  Several resource allocation problems involve multiple types of resources,
with a different agency being responsible for "locally" allocating the
resources of each type, while a central planner wishes to provide a guarantee
on the properties of the final allocation given agents' preferences. We study
the relationship between properties of the local mechanisms, each responsible
for assigning all of the resources of a designated type, and the properties of
a sequential mechanism which is composed of these local mechanisms, one for
each type, applied sequentially, under lexicographic preferences, a well
studied model of preferences over multiple types of resources in artificial
intelligence and economics. We show that when preferences are O-legal, meaning
that agents share a common importance order on the types, sequential mechanisms
satisfy the desirable properties of anonymity, neutrality, non-bossiness, or
Pareto-optimality if and only if every local mechanism also satisfies the same
property, and they are applied sequentially according to the order O. Our main
results are that under O-legal lexicographic preferences, every mechanism
satisfying strategyproofness and a combination of these properties must be a
sequential composition of local mechanisms that are also strategyproof, and
satisfy the same combinations of properties.
