The natural preference domain for many practical settings of the assignment problems is the one in which agents are allowed to be indifferent between objects, the weak preference domain. Most of the existing work on assignment problems assumes strict preferences. There are important exceptions, but they provide solutions only to the assignment problems with a social endowment, where agents own objects collectively and there are no private endowments. We consider the general class of assignment problems with private endowments and a social endowment. Our main contribution is a recursive solution for the weak preference domain. Our solution satisfies individual rationality, ordinal efficiency and a recently introduced fairness axiom, no justified-envy.
