Mean Field Games, Mean Field Type Control and Extensions

点击翻译按钮获取中文摘要

The term "Mean Field Games" was coined by P.L. Lions (Fields Medalist) and J.M. Lasry, a few years ago. It is a remarkable idea to transfer a well known approach of Physics to Social Sciences. The concept of Mean Field in physics attempts to describe the effect of the media on the motion of a particle, this media being composed of an infinite number of particles, similar to the individual one. Conversely in economics, models consider that agents interact through markets (real and financial) and equilibrium can be obtained through prices. It has been widely acknowledged that these types of models cannot realistically address all the phenomena that one can observe in real life, for instance the issue of systemic risk. Mean Field Games is a novel approach to understand what is missing in the models. It has been a spectacular success. Besides economics and finance, it has been very fruitful in many areas such as: traffic control, network analysis, as well as in understanding how technology can expand, and how environmental aspects impact growth. It turns out that the theory can handle also many new considerations of risk management. Independently of the applications, these concepts have completely changed control theory, differential games and introduced new types of partial differential systems. However, Mean Field Games is limited to agents who are identical, like particles. This is a serious limitation, since in social sciences, unlike in physics, the reality is more a situation of coalitions or dominant players. This is the major objective of this proposal: To study extensions to consider coalitions. One has to solve much more complex systems of partial differential equations. A second objective is to develop an Hamilton Jacobi Bellman equation approach to Mean Field Control Type problems, which has never been done before, because of a basic difficulty, called "Time inconsistency" inherent to Mean Field Control (different from Mean Field Games). A third objective is to develop ideas relevant to risk analysis, in which one cannot be satisfied in optimizing an average. Since risk aspects have become predominant in engineering as well as in economics, this direction can have very broad implications, in many areas of strategic importance.