Mean-Field Games: Theory, Numerics and Applications

2019-01-18T21:45:26Z (GMT) by Joao Saude
In dynamical systems with a large number of agents, competitive, and cooperative phenomena<br>occur in a broad range of designed and natural settings. Such as communications, environmental,<br>biological, transportation, trading, and energy systems, and they underlie much<br>economic and financial behavior. Analysis of such systems is intractable using the classical<br>finite N-players game theoretic methods is often intractable. The mean-field games (MFG)<br>framework was developed to study these large systems, modeling them as a continuum of<br>rational agents that interact in a non-cooperative way.<br>In this thesis, we address some theoretical aspects and propose a definition of relaxed<br>solution for MFG that allows establishing uniqueness under minimal regularity hypothesis.<br>We also propose a price impact model, that is a modification of the Merton’s portfolio problem<br>where we consider that assets’ transactions influence their prices.<br>We also study numerical methods for continuous time finite-state MFG that satisfy a<br>monotonicity condition, and for time-dependent first-order nonlocal MFG. MFG is determined<br>by a system of differential equations with initial and terminal boundary conditions. These<br>non-standard conditions make the numerical approximation of MFG difficult. Using the<br>monotonicity condition, we build a flow that is a contraction and whose fixed points solve<br>both for stationary and time-dependent MFG.<br>We also develop Fourier approximation methods for the solutions of first-order nonlocal<br>mean-field games (MFG) systems. Using Fourier expansion techniques, we approximate a<br>given MFG system by a simpler one that is equivalent to a convex optimization problem over<br>a finite-dimensional subspace of continuous curves. We solve this problem using a variant of<br>a primal-dual hybrid gradient method.<br>Finally, we introduce a price-formation model where a large number of small players can<br>store and trade electricity. Our model is a constrained MFG where the price is a Lagrange<br>multiplier for the supply versus demand balance condition. We establish the existence of a<br>unique solution using a fixed-point argument. Then, we study linear-quadratic models that<br>hold specific solutions, and we find that the dynamic price depends linearly on the instant<br>aggregated consumption.