  We study the set of incentive compatible and efficient two-sided matching
mechanisms. We classify all such mechanisms under an additional assumption --
"gender-neutrality" -- which guarantees that the two sides be treated
symmetrically. All group strategy-proof, efficient and gender-neutral
mechanisms are recursive and the outcome is decided in a sequence of rounds. In
each round, two agents are selected, one from each side. These agents are
either "matched-by-default" or "unmatched-by-default." In the former case
either of the selected agents can unilaterally force the other to match with
them while in the latter case they may only match together if both agree. In
either case, if this pair of agents is not matched together, each gets their
top choice among the set of remaining agents. As an important step in the
characterization, we first show that in one-sided matching all group
strategy-proof and efficient mechanisms are sequential dictatorships. An
immediate corollary is that there are no individually rational, group
strategy-proof and efficient one-sided matching mechanisms.
