两阶段Shapley-solidarity值的新刻画及其应用

doi:10.15960/j.cnki.issn.1007-6093.2026.02.013

摘要/Abstract

摘要： 2022年,Zou等提出了具有联盟结构合作对策的两阶段Shapley-solidarity值,该值中各参与者获得的分配不仅包含限制在其所属优先联盟子对策上的solidarity值,还包括其所属优先联盟分得的Shapley值与联盟自身效用间盈余部分在该联盟内所有参与者间平均分配的结果。本文提出了大联盟团结性公理,结合有效性、联盟间均衡贡献性和联盟内参与者团结性给出了两阶段Shapley-solidarity值的公理化刻画。研究指出利用有效性、联盟间对称性、联盟间边际性、联盟内参与者团结性和大联盟团结性,也可唯一刻画两阶段Shapley-solidarity值。最后,通过对应用案例分析,发现两阶段Shapley-solidarity值在保障联盟内公平性的同时更能照顾到联盟内的弱势参与者,体现了联盟内更高程度的团结。

关键词: 合作对策, 联盟结构, 两阶段Shapley-solidarity值

Abstract: In 2022, Zou et al. proposed a two-step Shapley-solidarity value for cooperative games with coalition structure, which distributes the total worth in two steps. Firstly, players within one union obtain the solidarity value in the subgame restricted to the corresponding union. Then, the surplus of the difference of between the Shapley value of the union obtained in the quotient game, and the worth of the union, will also be allocated to the players in the same union equally. This research proposes a new axiom called the grand coalition solidarity property, and proves that the two-step Shapley-solidarity value can be uniquely characterized by four axioms: efficiency, coalitional balanced contributions, population solidarity within unions and grand coalition solidarity property. Besides, this paper shows that the two-step Shapley-solidarity value is the only value that satisfies efficiency, coalitional symmetry, coalitional marginality, population solidarity within unions and grand union solidarity property. Finally, comparing the two-step Shapley-solidarity value with other values by a numerical example, this paper shows that the two-step Shapley-solidarity value can take care of the weak players better, while ensuring the fair allocation within unions. It turns out that the two-step Shapley-solidarity value reflects a higher degree of solidarity than those values.

Key words: TU-game, coalition structure, two-step Shapley-solidarity value

中图分类号:

F224.32

原萌, 刘涛, 单而芳. 两阶段Shapley-solidarity值的新刻画及其应用[J]. 运筹学学报（中英文）, 2026, 30(2): 169-178.

YUAN Meng, LIU Tao, SHAN Erfang. New characterizations of the two-step Shapley-solidarity value and its application[J]. Operations Research Transactions, 2026, 30(2): 169-178.

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