Generalized Shapley function for cooperative games with fuzzy payoffs

Based on the extended generalized Hukuhara difference, we introduce and study a new Shapley type of value for cooperative games with fuzzy payoffs. We first propose and characterize a new interval Shapley value for interval-valued cooperative games. Those results are then extended to cooperative games with fuzzy payoffs, and the generalized Shapley function is introduced. We characterize the generalized Shapley function using the properties of generalized efficiency, generalized dummy player, generalized symmetry, and generalized additivity. At the same time, the necessary and sufficient condition for the existence of the generalized Shapley function is given. This study also shows that the generalized Shapley function is a generalization of the Hukuhara-Shapley function defined by Yu and Zhang [22]. Meanwhile, an arbitrary cooperative game with payoffs of center triangular fuzzy numbers has a unique generalized Shapley function.