Strategic behavior in one-to-one matching markets without outside options

In two-sided one-to-one matching markets, each side of the market has a single stable mechanism that is strategy-proof for its members (Alcalde and Barberà, 1994). When agents may not declare potential partners inadmissible, this uniqueness result only holds for the short side, if there is one. Furthermore, among the stable mechanisms that are strategy-proof for the long side of the market, there is one that is less manipulable by coalitions of its members than the long-side optimal deferred acceptance mechanism. In general, Alcalde and Barberà's uniqueness result holds for one side of the market if and only if either at most one of its members may not declare inadmissibilities or there are fewer agents on that side than individuals without outside options on the other side of the market.