Mean Field Games with Monotonicity and Anti-monotonicity Conditions

Project: Research

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Initiated independently by Caines-Huang-Malhame and Lasry-Lions, Mean Field Games (MFGs) have attracted very strong attention in the past decade. The theory of MFGs has rapidly developed into one of the most significant tools towards the study of the Nash equilibrium behavior of large systems. Such problems consider limit behavior of large systems where the homogenous strategic players interact with each other in certain symmetric way. They have a wide variety of applications, including economics, engineering, finance, social science and many others. Unlike Mean Field Control (MFC) problems, a MFG problem is an infinite dimensional fixed point problem. The fixed point here stands for the Nash equilibrium of the MFG and the value function at the Nash equilibrium gives rise to the candidate solution to the master equation. The master equation, proposed by Lasry-Lions, serves as a powerful tool in the MFG theory, which decodes all the information on the MFG. Though many serious efforts on the study of the MFG theory have been made in the past years, the literature on volatility controlled MFGs still remains very few; it also misses general theories on MFGs with major and minor players. The well-studied category of MFG models is essentially limited to drift controlled MFGs with homogenous minor players. Apparently, many MFG models in economics and finance are beyond such well-studied MFGs. For instance, usually the real-world market does not consist of only tiny corporations. Some large corporations (oligopolists) can have a significant impact on all the tiny corporations while they are only affected by all the tiny corporations as a whole. Our current proposed project is devoted to the systematic study of volatility controlled MFGs and also MFGs with a major player through their master equations. The key is to derive certain monotonicity and anti-monotonicity conditions for the data. A MFG master equation can be described through a forward-backward system of mean field stochastic differential equations or stochastic partial differential equations. Certain monotonicity/anti-monotonicity is a crucial condition to ensure such system admits a unique solution. 


Project number9043379
Grant typeGRF
StatusNot started
Effective start/end date1/01/23 → …