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