Cooperative game with fuzzy coalition and payoff value in the generalized integral form

In this paper, a new class of games with fuzzy coalitions and fuzzy payoff value is proposed. This class of fuzzy games is based on the generalized integral form, which contains several kinds of other fuzzy games, such as fuzzy cooperative game with fuzzy payoff value, the multilinear extension game introduced by Owen, the game with proportional value proposed by Butnariu and the game with Choquet integral form given by Tsurumi et al. Also, the proposed fuzzy game is also further extension of the fuzzy game in Choquet integral form proposed by Yu et al., which is also is a kind of fuzzy games in the condition that coalition and fuzzy payoff value are both fuzzy information. The fuzzy Shapley value for this kind of fuzzy games is represented by Shapley value of corresponding fuzzy cooperative game with fuzzy payoff value. Based on Hukuhara-difference, we give the explicit Shapley value for the proposed fuzzy game. It has been seen that most of properties hold well in the proposed fuzzy game, which are processed by cooperation game with fuzzy payoff value, and fuzzy coalition game, respectively. Finally, a practical application of the proposed model is also provided.