  We formalize an allocation model under ordinal preferences that is more
general than the well-studied Shapley-Scarf housing market. In our model, the
agents do not just care which house or resource they get but also care about
who gets their own resource. This assumption is especially important when
considering temporary exchanges in which each resource is eventually returned
to the owner. We show that several positive axiomatic and computational results
that hold for housing markets do not extend to the more general setting. We
then identify natural restrictions on the preferences of agents for which
several positive results do hold. One of our central results is a general class
of algorithms that return any allocation that is individually rational and
Pareto optimal with respect to the responsive set extension.
