双边配给问题描述了现实生活中一类带有二部图结构的稀缺资源配置问题, 例如, 在自然灾害期间救援物资的配给; 电力和天然气等自然资源按需分配; 高校引进人才调配等。本文通过求解线性规划, 并从联盟边际贡献的角度出发定义了双边配给问题的一个Shapley解。之后, 通过合作对策模型和解的公理化方法说明新解的合理性。首先, 建立双边配给问题的合作对策模型, 论证了新解与双边配给合作对策的Shapley值一致; 其次, 证明了Shapley解是唯一满足优先一致性的有效配给方案。最后, 将Shapley解应用于博物馆通票问题的研究, 探讨了博物馆合作制定通票后所得单票和通票收益的分配方式。

Bipartite rationing problem was studied to divide a short supply between resource and sink nodes in a bipartite graph. It had been used to deal with numerous issues in the real world, such as, the allocations of aid relief during natural disasters, utilities like electricity and natural gas, and talents of different types to universities, etc. From the view of marginal contribution of coalitions, this paper proposed the Shapley solution of bipartite rationing problems calculated by linear programming, and characterized it by cooperative game and axiomatization. First, we defined the cooperative game of bipartite rationing problems, called the bipartite rationing game, and proved that the Shapley value of the corresponding cooperative game coincides with the solution we propose. Then, we showed that the Shapley solution is the unique feasible allocation satisfying priority-consistency. Finally, we considered the application of the Shapley solution to museum pass problems, and discussed the allocations of the revenue of single tickets and pass tickets of each participating museum.