A graph cooperative game with interval-valued payoffs and its simplified solving method

This paper proposes the concept of interval-valued average tree solution ("AT solution" for short) of graph cooperative games with interval-valued payoffs, and develop an effective and a direct simplified method for solving a subclass of interval-valued graph cooperative games. In this method, the interval-valued AT solution is proved to be a monotonic and non-decreasing function of coalitions' values under specific condition. Hence, the lower and upper bounds of interval-valued AT solutions of graph cooperative games can be obtained directly by using the lower and upper bounds of the interval-valued coalitions' payoffs, respectively. The proposed method gives better results than general interval subtraction and the partial subtraction operator methods. In addition, some important properties of the interval-valued AT solutions of interval-valued graph cooperative games are discussed. At last, the applicability and superiority of the proposed approach is demonstrated by comparing with other methods.