The core of housing markets from an agent's perspective: Is it worth sprucing up your home?
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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 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 where a receives either h, or a house that she 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.
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