The stability of many-to-many matching with max-min preferences

- The equivalence between the pairwise-stability and the setwise-stability is obtained.
- We show that the pairwise-stability implies the strong corewise-stability.
- We show that the strong core may be a proper subset of the core.
- We show that the deferred acceptance algorithm yields a pairwise-stable matching.

This paper investigates the two-sided many-to-many matching problem, where every agent has max-min preference. The equivalence between the pairwise-stability and the setwise-stability is obtained. It is shown that the pairwise-stability implies the strong corewise-stability and the former may be strictly stronger than the latter. We also show that the strong core may be a proper subset of the core. The deferred acceptance algorithm yields a pairwise-stable matching. Thus the set of stable matchings (in all four senses) is non-empty.