, Volume 2, Issue 1, pp 229-260
Date: 28 Mar 2007

Mean field games

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We survey here some recent studies concerning what we call mean-field models by analogy with Statistical Mechanics and Physics. More precisely, we present three examples of our mean-field approach to modelling in Economics and Finance (or other related subjects...). Roughly speaking, we are concerned with situations that involve a very large number of “rational players” with a limited information (or visibility) on the “game”. Each player chooses his optimal strategy in view of the global (or macroscopic) informations that are available to him and that result from the actions of all players. In the three examples we mention here, we derive a mean-field problem which consists in nonlinear differential equations. These equations are of a new type and our main goal here is to study them and establish their links with various fields of Analysis. We show in particular that these nonlinear problems are essentially well-posed problems i.e., have unique solutions. In addition, we give various limiting cases, examples and possible extensions. And we mention many open problems.

Communicated by: Toshiyuki Kobayashi
This article is based on the 1st Takagi Lectures that the second author delivered at Research Institue for Mathematical Sciences, Kyoto University on November 25 and 26, 2006.
Jean-Michel Lasry and Pierre-Louis Lions: work partially supported by the chair “Finance and sustainable development”