# Mean Field Games with monotonous interactions through the law of states and controls of the agents

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## A priori estimates on derivatives of u

Bernstein methods are useful tools when studying HJB equations or MFG systems. They allow one to obtain a priori estimates on rxu by considering the partial differential equations satisfied by some well-chosen functions depending on u and rxu. See for example the video of the lecture of P.L. Lions on November the 23rd 2018 , in which Bernstein estimates are derived for MFG systems without interactions through controls. More precisely, P.L. Lions used the function defined by jrxuj2 e􀀀u, for small . Here this method might work only if we knew a uniform estimates on kuk1 and if q = 2. After significant changes in the latter method, we can derive an estimate on u which is weaker than the one for MFG without interactions through controls. Namely, we state that krxuk1 is bounded by a quantity that depends linearly on kuk1 by studying the functions w and ‘ defined in (2.4.11) below. To our knowledge, such estimates for systems of MFG with nonlocal dependency on rxu (or more generally for MFG systems in which we do not have a uniform a priori estimate on u) are new in the literature. We believe that this result may hold for more general HJB equations with nonlocal dependency .

### Exhaustible ressource model with nonpositively correlated ressources

This model is often referred to as Bertrand and Cournot competition model for exhaustible ressources, introduced in the independent works of Cournot  and Bertrand ; its mean field game version in dimension one was introduced in  and numerically analyzed in ; for theoretical results see [25, 62, 73, 63]. We consider a continuum of producers selling exhaustible ressources. The production of a representative agent is (qt)t2[0;T ]; the agents differ in their production capacities Xt 2 T (the state variable), that satifies, dXt = 􀀀qtdt + p 2dWt.
where > 0 and W is a Brownian motion. Each producer is selling a different ressource and has her own consumers. However, the ressources are substitutable and any consumer may change her mind and buy from a competitor depending on the degree of competition in the game (which is characterized by  » in the linear demand case below for instance). Therefore, the selling price per unit of ressource that a producer can make when she sales q units of ressource, depends naturally on q and on the quantity produced by the other agents. The price satisfies a supply-demand relationship, and is given by P (q; q), where q is the aggregate demand which depends on the overall distribution of productions of the agents. A producer tries to maximize her profit, or equivalently to minimize the following quantity.

#### Price impact models with bid and ask prices

The price impact model without bid and ask prices is inspired by the Almgren and Chriss’s model , and was introduced in the MFG literature in  and  where existence and uniqueness results are proved when the admissible controls stay in a compact set. Here we consider an extension with bid and ask prices.
We suppose that a continuum of agents are trading an asset, the state of a representative agent is Xt the amount of this asset she owns. Her control is the quantity she buys (if 0) or sell (if < 0). The state space is the one-dimensional torus T, and Xt is given by, dXt = tdt + dWt. where W is a Brownian motion, and > 0 is a real constant. We define St as the asking price of the asset, and  » ((t)) as the difference between the bidding and asking prices, where (t) is the law of (Xt; t). The agent buys at the bidding price St + » (t), thus her cash is given by dKt = 􀀀(tSt + t » ((t)) + `(t)) dt.

First-order flocking model with velocity as controls

Cucker and Smale proposed a form of Vicseck model in  to illustrate the behavior of flocks of birds. This model is of second-order in the sense that the state of an agent is given by a couple (x; v) standing for her position and velocity respectively, and the equation of evolution of her state involves considering her acceleration.
A game version of this model in which an agent controls her acceleration has been introduced in , the authors derived a MFG formulation in the infinite horizon case.
Here we are interested in the finite horizon problem which was studied in [43, 41]. This model is still of second-order. More precisely the state of an agent is given by (Xt; Vt)t2[0;T ] respectively her position and velocity, two random processes which satisfy the following system of stochastic differential equations, dXt = Vtdt.

A brief discussion on the mathematical analysis of (3.1.1)

Recall that the Hamiltonian of the problem is (x; p; ) 7! H(x; p; ), (x; p; ) 2 Rd P 􀀀 Rd .
From the viewpoint of mathematical analysis, a priori estimates for (3.1.1) are more difficult to obtain than in the case when the agents interact only via the distribution of states m. Indeed, in the latter case, if for example the costs f and are uniformly bounded, then a priori estimates on kuk1 stem from the maximum principle for secondorder parabolic equations. By contrast, since the Hamiltonian in (3.1.1) depends nonlocally on rxu, the maximum principle applied to the HJB equation only permits to bound kuk1 by a quantity which depends (quadratically under standard assumptions on H) on krxuk1, and this information may be useless without additional arguments. If the agents interact only through the distribution of states and if the Hamiltonian depends separately on p and m, a natural assumption is that the latter is monotone with respect to m, see [83, 84, 85]; it implies existence and uniqueness of solutions, see in [87, 85]. Such an assumption is quite sensible in many situations, since it models the aversion of the agents to highly crowded regions of the state space. It is possible to extend these arguments to MFGCs, see  for a probabilistic point of view and  for a PDE point of view, and the monotonicity assumption then means that the agents favor controls that are opposite to the main stream. In  and in the present work, we prefer to avoid such an assumption, because it is generally not satisfied, at least in models of crowd motion: indeed, in models of traffic or pedestrian flows, a generic agent would rather try to adjust her speed (control) to the average speed in a neighborhood of her position.
The third equation in (3.1.1) can be seen as a fixed point problem for given u and m, which turns to be well-posed under the Lasry-Lions monotonicity assumption adapted to MFGC, provided that u and m are smooth enough. We shall replace this assumption by a new structural condition which has been introduced in , namely that Hp depends linearly on the variable and is a contraction with respect to (using a suitable distance on probability measures). In the context of crowd motion, this structural condition is satisfied if the representative agent targets controls that are proportional to an average of the controls chosen by the other agents nearby, with a positive proportionality coefficient smaller than one. Were this coefficient equal to or larger than one, it would be easy to cook up examples in which there is no solution to (3.1.1) or even to the N-agent game, see Remark 3.4.3 below.

