The Shapley function for fuzzy games with fuzzy characteristic functions

In this paper, we research fuzzy games with fuzzy characteristic functions, where the player participation levels are real numbers in [0, 1], and their payoffs are fuzzy numbers. Based on the Hukuhara difference between fuzzy numbers and the calculating formula of the Shapley function, we discuss a general form of fuzzy games with fuzzy characteristic functions, and study the Shapley function for this kind of fuzzy games. When the given fuzzy game is convex, we research some properties of fuzzy games with fuzzy characteristic functions. Furthermore, we research a special kind of fuzzy games with fuzzy characteristic functions, which is can be seen as an extension of fuzzy games with multilinear extension form. Finally, a numerical example is given.