Impossibility of Stable and Non-damaging bossy Matching Mechanism

In this paper we prove the impossibility of stability rules that satisfy a concept weaker than nonbossiness. Stability and nonbossiness are essential to matching theory. However, Kojima Fuhito(2010) shows that a matching mechanism that is both stable and nonbossy dose not exist. We define a new concept that is weaker than nonbossiness and consider whether or not stability and the new concept are compatible. Unfortunately, we show that these properties are incompatible.