发表机构
Central Institute of Economics and Mathematics of the RAS（俄罗斯科学院中央经济数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对整数选择函数的稳定广义分配问题，提出分裂技术将解格嵌入稳定匹配或稳定分配格，推广Fleiner结果。
AI 中文摘要
我们考虑Alkan--Gale在双边市场中稳定性模型的整数版本，称为稳定广义分配。该模型由三元组$(G,b,C)$给出，其中$G=(V,E)$是有限二分图，边$e\in E$具有非负整数容量$b(e)\in{\mathbb Z}_+$，且对于每个顶点（“智能体”）$v\in V$，其对关联边集$E_v$的偏好依赖于选择函数$C_v$。后者作用于由容量约束的${\mathbb Z}_+^{E_v}$中的向量集，并满足可替代性和规模单调性的标准公理。Alkan--Gale的著名定理表明，在这种情况下，稳定性问题总是存在稳定解$x\in{\mathbb Z}_+^E$，而且这些解的集合${\cal S}_{G,b,C}$（“稳定广义分配”）构成一个分配格。然而，这个格构造和操作起来相当复杂，我们想知道它是否可以通过一个“更简单”的稳定性模型来表示。针对这个问题，我们安排了一种分裂技术，将${\cal S}_{G,b,C}$作为子格嵌入到稳定匹配的格中，并更紧凑地嵌入到稳定分配的格中（如Baiou--Balinski的稳定性模型）。这推广了Fleiner关于所有单位容量特殊情况下分离的结果。关键词：稳定婚姻，稳定分配，选择函数，旋转，分配格，偏序集表示。
英文摘要
We consider the integer version of Alkan--Gale's model on stability in a two-sided market, called the stable generalized allocation one. It is given by a triple $(G,b,C)$, where $G=(V,E)$ is a finite bipartite graph with nonnegative integer capacities $b(e)\in{\mathbb Z}_+$ of edges $e\in E$, and for each vertex (``agent'') $v\in V$, the preferences on the set $E_v$ of its incident edges depend on a choice function $C_v$. The latter acts on the set of vectors in ${\mathbb Z}_+^{E_v}$ bounded by the capacities and obeys the standard axioms of substitutability and size monotonicity. Alkan--Gale's prominent theorem implies that the stability problem in this case always has a stable solution $x\in{\mathbb Z}_+^E$ and, moreover, the set ${\cal S}_{G,b,C}$ of these solutions (``stable generalized allocations'') forms a distributive lattice. However, this lattice is rather intricate to construct and work with, and we wonder whether it can be represented via a ``simpler'' stability model. Answering this issue, we arrange a sort of splitting techniques to embed ${\cal S}_{G,b,C}$, as a sublattice, in the lattice of stable matchings and, more compactly, in the lattice of stable allocations (as in Baiou--Balinski's stability model). This generalizes Fleiner's result on a detachment in the special case with all-unit capacities. Keywords: stable marriage, stable allocation, choice function, rotation, distributive lattice, poset representation
Comments19 pages
