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The Strong Core of Housing Markets with Partial Order Preferences

Ildikó Schlotter, Lydia Mirabel Mendoza-Cadena

TL;DR

This paper investigates the strong core of housing markets when agents express preferences as partial orders, filling a gap between strict and weak-order analyses. It provides a peak-set based structural characterization and a polynomial-time algorithm (SCFA) that finds a strong-core allocation or detects emptiness, even with forbidden and forced arcs, and proves group-strategyproofness. The authors extend known weak-order results to partial orders, showing the strong core respects improvement, and they demonstrate that an ILP for the strong core remains sound under partial orders. They also discuss the capacity to enumerate allocations in the strong core and highlight limitations of extending Quint–Wako fully to partial orders. The work advances practical matching in kidney-exchange-inspired settings where incomparability between options is natural and supports robust, strategyproof allocations.

Abstract

We study the strong core of housing markets when agents' preferences over houses are expressed as partial orders. We provide a structural characterization of the strong core, and propose an efficient algorithm that finds an allocation in the strong core or decides that it is empty, even in the presence of forced and forbidden arcs. The algorithm satisfies the property of group-strategyproofness. Additionally, we show that certain results known for the strong core in the case when agents' preferences are weak orders can be extended to the setting with partial order preferences; among others, we show that the strong core in such housing markets satisfies the property of respecting improvements.

The Strong Core of Housing Markets with Partial Order Preferences

TL;DR

This paper investigates the strong core of housing markets when agents express preferences as partial orders, filling a gap between strict and weak-order analyses. It provides a peak-set based structural characterization and a polynomial-time algorithm (SCFA) that finds a strong-core allocation or detects emptiness, even with forbidden and forced arcs, and proves group-strategyproofness. The authors extend known weak-order results to partial orders, showing the strong core respects improvement, and they demonstrate that an ILP for the strong core remains sound under partial orders. They also discuss the capacity to enumerate allocations in the strong core and highlight limitations of extending Quint–Wako fully to partial orders. The work advances practical matching in kidney-exchange-inspired settings where incomparability between options is natural and supports robust, strategyproof allocations.

Abstract

We study the strong core of housing markets when agents' preferences over houses are expressed as partial orders. We provide a structural characterization of the strong core, and propose an efficient algorithm that finds an allocation in the strong core or decides that it is empty, even in the presence of forced and forbidden arcs. The algorithm satisfies the property of group-strategyproofness. Additionally, we show that certain results known for the strong core in the case when agents' preferences are weak orders can be extended to the setting with partial order preferences; among others, we show that the strong core in such housing markets satisfies the property of respecting improvements.
Paper Structure (25 sections, 19 theorems, 5 equations, 4 figures, 2 tables)

This paper contains 25 sections, 19 theorems, 5 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Characterizing the strong core
  4. The Quint--Wako Characterization Fails for Partial Orders
  5. A Characterization Through Peak Sets
  6. Finding an allocation in the strong core
  7. Algorithm for SCFA
  8. Correctness of Algorithm \ref{['alg:SCFA']}
  9. Extension to Forbidden and Forced Arcs
  10. Strategyproofness
  11. Finding All Allocations in the Strong Core
  12. Properties of the strong core
  13. Conclusion
  14. Basics on graphs
  15. Additional material for Section \ref{['sec:characterization']}
  16. ...and 10 more sections

Key Result

theorem 1

Suppose that $H=(N,\{\succ_a:a \in N\})$ is a housing market where each $\succ_a$ is a weak order. Let $U$ be the set of undominated arcs in $D^H$, and $S$ an absorbing set in $D^H[U]$. An allocation $X$ for $H$ is in the strong core of $H$ if and only if it can be partitioned into sets $X_S$ and $X

Figures (4)

  • Figure 1: The underlying graph of housing market $H^1$ defined in Example \ref{['ex1']}. Henceforth, loops are omitted, and undominated arcs are shown in blue, the strongly connected components they form are shown as dashed polygons. Single and double line markings $\boldsymbol{|}$ and $\boldsymbol{||}$ convey domination: an arc marked with $\boldsymbol{||}$ dominates an arc marked with $\boldsymbol{|}$ and leaving the same agent; e.g., $(a,c)$ dominates $(a,d)$ but not $(a,b)$, and neither does $(a,b)$ dominate $(a,d)$.
  • Figure 2: The housing market $H^2$ defined in Example \ref{['ex3']}.
  • Figure 3: Illustration for Example \ref{['ex5']}.
  • Figure 5: The underlying graph of housing market $H^6$ defined in Example \ref{['ex4']}.

Theorems & Definitions (24)

  • Remark 1
  • theorem 1: Quint--Wako characterization WakoQuint
  • definition 1
  • lemma 1: $\star$
  • theorem 2
  • lemma 2
  • theorem 3: $\star$
  • corollary 1
  • corollary 2
  • definition 2
  • ...and 14 more