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

Alexander V. Karzanov

TL;DR

The paper extends the stable allocation framework to generalized allocations (g-allocations) where workers use fixed linear preferences and firms follow integer-valued choice functions satisfying natural axioms. It develops a rotation-based representation, aligning the lattice of stable g-allocations with a poset of rotations and a corresponding lattice of closed functions, under a gapless condition that keeps the rotation set tractable. The authors prove polynomial bounds on the number and length of rotations, construct the rotation poset, and show that minimum-cost stable g-allocations can be solved via min-cut reductions, all in weakly polynomial time. They also demonstrate that removing the gapless condition can lead to exponentially large rotational structures, highlighting the condition’s significance for computational efficiency and representation compactness.

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.

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

TL;DR

The paper extends the stable allocation framework to generalized allocations (g-allocations) where workers use fixed linear preferences and firms follow integer-valued choice functions satisfying natural axioms. It develops a rotation-based representation, aligning the lattice of stable g-allocations with a poset of rotations and a corresponding lattice of closed functions, under a gapless condition that keeps the rotation set tractable. The authors prove polynomial bounds on the number and length of rotations, construct the rotation poset, and show that minimum-cost stable g-allocations can be solved via min-cut reductions, all in weakly polynomial time. They also demonstrate that removing the gapless condition can lead to exponentially large rotational structures, highlighting the condition’s significance for computational efficiency and representation compactness.

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 with capacities on the edges (``contracts'') and quotas on the vertices (``agents''). Each vertex is endowed with a linear order on the set of edges incident to , which generates preference relations among functions (``contract intensities'') on , giving rise to a model of \it{stable allocations} for . 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'' are given via a choice function that acts on the box 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 associated with immediately preceding relations in the lattice 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 , obtaining a ``compact'' representation of the latter.
Paper Structure (13 sections, 20 theorems, 30 equations)

This paper contains 13 sections, 20 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Definitions and settings
  3. Active graph and rotations
  4. Active graph
  5. Rotations
  6. Additional properties of rotations
  7. Poset of rotations
  8. Efficient constructions
  9. Finding the preceding relation $\lessdot$ on rotations.
  10. Finding the initial stable g-allocation $x^{\rm min}$.
  11. Stage I: Finding a stable g-allocation
  12. Stage II: Transforming $x\in{\cal S}$ into $x^{\rm min}$
  13. Appendix. Withdrawal of property (C) and large rotational posets

Key Result

Lemma 2.1

Let $v\in V$, $z,z'\in {\cal A}_v$ and $z\prec_v z'$. Let $a\in E_v$ be such that $z(a)\le z'(a)$ and $a$ is not interesting for $v$ under $z$. Then $a$ is not interesting under $z'$ as well.

Theorems & Definitions (20)

  • Lemma 2.1
  • Lemma 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Lemma 3.4
  • Proposition 4.1
  • Corollary 4.2
  • Theorem 4.3
  • Lemma 5.1
  • Theorem 5.2
  • ...and 10 more