Abstract: The most critical factor affecting alliance enterprise cooperation is how to distribute the benefits obtained from forming alliances reasonably and fairly. The rationality of interest distribution directly affects the sustainability and stability of alliance innovation. How to effectively solve the problem of distribution of income within a cooperative alliance has become an important topic at domestic and abroad. The traditional Shapley value method assumes that each member of the cooperative alliance has the same marginal contribution as the premise. However, in the actual situation, due to the limitations of the self-generated technology and the uncertainty of the industrial cluster environment, the enterprise can only play a part of their capacities. At the same time, it is difficult for enterprises to fully identify the exact value of information. In order to solve the above problems, the restricted cooperation games is studied in this paper, in which the cooperation ability is the gray information and the dependency relationships among the enterprises that restrict their capacity to cooperate within some coalitions. First of all, the gray authorization operator based on the grey system theory is defined. Secondly, the Choquet integrals are used to integrate the dependencies between enterprises. And then, this dependency information is combined with the Shapley model to establish a restricted cooperative game model with gray authorization mechanism. And it is proved that the model satisfies the efficiency, symmetry, additivity and dummy player axioms. Finally, an example is given to illustrate the feasibility and practicality of the model.

Key words: industrial cluster, cooperative game, grey authorization operator, Choquet integral, Shapley value