Axiomatic characterizations of the Shapley function on a class of cooperative fuzzy games

We have given an axiomatic definition of the Shapley function deriving the Shapley value from a fuzzy game and a fuzzy coalition. An explicit form of the Shapley function on a natural class of fuzzy games has been given. In this paper, we show the uniqueness of the Shapley function on this class. Furthermore, we give another explicit form of the Shapley function. We extend the strong monotonicity to fuzzy games. It can be seen that the Shapley function is strongly monotone. Finally, we prove that the Shapley function is the unique function satisfying efficiency, symmetry and strong monotonicity.