The Shapley value for fuzzy bi-cooperative games

Abstract

In this article, we investigated a Shapley function on the class of crisp bi-cooperative games Firstly, we redefine bi-cooperative games axiom which was introduced by Grabisch and proposed the Shapley function that satisfies the four axioms. Then, the concepts related to a Shapley function have been extended to the case of fuzzy bi-cooperative games by choquet integral. Similarly, we define the four fuzzy Shapley axioms which correspond to four axioms crisp bi-cooperative Shapley, respectively. Finally, for the purpose of bridging the results to a real world problem, we gave a concrete example.