Mean field games were introduced independently by J-M. Lasry and P-L. Lions, and by M. Huang, R.P. Malham\'e and P. E. Caines, in order to bring a new approach to optimization problems with a large number of interacting agents. The description of such models split in two parts, one describing the evolution of the density of players in some parameter space, the other the value of a cost functional each player tries to minimize for himself, anticipating on the rational behavior of the others. Quadratic Mean Field Games form a particular class among these systems, in which the dynamics of each player is governed by a controlled Langevin equation with an associated cost functional quadratic in the control parameter. In such cases, there exists a deep relationship with the non-linear Schr\"odinger equation in imaginary time, connexion which lead to effective approximation schemes as well as a better understanding of the behavior of Mean Field Games. The aim of this paper is to serve as an introduction to Quadratic Mean Field Games and their connexion with the non-linear Schr\"odinger equation, providing to physicists a good entry point into this new and exciting field.

中文翻译：

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二次平均场游戏

平均场博弈是由 JM 独立引入的。拉斯里和 PL。Lions 以及 M. Huang、RP Malham\'e 和 PE Caines，旨在为具有大量交互代理的优化问题带来一种新方法。这种模型的描述分为两部分，一部分描述了某个参数空间中玩家密度的演变，另一部分描述了每个玩家试图为自己最小化的成本函数的值，并预测其他人的理性行为。二次平均场博弈在这些系统中形成了一个特定的类，其中每个玩家的动态由受控朗之万方程控制，控制参数中带有相关的成本函数二次方程。在这种情况下，与虚时间的非线性 Schr\"odinger 方程存在很深的关系，连接导致有效的近似方案以及更好地理解平均场游戏的行为。本文的目的是介绍二次平均场博弈及其与非线性 Schr\"odinger 方程的联系，为物理学家提供进入这个令人兴奋的新领域的良好切入点。