Mean Field Games, Mean Field Type Control and Extensions

几年前，P.L.Lions(菲尔兹奖牌获得者)和J.M.Lasry(J.M.Lasry)首创了Mean field Games(Mean Field Games)一词。将一种著名的物理学方法转化为社会科学是一个了不起的想法。物理学中平均场的概念试图描述介质对粒子运动的影响，这种介质由无数个粒子组成，类似于单个粒子。相反，在经济学中，模型认为代理人通过市场(真实的和金融的)相互作用，均衡可以通过价格获得。人们普遍认为，这些类型的模型不能现实地解决人们在现实生活中可以观察到的所有现象，例如系统性风险问题。平均场游戏是一种新的方法来理解模型中缺失的东西。这是一次惊人的成功。除了经济和金融，它在许多领域都取得了非常丰硕的成果，例如：交通控制、网络分析，以及在理解技术如何扩展以及环境因素如何影响增长方面。事实证明，该理论还可以处理风险管理的许多新考虑。与应用无关，这些概念完全改变了控制理论和微分对策，并引入了新类型的偏微分系统。然而，平均场游戏仅限于相同的代理，如粒子。这是一个严重的限制，因为在社会科学中，与物理学不同，现实更多的是联盟或主导参与者的情况。这是这项提案的主要目标：研究扩展以考虑联盟。人们不得不解更复杂的偏微分方程组。第二个目标是开发一种汉密尔顿·雅各比·贝尔曼方程方法来解决平均场控制类型的问题，这是以前从未做过的，因为平均场控制(不同于平均场游戏)固有的一个基本困难，称为“时间不一致”。第三个目标是开发与风险分析相关的想法，在这种想法中，人们不能满足于优化平均值。由于风险因素已成为工程学和经济学中的主导因素，这一方向在许多具有战略重要性的领域可能具有非常广泛的影响。

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.