Dynamic Set Values for Nonzero-Sum Games with Multiple Equilibriums
针对非零和博弈存在多个纳什均衡且值不同的情况，研究所有均衡值的集合（集合值），证明其满足动态规划原理，并给出离散和连续时间模型下的计算方法。
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 [Formula: see text]). 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); and (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.