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

TL;DR

This paper examines the computational complexity of core allocations in housing markets, showing NP-hardness for certain allocation questions and demonstrating that core stability respects improvements in house value, with efficient solutions.

Contribution

It establishes NP-hardness results for core-related questions in housing markets and proves that core stability respects improvements in house value, including efficient algorithms for certain cases.

Findings

NP-hardness of finding specific core allocations

Core stability respects house value improvements

Efficient algorithms for certain core allocation adjustments

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.