Alpar Richard Meszaros: Mean field games and master equations
The theory of mean field games (MFG) was initiated around 2006 by two groups of scientists (Lasry—Lions on the one hand and Huang—Malhamé—Caines on the other hand). The main goal of both groups was to characterize limits of Nash equilibria of stochastic differential games, when the number of agents tends to infinity. In the past decade this theory has found great applications in various fields, including machine learning, mathematical biology, mathematical finance and others. After a short introduction to the theory, we will present the so-called master equation (formulated first by Lions), which is a PDE of hyperbolic nature, set on the space of probability measures. We will discuss about various notions of monotonicity conditions which yield the global in time well-posedness of this PDE, and provides criteria for the uniqueness of MFG Nash equilibria. One of such conditions is deeply related to the concept of displacement convexity developed in the framework of optimal mass transportation. The second part of the talk will be based on several works in collaboration with W. Gangbo, C. Mou and J. Zhang.
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