On one generalization of stable allocations in a two-sided market

Abstract

In the stable allocation problem on a two-sided market introduced and studied by Baiou and Balinski in the early 2000's, one is given a bipartite graph $G=(V,E)$ with capacities $b$ on the edges (``contracts'') and quotas $q$ on the vertices (``agents''). Each vertex $v\in V$ is endowed with a linear order on the set $E_v$ of edges incident to $v$, which generates preference relations among functions (``contract intensities'') on $E_v$, giving rise to a model of \it{stable allocations} for $G$. This is a special case of Alkan-Gale's stability model for a bipartite graph with edge capacities in which, instead of linear orders, the preferences of each ``agent'' $v$ are given via a choice function that acts on the box $\{z\in{\mathbb R}_+^{E_v}\colon z(e)\le b(e),\, e\in E_v\}$ or a closed subset in it and obeys the (well motivated) axioms of consistence, substitutability and cardinal monotonicity. By central results in Alkan-Gale's theory, the set of stable assignments generated by such choice functions is nonempty and forms a distributive lattice. In this paper, being in frameworks of Alkan-Gale's model and generalizing the stable allocation one, we consider the situation when the preferences of ``agents'' of one side (``workers'') are given via linear orders, whereas the ones of the other side (``firms'') via integer-valued choice functions subject to the three axioms as above, thus introducing the model of \it{generalized allocations}, or g-allocations for short. Our main aims are to characterize and efficiently construct rotations, functions on $E$ associated with immediately preceding relations in the lattice $(S,\prec)$ of stable g-allocations, and to estimate the complexity of constructing a poset generated by rotations for which the lattice of closed functions is isomorphic to $(S,\prec)$, obtaining a ``compact'' representation of the latter.