An approach to computing interval-valued discounted Shapley values for a class of cooperative games under interval data

ABSTRACT

Through adding a discounted factor, Shapley values of cooperative games are extended to interval-valued (IV) cooperative games. The main purpose of this paper is to propose the concept of IV discounted Shapley values and hereby develop a simplified method for computing IV discounted Shapley values for a class of IV cooperative games. In this method, we introduce the concept of discounted Shapley values of $\alpha$-cooperative games associated with IV cooperative games for any attitude factor $\alpha \in [0,1]$ and find a weaker coalition monotonicity-like condition so that the defined discounted Shapley values possess the monotonicity. Hence we can obtain the analytical formula of IV discounted Shapley value of any IV cooperative game with the coalition monotonicity-like through utilizing only the lower and upper bounds of coalitions’ IV values. The proposed method utilizes the weaker coalition monotonicity-like condition rather than the irreversible interval subtraction and hereby can effectively overcome some irrational issues.

KEYWORDS:

• Uncertainty
• game theory
• interval-valued cooperative game
• Shapley value
• interval computing

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research for this article funded by the Key Program of the National Natural Science Foundation of China [NO. 71231003].

Notes on contributors

Wei Fei

Wei Fei was born in 1965. She received the B.Sc. degree in architecture from the Jilin Architecture and Civil Engineering Institute, Changchun, People’s Republic of China, in 1988. She is currently an associate professor with the School of Architecture, Fuzhou University, Fuzhou, People’s Republic of China. She has authored or coauthored over 10 journal and conference papers. Her current research interests include project management and planning, tourism management, game theory and system analysis.

Deng-Feng Li

Deng-Feng Li was born in 1965. He received the B.Sc. and M.Sc. degrees in applied mathematics from the National University of Defense Technology, Changsha, People’s Republic of China, in 1987 and 1990, respectively, and the Ph.D. degree in system science and optimization from the Dalian University of Technology, Dalian, People’s Republic of China, in 1995.

From 2003 to 2004, he was a Visiting Scholar with the School of Management, University of Manchester Institute of Science and Technology, Manchester, UK. He is currently the Cheung Kong Scholars Programme, Ministry of Education of China and a Distinguished Professor with the School of Economics and Management, Fuzhou University, Fuzhou, China. He has authored or coauthored more than 300 journal papers and 8 monographs. He has co-edited two proceedings of the international conference and won 26 academic achievements and awards such as Chinese State Natural Science Award and 2013 IEEE Transactions on Fuzzy Systems Outstanding Paper Award of the IEEE Computational Intelligence Society. His current research interests include fuzzy sets and system, fuzzy game theory, fuzzy decision analysis, fuzzy group decision making, supply chain, fuzzy optimization, and differential game. He is the Editor-in-chief (or Associate Editor) and Editor of several international journals.

Yin-Fang Ye

Yin-Fang Ye was born in 1988. She received the B.Sc. degree in mathematics and applied mathematics from Zhangzhou Normal University, Zhangzhou, People’s Republic of China, in 2011, and the M.Sc. degree in applied mathematics from Fuzhou University, Fuzhou, People’s Republic of China, in 2014. She is currently working for the doctoral degree with the School of Economics and Management, Fuzhou University, Fuzhou, People’s Republic of China. She has authored or coauthored about ten journal papers. Her research interests include fuzzy game theory and supply chain management.