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Networks & Heterogeneous Media

2012 , Volume 7 , Issue 2

Special Issue
on Mean Field Games

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Preface
Fabio Camilli, Italo Capuzzo Dolcetta and Maurizio Falcone
2012, 7(2): i-ii doi: 10.3934/nhm.2012.7.2i +[Abstract](225) +[PDF](101.9KB)
Abstract:
The theory of Mean Field Games (MFG, in short) is a branch of the theory of Differential Games which aims at modeling and analyzing complex decision processes involving a large number of indistinguishable rational agents who have individually a very small influence on the overall system and are, on the other hand, influenced by the mass of the other agents. The name comes from particle physics where it is common to consider interactions among particles as an external mean field which influences the particles. In spite of the optimization made by rational agents, playing the role of particles in such models, appropriate mean field equations can be derived to replace the many particles interactions by a single problem with an appropriately chosen external mean field which takes into account the global behavior of the individuals.

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Iterative strategies for solving linearized discrete mean field games systems
Yves Achdou and Victor Perez
2012, 7(2): 197-217 doi: 10.3934/nhm.2012.7.197 +[Abstract](263) +[PDF](344.1KB)
Abstract:
Mean fields games (MFG) describe the asymptotic behavior of stochastic differential games in which the number of players tends to $+\infty$. Under suitable assumptions, they lead to a new kind of system of two partial differential equations: a forward Bellman equation coupled with a backward Fokker-Planck equation. In earlier articles, finite difference schemes preserving the structure of the system have been proposed and studied. They lead to large systems of nonlinear equations in finite dimension. A possible way of numerically solving the latter is to use inexact Newton methods: a Newton step consists of solving a linearized discrete MFG system. The forward-backward character of the MFG system makes it impossible to use time marching methods. In the present work, we propose three families of iterative strategies for solving the linearized discrete MFG systems, most of which involve suitable multigrid solvers or preconditioners.
From discrete to continuous Wardrop equilibria
Jean-Bernard Baillon and Guillaume Carlier
2012, 7(2): 219-241 doi: 10.3934/nhm.2012.7.219 +[Abstract](283) +[PDF](454.1KB)
Abstract:
The notion of Wardrop equilibrium in congested networks has been very popular in congested traffic modelling since its introduction in the early 50's, it is also well-known that Wardrop equilibria may be obtained by some convex minimization problem. In this paper, in the framework of $\Gamma$-convergence theory, we analyze what happens when a cartesian network becomes very dense. The continuous model we obtain this way is very similar to the continuous model of optimal transport with congestion of Carlier, Jimenez and Santambrogio [6] except that it keeps track of the anisotropy of the network.
Explicit solutions of some linear-quadratic mean field games
Martino Bardi
2012, 7(2): 243-261 doi: 10.3934/nhm.2012.7.243 +[Abstract](441) +[PDF](433.5KB)
Abstract:
We consider $N$-person differential games involving linear systems affected by white noise, running cost quadratic in the control and in the displacement of the state from a reference position, and with long-time-average integral cost functional. We solve an associated system of Hamilton-Jacobi-Bellman and Kolmogorov-Fokker-Planck equations and find explicit Nash equilibria in the form of linear feedbacks. Next we compute the limit as the number $N$ of players goes to infinity, assuming they are almost identical and with suitable scalings of the parameters. This provides a quadratic-Gaussian solution to a system of two differential equations of the kind introduced by Lasry and Lions in the theory of Mean Field Games [22]. Under a natural normalization the uniqueness of this solution depends on the sign of a single parameter. We also discuss some singular limits, such as vanishing noise, cheap control, vanishing discount. Finally, we compare the L-Q model with other Mean Field models of population distribution.
A semi-discrete approximation for a first order mean field game problem
Fabio Camilli and Francisco Silva
2012, 7(2): 263-277 doi: 10.3934/nhm.2012.7.263 +[Abstract](380) +[PDF](437.1KB)
Abstract:
