Core stability of the Shapley value for cooperative games
Core stability of the Shapley value for cooperative games
复制标题
合作博弈Shapley值的核心稳定性
DOI：
10.1007/s00355-022-01432-4
复制
发表时间：
2022
影响因子：
0.9
通讯作者：
Nakada Satoshi
中科院分区：
文献类型：
--
作者：
Abe Takaaki;Nakada Satoshi
我们的目标是分析Shapley值与具有可转移效用的合作博弈的核心之间的关系。我们首先描述均衡博弈，即，通过几何性质，具有非空核心的博弈集合。我们证明了一组平衡的游戏生成一个多面体锥，一个游戏是平衡的当且仅当它是一个非负的线性组合的一些简单的游戏。此外，我们表明，一组游戏的Shapley值位于核心也产生一个多面体锥，游戏服从这个属性当且仅当它是一个非负的线性组合的简单游戏满足某些性质。副产品，我们还表明，对应于多面体的极端射线的游戏的数量与最小的平衡集合的数量相一致。
Our objective is to analyze the relationship between the Shapley value and the core of cooperative games with transferable utility. We first characterize balanced games, i.e., the set of games with a nonempty core, through geometric properties. We show that the set of balanced games generates a polyhedral cone and that a game is balanced if and only if it is a nonnegative linear combination of some simple games. Moreover, we show that the set of games whose Shapley value lies in the core also yields a polyhedral cone and that a game obeys this property if and only if it is a nonnegative linear combination of simple games satisfying certain properties. By-products, we also show that the number of games that correspond to the extreme rays of the polyhedron coincides with the number of minimal balanced collections.