On non-bossy matching rules in two-sided matching problems

It is well known that in two-sided matching problems, stability is incompatible with non-bossiness. We extensively study whether there exists a non-bossy matching rule that satisfies certain properties weaker than stability. Results demonstrate that weak stability and respect for recursive unanimity are incompatible with non-bossiness; however, paired individual rationality and efficiency, and respect for 2-unanimity, are compatible with non-bossiness. Thus, a clear contrast exists between strategy-proofness, which is incompatible with all three aforementioned properties and the paired properties, and non-bossiness, which is incompatible with only two of them.