  We consider multiple-type housing markets (Moulin, 1995), which extend
Shapley-Scarf housing markets (Shapley and Scarf, 1974) from one dimension to
higher dimensions. In this model, Pareto efficiency is incompatible with
individual rationality and strategy-proofness (Konishi et al., 2001).
Therefore, we consider two weaker efficiency properties: coordinatewise
efficiency and pairwise efficiency. We show that these two properties both (i)
are compatible with individual rationality and strategy-proofness, and (ii)
help us to identify two specific mechanisms. To be more precise, on various
domains of preference profiles, together with other well-studied properties
(individual rationality, strategy-proofness, and non-bossiness), coordinatewise
efficiency and pairwise efficiency respectively characterize two extensions of
the top-trading-cycles mechanism (TTC): the coordinatewise top-trading-cycles
mechanism (cTTC) and the bundle top-trading-cycles mechanism (bTTC). Moreover,
we propose several variations of our efficiency properties, and we find that
each of them is either satisfied by cTTC or bTTC, or leads to an impossibility
result (together with individual rationality and strategy-proofness).
Therefore, our characterizations can be primarily interpreted as a
compatibility test: any reasonable efficiency property that is not satisfied by
cTTC or bTTC could be considered incompatible with individual rationality and
strategy-proofness. The external validity of our results in the context of
general environments is also discussed. For multiple-type housing markets with
strict preferences, our characterization of bTTC constitutes the first
characterization of an extension of the prominent TTC mechanism
