Nonlinear programming method for interval-valued n-person cooperative games

The aim of this paper is to develop a nonlinear programming method for computing the elements of the interval-valued cores of n-person cooperative games in which coalitions’ values are expressed with intervals, which are often called interval-valued n-person cooperative games for short. With finding out the maximum satisfactory degree in the situation with the features of inclusion and/or overlap relations between intervals, this paper tries to explore the cooperation chance in this type of cooperative games. Firstly, we define the concept of interval-valued cores of interval-valued n-person cooperative games and satisfactory degrees (or ranking indexes) of comparing intervals with the features of inclusion and/or overlap relations. Hereby, we propose the auxiliary nonlinear programming model and method for solving interval-valued cores of any interval-valued n-person cooperative games. The developed method can provide cooperative chance under the situation of inclusion and/or overlap relations between intervals, while the traditional interval ranking method may not assure that the interval-valued cores exist. This method is a complement to the traditional methods rather than the alternative one. The feasibility and applicability of the model and method proposed in this paper are illustrated with a numerical example.