Simplification of Shapley value for cooperative games via minimum carrier

• Published in: Control theory and technology Vol. 19; no. 2; pp. 157 - 169
• Main Authors: Li, Haitao, Wang, Shuling, Liu, Aixin, Xia, Meixia
• Format: Journal Article
• Language: English
• Published: Guangzhou South China University of Technology and Academy of Mathematics and Systems Science, CAS 01.05.2021; Springer Nature B.V; School of Mathematics and Statistics,Shandong Normal University,Jinan 250014,Shandong,China
• Subjects: Algorithms; Complexity; Computational Intelligence; Control; Control and Systems Theory; Engineering; Expected values; Games; Mathematical analysis; Mechatronics; Optimization; Players; Research Article; Robotics; Systems Theory; Tensors; Shapley value; Carrier; Algebraic form; Semi-tensor product of matrices; Cooperative game
• ISSN: 2095-6983; 2198-0942
• DOI: 10.1007/s11768-020-00003-1

Shapley value is one of the most fundamental concepts in cooperative games. This paper investigates the calculation of the Shapley value for cooperative games and establishes a new formula via carrier. Firstly, a necessary and sufficient condition is presented for the verification of carrier, based on which an algorithm is worked out to find the unique minimum carrier. Secondly, by virtue of the properties of minimum carrier, it is proved that the profit allocated to dummy players (players which do not belong to the minimum carrier) is zero, and the profit allocated to players in minimum carrier is only determined by the minimum carrier. Then, a new formula of the Shapley value is presented, which greatly reduces the computational complexity of the original formula, and shows that the Shapley value only depends on the minimum carrier. Finally, based on the semi-tensor product (STP) of matrices, the obtained new formula is converted into an equivalent algebraic form, which makes the new formula convenient for calculation via MATLAB.