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1 Introduction
1.1 La théorie des jeux à champ moyen
1.1.1 Le jeu à N joueurs
1.1.2 Le système de jeu à champ moyen
1.1.3 Jeux à champ moyen de contrôle
1.2 Organisation de la thèse
1.2.1 À propos des solutions classiques du système de jeu à champ moyen de contrôle
1.2.2 Approximation par la méthode des différences finies du système de jeu à champ moyen de contrôle
1.2.3 Jeux à champ moyen avec interactions monotones par la loi des états et des contrôles des joueurs
1.2.4 Un algorithme primal-dual pour des jeux à champ moyen dynamique du second-ordre et à couplage local
1.2.5 Perspectives et travaux futurs
2 On Classical Solutions to the Mean Field Game System of Controls
2.1 Introduction
2.2 Notations and assumptions
2.2.1 Notations and definitions
2.2.2 Assumptions
2.2.3 Main results
2.3 The fixed point relation in and the proof of Lemma 2.2.3
2.4 A priori estimates and the proof of Lemma 2.2.4
2.4.1 A priori estimates on u
2.4.2 A priori estimates on m
2.4.3 A priori estimates on derivatives of u
2.5 Existence and uniqueness results under additional assumptions
2.5.1 Solving the MFGC systems for M < 1
2.5.2 Existence results when q0 q0
2.5.3 Existence results which do not need the assumption q0 < q0
2.5.4 Existence and uniqueness results with a short-time horizon assumption
2.6 Applications
2.6.1 Exhaustible ressource model with nonpositively correlated ressources
2.6.2 Price impact models with bid and ask prices
2.6.3 First-order flocking model with velocity as controls
2.6.4 A model of crowd motion
3 Mean Field Games of Controls: Finite Difference Approximations
3.1 Introduction
3.1.1 A brief discussion on the mathematical analysis of (3.1.1)
3.1.2 A more detailed description of the considered class of MFGCs
3.1.3 Organization of the paper
3.2 Finite difference methods
3.2.1 Notations and definitions
3.2.2 The scheme
3.2.3 Solving the discrete version of the Hamilton-Jacobi-Bellman equation
3.2.4 Solving the discrete version of the Fokker-Planck-Kolmogorov equation
3.3 Newton algorithms for solving the whole system (3.2.8)-(3.2.17)
3.3.1 The coupling cost and the average drift
3.3.2 The linearized operators
3.3.3 The algorithm for solving (3.2.8)-(3.2.17)
3.4 Numerical simulations
3.4.1 First example
3.4.2 Second example
4 Mean Field Games with monotonous interactions through the law of states and controls of the agents
4.1 Introduction
4.2 Assumptions
4.2.1 Notations
4.2.2 Hypotheses
4.2.3 Main results
4.2.4 Properties of the Lagrangian and the Hamiltonian in (4.2.3) and (4.2.5)103
4.3 Applications
4.3.1 Exhaustible ressource model
4.3.2 A model of crowd motion
4.4 The fixed point (4.2.5c) and the proof of Lemma 4.2.4
4.4.1 Leray-Schauder Theorem for solving the fixed point in
4.4.2 The continuity of the fixed point in time
4.5 A priori estimates for the solutions to (4.2.5)
4.6 Existence and Uniqueness Results
4.6.1 Proof of Theorem 4.2.6: existence of solutions to (4.2.5)
4.6.2 Proof of Theorem 4.2.7: passing from the torus to Rd
4.6.3 Proof of Theorem 4.2.8: uniqueness of the solutions to (4.2.3) and (4.2.5)
4.6.4 Theorems 4.2.2 and 4.2.3: existence and uniqueness of the solution to (4.1.6)
5 On the implementation of a primal-dual algorithm for second order timedependent mean field games with local couplings
5.1 Introduction
5.2 Preliminaries and the finite difference scheme
5.3 The finite dimensional variational problem and the discrete MFG system .
5.4 A primal-dual algorithm
5.5 Preconditioning strategies
5.5.1 Multigrid preconditioner
5.5.2 Numerical Tests

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