem in the case of a large class of Hamiltonians with arbitrary superlinear order of growth at infinity and local coupling functions. We require the initial and final measures to be merely summable. At the same time (relying on the techniques developed recently by Graber and Mészáros), under stronger monotonicity and convexity conditions on the data, we obtain Sobolev estimates on the solutions of the planning problem both for space and time derivatives.

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arXiv2019

Stochastic Control on Space of Random Variables

By extending \cite{bensoussan2015control}, we implement the proposal of Lions \cite{lions14} on studying mean field games and their master equations via certain control problems on the Hilbert space of square integrable random variables. In \cite{bensoussan2015control}, the Hilbert space could be quite general in the face of the "deterministic control problem" due to the absence of additional randomness; while the special case of $L^2$ space of square integrable random variables was brought in at the interpretation stage. The effectiveness of the approach was demonstrated by deriving Bellman equations and the first order master equations through control theory of dynamical systems valued in the Hilbert space. In our present problem for second order master equations, it connects with a stochastic control problem over the space of random variables, and it possesses an additional randomness generated by the Wiener process which cannot be detached from the randomness caused by the elements in the Hilbert space. Nevertheless, we demonstrate how to tackle this difficulty, while preserving most of the efficiency of the approach suggested by Lions \cite{lions14}.

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arXiv2019

On mean field games models for exhaustible commodities trade

We investigate a mean field game model for the production of exhaustible resources. In this model, firms produce comparable goods, strategically set their production rate in order to maximise profit, and leave the market as soon as they deplete their capacities. We examine the related Mean Field Game system and prove well-posedness for initial measure data by deriving suitable a priori estimates. Then, we show that feedback strategies which are computed from the Mean Field Game system provide $\varepsilon$-Nash equilibria to the corresponding $N$-Player Cournot game, for large values of $N$. This is done by showing tightness of the empirical process in the Skorokhod ${\rm M 1}$ topology, which is defined for distribution-valued processes.

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arXiv2018

Sobolev regularity for first order Mean Field Games

In this paper we obtain Sobolev estimates for weak solutions of first oder variational Mean Field Game systems with coupling terms that are local function of the density variable. Under some coercivity condition on the coupling, we obtain first order Sobolev estimates for the density variable, while under similar coercivity condition on the Hamiltonian we obtain second order Sobolev estimates for the value function. These results are valid both for stationary and time-dependent problems. In the latter case the estimates are fully global in time, thus we resolve a question which was left open in [PS17]. Our methods apply to a large class of Hamiltonians and coupling functions.

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