In this article we consider a model first order mean field game problem, introduced by J.M. Lasry and P.L. Lions in [18]. Its solution $(v,m)$ can be obtained as the limit of the solutions of the second order mean field game problems, when the noise parameter tends to zero (see [18]). We propose a semi-discrete in time approximation of the system and, under natural assumptions, we prove that it is well posed and that it converges to $(v,m)$ when the discretization parameter tends to zero.
Long time average of mean field games
Pierre Cardaliaguet, Jean-Michel Lasry, Pierre-Louis Lions and Alessio Porretta
2012, 7(2): 279-301 doi: 10.3934/nhm.2012.7.279 +[Abstract](215) +[PDF](429.4KB)
Abstract:
We consider a model of mean field games system defined on a time interval $[0,T]$ and investigate its asymptotic behavior as the horizon $T$ tends to infinity. We show that the system, rescaled in a suitable way, converges to a stationary ergodic mean field game. The convergence holds with exponential rate and relies on energy estimates and the Hamiltonian structure of the system.
A-priori estimates for stationary mean-field games
Diogo A. Gomes, Gabriel E. Pires and Héctor Sánchez-Morgado
2012, 7(2): 303-314 doi: 10.3934/nhm.2012.7.303 +[Abstract](284) +[PDF](366.7KB)
Abstract:
In this paper we establish a new class of a-priori estimates for stationary mean-field games which have a quasi-variational structure. In particular we prove $W^{1,2}$ estimates for the value function $u$ and that the players distribution $m$ satisfies $\sqrt{m}\in W^{1,2}$. We discuss further results for power-like nonlinearities and prove higher regularity if the space dimension is 2. In particular we also obtain in this last case $W^{2,p}$ estimates for $u$.
New numerical methods for mean field games with quadratic costs
Olivier Guéant
2012, 7(2): 315-336 doi: 10.3934/nhm.2012.7.315 +[Abstract](262) +[PDF](1602.8KB)
Abstract:
Mean field games have been introduced by J.-M. Lasry and P.-L. Lions in [13, 14, 15] as the limit case of stochastic differential games when the number of players goes to $+\infty$. In the case of quadratic costs, we present two changes of variables that allow to transform the mean field games (MFG) equations into two simpler systems of equations. The first change of variables, introduced in [11], leads to two heat equations with nonlinear source terms. The second change of variables, which is introduced for the first time in this paper, leads to two Hamilton-Jacobi-Bellman equations. Numerical methods based on these equations are presented and numerical experiments are carried out.
A modest proposal for MFG with density constraints
Filippo Santambrogio
2012, 7(2): 337-347 doi: 10.3934/nhm.2012.7.337 +[Abstract](224) +[PDF](354.6KB)
Abstract:
We consider a typical problem in Mean Field Games: the congestion case, where in the cost that agents optimize there is a penalization for passing through zones with high density of agents, in a deterministic framework. This equilibrium problem is known to be equivalent to the optimization of a global functional including an $L^p$ norm of the density. The question arises as to produce a similar model replacing the $L^p$ penalization with an $L^\infty$ constraint, but the simplest approaches do not give meaningful definitions. Taking into account recent works about crowd motion, where the density constraint $\rho\leq 1$ was treated in terms of projections of the velocity field onto the set of admissible velocity (with a constraint on the divergence) and a pressure field was introduced, we propose a definition and write a system of PDEs including the usual Hamilton-Jacobi equation coupled with the continuity equation. For this system, we analyze an example and propose some open problems.
Liquidity generated by heterogeneous beliefs and costly estimations
Min Shen and Gabriel Turinici
2012, 7(2): 349-361 doi: 10.3934/nhm.2012.7.349 +[Abstract](298) +[PDF](359.4KB)
Abstract:
We study the liquidity, defined as the size of the trading volume, in a situation where an infinite number of agents with heterogeneous beliefs reach a trade-off between the cost of a precise estimation (variable depending on the agent) and the expected wealth from trading. The "true" asset price is not known and the market price is set at a level that clears the market. We show that, under some technical assumptions, the model has natural properties such as monotony of supply and demand functions with respect to the price, existence of an equilibrium and monotony with respect to the marginal cost of information. We also situate our approach within the Mean Field Games (MFG) framework of Lions and Lasry which allows to obtain an interpretation as a limit of Nash equilibrium for an infinite number of agents.

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