"Safe-setwise stability and complementarities in matching markets".
I study stability of matching in two-sided markets with non-transferable utility, and propose a notion of stability that reflects a requirement that blocks be robust to partial execution. To execute a particular block in which a set of agents stand to gain, agents may have to change multiple independent relationships. The block is only partially executed if some agents do not follow the block's prescription, which may leave some blocking agents worse of than in the original matching. I define a block of a matching to be safe if the gain from participating in the block is robust to partial execution.
I show that safe-setwise stable matchings, those that admit no safe blocks, exist for markets where preferences can feature some types of complementarities. In particular, I identify that conditional complementarities that are symmetric do not pose a problem for existence. With many previous notions of stability, some such markets admit no stable matching. For such markets with a decentralized random meeting process, I then show that from any initial matching a safe-setwise stable matching is eventually reached. For centralized matching, I provide a sufficient, and in a sense necessary, condition on preferences for a deferred acceptance procedure to produce a safe-setwise stable matching.