Nash and Stackelberg Differential Games

• Published in: Chinese annals of mathematics. Serie B Vol. 33; no. 3; pp. 317 - 332
• Main Authors: Bensoussan, Alain, Frehse, Jens, Vogelgesang, Jens
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.05.2012
• Subjects: Applications of Mathematics; Mathematics; Mathematics and Statistics; 主从; 偏微分方程; 反馈策略; 哈密顿结构; 微分对策; 状态方程; 纳什; 规划参数; Stochastic games; Bellman equation; Nonlinear elliptic and parabolic equations; 35J60; Hamiltonians; 35B65; Stochastic differential games; 35K55; 35J55
• ISSN: 0252-9599; 1860-6261
• DOI: 10.1007/s11401-012-0716-1

A large class of stochastic differential games for several players is considered in this paper. The class includes Nash differential games as well as Stackelberg differential games. A mix is possible. The existence of feedback strategies under general conditions is proved. The limitations concern the functionals in which the state and the controls appear separately. This is also true for the state equations. The controls appear in a quadratic form for the payoff and linearly in the state equation. The most serious restriction is the dimension of the state equation, which cannot exceed 2. The reason comes from PDE （partial differential equations） techniques used in studying the system of Bellman equations obtained by Dynamic Programming arguments. In the authors＇ previous work in 2002, there is not such a restriction, but there are serious restrictions on the structure of the Hamiltonians, which are violated in the applications dealt with in this article.