On splitting properties of the stability problem with integer choice functions

Abstract

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