Stability and efficiency of partitions in matching problems

We define two versions of stability and efficiency of partitions and analyze their relationships for some matching rules. The stability and efficiency of a partition depends on the matching rule f. The results are stated under various membership property rights axioms. It is shown that in a world where agents can freely exit from and enter coalitions, whenever the matching rule is individually rational and Pareto optimal, the set of f-stable and f-efficient partitions coincide and it is unique: the grand coalition. Then we define a weaker version of stability and efficiency, namely spcific to a given preference profile and find some negative results for stable matching rules.