On Quasi-stationary Mean Field Games Models

Abstract

We explore a mechanism of decision-making in Mean Field Games with myopic players. At each instant, agents set a strategy which optimizes their expected future cost by assuming their environment as immutable. As the system evolves, the players observe the evolution of the system and adapt to their new environment without anticipating. With a specific cost structures, these models give rise to coupled systems of partial differential equations of quasi-stationary nature. We provide sufficient conditions for the existence and uniqueness of classical solutions for these systems, and give a rigorous derivation of these systems from N -players stochastic differential games models. Finally, we show that the population can self-organize and converge exponentially fast to the ergodic Mean Field Games equilibrium, if the initial distribution is sufficiently close to it and the Hamiltonian is quadratic.

A continuous dependence estimate for viscous Hamilton-Jacobi equations on networks with applications

“…Quasi-stationary Mean Field Games, introduced in [21] (see also [11]), modelize the case when the agent cannot predict the evolution of the population in the future, as in the classical MFG theory, but, at each instant, it decides its behaviour only on the basis of the information available at the current time. This feature leads to systems given by an evolutive Fokker-Planck equation and a stationary HJ equation (which in fact depends on time through the cost).…”

“…In the classical formulation, MFG lead to the study of a coupled system of two evolutive PDEs, a backward HJ equation for the value function of the representative agent, a forward Fokker-Planck (FP for short) equation for the distribution of the agents. Recently, a different strategy mechanisms from classical MFG theory has been proposed in [21] (see also [11]): the agents are myopic and choose their strategy only according to the information available at present time, without forecasting the future evolution. In this case, the Nash equilibria for the distribution of the agents are characterized by a quasi-stationary MFG system, which is composed of a stationary HJ equation and a evolutive Fokker-Planck equation.…”

On quasi-stationary Mean Field Games of Controls

“…In this theory, the individual is assumed to be able to forecast the behaviour of the population at any later time, a somewhat restrictive assumption for some models such as pedestrian motion. In [17], it is considered a different strategy mechanism: the agent assumes that the environment is immutable and, at each instant, it decides its behaviour only on the basis of the information available at the current time without anticipating the future. This leads to the study a class of quasi-stationary MFG systems, where a stationary HJB equation is coupled with an evolutive Fokker-Planck equation.…”

“…The corresponding MFG system, with respect to the classical one, involves an additional fixed-point equation for the joint distribution of agent state and control. Aim of this paper it to extend the model developed in [17] to Mean Field Games of Controls. Hence we deal with the following system…”

“…In the classical setting, this regularity follows by the parabolicity of the HJB equation. In the quasi-stationary one [17], it is first obtained, via a continuous dependence estimate, a bound on the C 2 -norm of u(t)−u(s) by means of the Wasserstein distance d 1 (m(t), m(s)).…”