Models and Algorithms for Least Square Interval-Valued Nucleoli of Cooperative Games with Interval-Valued Payoffs

The aim of this paper is to develop a new method for computing least square interval-valued nucleoli of cooperative games with interval-valued payoffs, which usually are called interval-valued cooperative games for short. In this methodology, based on the square excess which can be intuitionally interpreted as a measure of the dissatisfaction of the coalitions, we construct a quadratic programming model for least square interval-valued prenucleolus of any interval-valued cooperative game and obtain its analytical solution, which is used to determine players' interval-valued imputations via the designed algorithms that ensure the nucleoli always satisfy the individual rationality of players. Hereby the least square interval-valued nucleoli of interval-valued cooperative games are determined in the sense of minimizing the difference of the square excesses of the coalitions. Moreover, we discuss some useful and important properties of the least square interval-valued nucleolus such as its existence and uniqueness, efficiency, individual rationality, additivity, symmetry, and anonymity.