Mean-Field Games: Theory, Numerics and Applications

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