Decision-implementation complexity of cooperative game systems

• Published in: Science China. Information sciences Vol. 60; no. 11; pp. 176 - 193
• Main Authors: Xu, Changbao, Zhao, Yanlong, Zhang, Ji-Feng
• Format: Journal Article
• Language: English
• Published: Beijing Science China Press 01.11.2017; Springer Nature B.V
• Subjects: Algorithms; Complexity; Computer Science; Control theory; DIC; Game theory; Games; Information Systems and Communication Service; Research Paper; 决策; 合作博弈; 复杂性; 执行系统; 控制协议; 解析表达式; 计算复杂度; optimal control; cooperative game; twophase protocol; distributed protocol; decision-implementation complexity
• ISSN: 1674-733X; 1869-1919
• DOI: 10.1007/s11432-017-9117-0

This paper focuses on the decision-implementation complexity （DIC） of cooperative games. The complexity of a control law is a fundamental issue in practice because it is closely related to control cost. First, we formulate implementation measures of strategies as system control protocols. Then, for a class of cooperative games, a decision-implementation system model is established, and an energy-based DIG index is given as the energy expectation under Nash equilibrium strategies. A definition of DIC is presented to describe the optimal values of the DIC index. DIC is calculated by the exhaust algorithm in some specific cases, whereas the one for general cases is too complex to be calculated. In order to obtain a general calculation method, the problem is described in the form of matrices; an analytical expression describing DIC is obtained by using the properties of matrix singular values. Furthermore, when only partial information of actions is shared among players, DIC can be reduced and an improved protocol can be designed as a two-phase protocol. A numerical example is given to show that the DIC obtained in this study is the same as the one obtained by the exhaust algorithm, and that the calculation complexity of the proposed algorithm is much lower.