控制的平均场对策：Finite Difference近似
Mean Field Games of Controls: Finite Difference Approximations
摘要：
我们考虑一类平均场游戏，其中代理通过他们的状态和控制进行交互，我们专注于通用代理试图将她的速度（控制）调整为平均速度（平均值是在状态空间的邻域中）的情况。在这种情况下，在平均场博弈理论中经常做出的单调性假设并不成立，并且通常不能期望唯一性。这样的模型导致系统的向前向后非线性非局部抛物型方程，后者补充了各种边界条件，特别是Neumann的边界条件，来自反射条件的基础controled随机过程。本工作涉及上述系统的数值逼近。在描述了有限差分格式后，我们提出了一种迭代法，用于求解在离散设置中出现的非线性方程组的系统，它结合了连续方法，牛顿迭代和内循环的双梯度求解器。用数值方法对两个算例进行了数值模拟。我们还做了实验的行为时，模型的参数变化的迭代算法。平均场博弈理论（Mean Field Game，简称MFG）是研究在博弈主体数量趋于无穷大时的确定性或随机性微分对策（Nash equilibrium）的理论。它假设理性主体是不可区分的，并且单独对游戏的影响可以忽略不计，并且每个单独的策略都受到一些数量平均值的影响，这些数量平均值取决于其他主体的状态（或本工作中的控制）。MFG已经在J-M的开创性工作中引入。拉斯里和P-L。狮子[17，18，19]。独立地，大约在同一时间，平均场游戏的概念出现在工程文献中，见M.Y. Huang，P.E. Caines和R.Malham{é} [14，15].目前的工作涉及平均场游戏的数值逼近，其中代理人通过他们的状态和控制进行交互;它遵循第二作者的更多理论工作，[16]，致力于非局部偏微分方程的相关系统的数学分析。没有太多关于MFG的文献，其中代理人也通过其控制相互作用，见[13，12，8，10，7，16]。为了强调考虑的是后一种情况，我们有时会使用“控制的平均场博弈”这个术语和MFGC这个缩写。
英文摘要：
We consider a class of mean field games in which the agents interact through both their states and controls, and we focus on situations in which a generic agent tries to adjust her speed (control) to an average speed (the average is made in a neighborhood in the state space). In such cases, the monotonicity assumptions that are frequently made in the theory of mean field games do not hold, and uniqueness cannot be expected in general. Such model lead to systems of forward-backward nonlinear nonlocal parabolic equations; the latter are supplemented with various kinds of boundary conditions, in particular Neumann-like boundary conditions stemming from reflection conditions on the underlying controled stochastic processes. The present work deals with numerical approximations of the above mentioned systems. After describing the finite difference scheme, we propose an iterative method for solving the systems of nonlinear equations that arise in the discrete setting; it combines a continuation method, Newton iterations and inner loops of a bigradient like solver. The numerical method is used for simulating two examples. We also make experiments on the behaviour of the iterative algorithm when the parameters of the model vary. The theory of mean field games, (MFGs for short), aims at studying deterministic or stochastic differential games (Nash equilibria) as the number of agents tends to infinity. It supposes that the rational agents are indistinguishable and individually have a negligible influence on the game, and that each individual strategy is influenced by some averages of quantities depending on the states (or the controls as in the present work) of the other agents. MFGs have been introduced in the pioneering works of J-M. Lasry and P-L. Lions [17, 18, 19]. Independently and at approximately the same time, the notion of mean field games arose in the engineering literature, see the works of M.Y. Huang, P.E. Caines and R.Malham{é} [14, 15]. The present work deals with numerical approximations of mean field games in which the agents interact through both their states and controls; it follows a more theoretical work by the second author, [16], which is devoted to the mathematical analysis of the related systems of nonlocal partial differential equations. There is not much literature on MFGs in which the agents also interact through their controls, see [13, 12, 8, 10, 7, 16]. To stress the fact that the latter situation is considered, we will sometimes use the terminology mean field games of control and the acronym MFGC.
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