论文标题
控制的平均野外游戏：有限差近似
Mean Field Games of Controls: Finite Difference Approximations
论文作者
论文摘要
我们考虑了一类平均野外游戏，其中代理通过其状态和控件进行互动，我们专注于通用代理试图将其速度（控制）调整为平均速度的情况（该速度是在州空间中的一个社区中制成的）。在这种情况下，在平均野外游戏理论中经常做出的单调性假设并不成立，并且总体上不能期望唯一性。这种模型导致前向后非线性非局部抛物线方程的系统；后者补充了各种边界条件，特别是由于基础控制的随机过程的反射条件而导致的类似诺伊曼的边界条件。本工作涉及上述系统的数值近似值。在描述了有限差方案之后，我们提出了一种迭代方法，用于求解离散设置中出现的非线性方程系统。它结合了延续方法，牛顿迭代和像求解器这样的大型求解的内部回路。数值方法用于模拟两个示例。当模型的参数变化时，我们还对迭代算法的行为进行实验。平均野外游戏的理论（简称MFG）旨在研究确定性或随机差异游戏（NASH Equilibria），因为代理的数量倾向于无穷大。它假设理性的代理人是无法区分的，并且对游戏的影响很小，并且每种策略都受到某些数量的平均数量的影响，具体取决于其他代理的状态（或目前的工作中的控制）。 MFG已在J-M的开创性工作中引入。 LASRY和P-L。狮子[17、18、19]。在工程文献中出现了平均野外游戏的概念，在大约同一时间独立，大约同时，请参阅M.Y.的作品。黄，体育Caines和R.Malham {é} [14，15]。目前的工作涉及平均野外游戏的数值近似，在这些游戏中，代理通过其状态和控件进行互动。它遵循第二作者[16]的更理论工作，该工作致力于对非局部偏微分方程相关系统的数学分析。关于MFG的文献并不多，其中代理也通过其控件相互作用，请参见[13、12、8、10、7、16]。为了强调考虑后一种情况的事实，我们有时会使用控制的术语平均野外游戏和首字母缩写词MFGC。
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.
