On Some Generalized Mean Field Games

The Mean Field Game (MFG) theory was introduced independently by Caines-Huang-Malhame and Lasry-Lions. It serves as a systematic framework for characterizing the continuum limits of N-player stochastic games, which are often simpler to analyze. The MFG theory has developed a rich mathematical theory and a wide variety of applications, including economics, engineering, finance, social science and many others. Though many efforts have been spent on the MFG theory in the past decade, the MFG framework is fundamentally limited to the games with homogeneous minor players where the mean field interaction is only via the distribution of their states. Apparently, many MFG models in the reality are beyond such well-studied ones. For instance, the real-world stock market consists of not only individual traders (with a small amount of money) but also fund companies (with a large amount of money). On one hand, a fund company itself can have a significant impact on the prices of stocks they trade and thus affect the individual traders' decisions. On the other hand, an individual trader has little influence on the prices of the traded stocks however all the individual traders as a whole can have an impact on the prices. This is a typical model of MFGs with major and minor players. Again take the stock market as an example. It is common sense that it is not good to buy a particular stock while other traders are buying but it is good to buy while others are selling. In such configuration, the traders act strategically, taking into account all the information they have (not just others' wealth). This leads to a Nash equilibrium, in which the mean field interaction depends on the others' strategies. This type of MFGs is called the MFG of controls. Our current proposed project is devoted to the systematic studies on the above two types of the MFG models: MFGs with a major player, and MFGs of controls. Our studies are expected through their master equations. The master equations serve as a powerful tool in the MFG framework and it decodes all the information on the MFG. One of the main methods to solve the master equations is via the corresponding mean field forward-backward stochastic differential equations.