House allocation with fractional endowments

We study a generalization of the well known house allocation problem in which agents may own fractions of different houses summing to an arbitrary quantity, but have use for only the equivalent of one unit of a house. We also depart from the classical model by assuming that arbitrary quantities of each house may be available to the market. Justified envy considerations arise when two agents have the same initial endowment, or when an agent is in some sense disproportionately rewarded in comparison to her peers. For this general model, we design an algorithm to find a fractional allocation of houses to agents that satisfies ordinal efficiency, individual rationality, and no justified envy. Our results extend to the full preference domain. We show that individual rationality, ordinal efficiency, and no justified envy conflict with weak strategyproofness. We also show that individual rationality, ordinal efficiency and strategyproofness are incompatible. Finally, we prove that two reasonable notions of envyfreeness, no justified envy and equal-endowment no envy, conflict in the presence of ordinal efficiency and individual rationality. All our impossibility results hold in the strict preference domain.