Quasilinear Rental Harmony
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Rental Harmony is the problem of assigning rooms in a rented house to tenants with different preferences, and simultaneously splitting the rent among them, such that no tenant envies the bundle (room+price) given to another tenant. Different papers have studied this problem under two incompatible assumptions: the miserly tenants assumption is that each tenant prefers a free room to a non-free room; the quasilinear tenants assumption is that each tenant attributes a monetary value to each room, and prefers a room of which the difference between value and price is maximum. This note shows how to adapt the main technique used for rental harmony with miserly tenants, using a variant of Sperner's lemma, to rental harmony with quasilinear tenants. This implies that some recent results derived for miserly tenants apply to quasilinear tenants too. Moreover, the proof is valid even for some classes of non-linear preferences.
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