Stability may be hard to achieve in decentralized matching

So says a paper by Muriel Niederle and Leeat Yariv: Decentralized Matching with Aligned Preferences

Abstract. We study a simple model of a decentralized market game in which firms make directed offers to workers. We focus on markets in which agents have aligned preferences. When agents have complete information or when there are no frictions in
the economy, there exists an equilibrium that yields the stable match. In the presence of market frictions and preference uncertainty, harsher assumptions on the richness of the economy have to be made in order for decentralized markets to generate stable outcomes in equilibrium.