Mean field games with imperfect information

Abstract

In this thesis, three topics in Mean Field Games in the absence of complete information have been studied.

The first part of the thesis focus on Mean Field Stackelberg Games between a large group of followers and a leader, in such a way that each follower is subject to a delay effect inherited from the leader. The case with delays being identical among the followers in the population is first considered. Under mild assumptions of regular enough coefficients, the whole Stackelberg game problem is solved via stochastic maximum principle. The solution could be represented by a system of six coupled forward backward stochastic differential equations. A comprehensive study on the particular Linear Quadratic case has been provided. By considering the corresponding linear functional, the time-independent sufficient condition which warrants the unique existence of the solution of the whole Stackelberg game is obtained. Several numerical examples are also demonstrated.

The second work studies another class of Stackelberg games, under a Linear Quadratic setting, in the presence with an additional leader. Given the trajectories of the mean field term and two leaders, the follower's optimal control problem is first solved. Depending on whether or not the leaders cooperate, the solutions of the respective Pareto and Nash games between the leaders are obtained, which can be represented by systems of forward backward stochastic functional differential equations. To numerically implement the obtained results, explicit expression of solutions of the whole problem: Mean Field Game among the followers and Nash (and Pareto) Game between the leaders, are provided. Finally, several examples are given to study the impact of different games on the cost functionals of the followers. An interesting example shows that the population are worse off as the leaders cooperate.

The last part of the thesis studies discrete time partially observable mean field systems in the presence of a common noise. Each player makes decision solely based on the observable processes but not the common noise. Both the mean field game and the associated mean field type stochastic control problem are formulated. The mean field type control problem is solved by adopting the classical discrete time Kalman filter with notable modifications; indeed, the unique existence of the resulting forward-backward stochastic difference system is then established by Separation Principle. The mean field game problem is also solved via an application of stochastic maximum principle, while the existence of the mean field equilibrium is shown by the Schauder's fixed point theorem.