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We study housing markets as introduced by Shapley and Scarf (1974). Weinvestigate the computational complexity of various questions regarding thesituation 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 housingmarkets respects improvement in the following sense: given an allocation in thecore of $H$ where agent $a$ receives a house $h$, if the value of the houseowned by $a$ increases, then the resulting housing market admits an allocationin 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 ananalogous result in the Stable Roommates setting by proving that stablematchings in a one-sided market also respect improvement.

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