论文标题
具有多个平衡的非零和游戏的动态设置值
Dynamic Set Values for Nonzero Sum Games with Multiple Equilibriums
论文作者
论文摘要
非零总和游戏通常具有多个NASH平衡（或没有平衡），并且与零和案例不同，它们在不同的平衡下可能具有不同的值。我们没有专注于单个平衡的存在，而是研究所有平衡的值集，我们称之为游戏的设定值。设定值本质上是唯一的，并且始终存在（可能具有$ \ emptyset $）。与控制文献中的标准值函数相似，它具有许多不错的特性，例如规律性，稳定性，更重要的是动态编程原理。为了获得动态编程原理，有两个主要功能：（i）我们必须使用闭环控件（而不是开环控件）； （ii）即使问题处于状态依赖（马尔可夫）设置，我们也必须允许依赖路径依赖的控件。我们将在有限的时间范围内考虑离散和连续的时间模型。对于后者，我们还将通过某些标准PDE（或依赖路径依赖性PDE）提供双重性方法，这对于数值计算游戏的设定值非常有效。
Nonzero sum games typically have multiple Nash equilibriums (or no equilibrium), and unlike the zero sum case, they may have different values at different equilibriums. Instead of focusing on the existence of individual equilibriums, we study the set of values over all equilibriums, which we call the set value of the game. The set value is unique by nature and always exists (with possible value $\emptyset$). Similar to the standard value function in control literature, it enjoys many nice properties such as regularity, stability, and more importantly the dynamic programming principle. There are two main features in order to obtain the dynamic programming principle: (i) we must use closed-loop controls (instead of open-loop controls); (ii) we must allow for path dependent controls, even if the problem is in a state dependent (Markovian) setting. We shall consider both discrete and continuous time models with finite time horizon. For the latter we will also provide a duality approach through certain standard PDE (or path dependent PDE), which is quite efficient for numerically computing the set value of the game.
