We study the problem of allocating objects among people. We consider cases where each object is initially owned by someone, no object is initially owned by anyone, and combinations of the two. The problems we look at are those where each person has a need for exactly one object and initially owns at most one object (also known as house allocation with existing tenants"). We split with most of the existing literature on this topic by dropping the assumption that people can always strictly rank the objects. We show that, without this assumption, problems in which either some or all of the objects are not initially owned are equivalent to problems where each object is initially owned by someone. Thus, it suffices to study problems of the latter type. We ask if there are efficient rules that provide incentives for each person not only to participate (rather than stay home with what he owns), but also to state his preferences honestly. Our main contribution is to show that the answer is positive. The intuitive "top trading cycles" algorithm provides the only such rule for environments where people are never indifferent (Ma 1994). We generalize this algorithm in a way that allows for indifference without compromising on efficiency and incentives."
