The core of housing markets from an agent's perspective: Is it worth sprucing up your home?

TL;DR

The paper investigates core allocations in Shapley–Scarf housing markets with partial-order preferences, addressing which outcomes an agent can expect and how improving an agent’s initial endowment affects possibilities. It proves NP-hardness for core-arc questions and introduces HM-Improve, a linear-time algorithm that preserves improvement (RI-best) in the core under endowment upgrades, with an analogous result for Stable Roommates under certain conditions. It also establishes that some RI-related questions remain intractable under the core (and related strict contexts) but shows that the strict core admits a polynomial-time arc-restriction solution, highlighting a computational gap between core and strict core. The work further develops TTC adaptations for partial orders, analyzes stability dynamics in Stable Roommates under improvements, and discusses Max Core hardness and approximation limits, with practical implications for kidney-exchange design and centralized housing markets.

Abstract

We study housing markets as introduced by Shapley and Scarf (1974). We investigate the computational complexity of various questions regarding the situation of an agent $a$ in a housing market $H$: we show that it is $\mathsf{NP}$-hard to find an allocation in the core of $H$ where (i) $a$ receives a certain house, (ii) $a$ does not receive a certain house, or (iii) $a$ receives a house other than her own. We prove that the core of housing markets respects improvement in the following sense: given an allocation in the core of $H$ where agent $a$ receives a house $h$, if the value of the house owned by $a$ increases, then the resulting housing market admits an allocation in its core in which $a$ receives either $h$, or a house that $a$ prefers to $h$; moreover, such an allocation can be found efficiently. We further show an analogous result in the Stable Roommates setting by proving that stable matchings in a one-sided market also respect improvement.

Figures (12)

• Figure 1: Illustration of the housing market $H$ constructed in the $\mathsf{NP}$-hardness proof for Arc in Core. Here and everywhere else we depict markets through their acceptability graphs with all loops omitted. Preferences are indicated by numbers along the arcs; the symbol $\infty$ indicates the least-preferred choice of an agent. The example assumes that $(v_1,v_2)$ and $(v_n,v_2)$ are arcs of the directed input graph $D$, as indicated by the dashed arcs.
• Figure 2: The housing markets $H$ and $H'$ in the proof of Proposition \ref{['prop:core-worst-RI-fails']}. For both $H$ and $H'$, the allocation represented by bold (and blue) arcs yields the worst possible outcome for $p$ in any core allocation of the given market.
• Figure 3: Illustration for the concept of sub-allocation. The arcs sets $Y_1$ and $Y_2$, shown with bold, teal lines, are both sub-allocations from $\{p\}$ to $\{a\}$ in the depicted housing market (as usual, loops are omitted). Source and sink vertices of $Y$ are depicted with a white black diamond, respectively. For each of $Y_1$ and $Y_2$, we show the corresponding envy arcs (i.e., the arcs in the corresponding envy graphs) with dashed, red lines; as can be seen, $Y_1$ is stable while $Y_2$ is not.
• Figure 4: The housing market $H$ of Example \ref{['ex:HMalgo']} and the modified housing market $\widetilde{H}$ constructed by Algorithm HM-Improve based on the $p$-improvement of $H$ where $q_1$ and $q_2$ change their preferences so that $q_1$ comes to prefer $p$ to $a$, and $q_2$ comes to prefer $p$ to $d$. We depicted the core allocation $X$ for $H$ using blue lines, and we depicted the corresponding sub-allocation $Y$, as constructed by Algorithm HM-Improve in its initialization step, using teal lines. Sub-allocation $Y$ has two sources, $a$ and $d$, highlighted by diamonds, and two sinks, $\widetilde{q}_1$ and $\widetilde{q}_2$. Envy arcs for both the original allocation $X$ in $H$ and the sub-allocation $Y$ in $\widetilde{H}$ are shown using red, dashed lines.
• Figure 5: Illustration of the possible steps performed during the iteration by HM-Improve. The edges of the current sub-allocation $Y$ are depicted using bold, teal lines, while edges of the envy graph $\widetilde{G}_{Y \prec}$ are shown by dashed, red lines. As in Figure \ref{['fig:sub-allocation']}, source and sink vertices of $Y$ are depicted with a white black diamond, respectively. Vertices of $R$ as well as all edges incident to them are shown in grey.

Theorems & Definitions (40)

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