Interval-Valued Least Square Prenucleolus of Interval-Valued Cooperative Games with Fuzzy Coalitions

In this paper, an important solution concept of interval-valued (IV) cooperative games with fuzzy coalitions, called the IV least square prenucleolus, is proposed. Firstly, we determine the fuzzy coalitions' values by using Choquet integral and hereby obtain the IV cooperative games with fuzzy coalitions in Choquet integral forms. Then, we develop a simplified method to compute the IV least square prenucleolus of a special subclass of IV cooperative games with fuzzy coalitions in Choquet integral forms. In this method, we give some weaker coalition size monotonicity-like conditions, which can always ensure that the least square prenucleolus of our defined cooperative games with fuzzy coalitions in Choquet integral form are monotonic and non-decreasing functions of fuzzy coalitions' values. Hereby, the lower and upper bounds of the proposed IV least square prenucleolus can be directly obtained via utilizing the lower and upper bounds of the IV coalitions values, respectively. In addition, we investigate some important properties of the IV least square prenucleolus. The feasibility and applicability of the method proposed in this paper are illustrated with numerical examples.