A Shapley function on a class of cooperative fuzzy games

In this paper, we make a study of the Shapley values for cooperative fuzzy games, games with fuzzy coalitions, which admit the representation of rates of players' participation to each coalition. A Shapley function has been introduced by another author as a function which derives the Shapley value from a given pair of a fuzzy game and a fuzzy coalition. However, the previously proposed axioms of the Shapley function can be considered unnatural. Furthermore, the explicit form of the function has been given only on an unnatural class of fuzzy games. We introduce and investigate a more natural class of fuzzy games. Axioms of the Shapley function are renewed and an explicit form of the Shapley function on the natural class is given. We make sure that the obtained Shapley value for a fuzzy game in the natural class has several rational properties. Finally, an illustrative example is given.