Matching is the part of economics that answers “Who gets what?”, but what if there exists an agent that is unmatched in the marriage problem?
First of all I found that Mcvitie and Wilson 1970 theorem aims that “The set of agents who are matched is the same for all stable matchings” … so if you are matched, it must be that you will be matched in every single stage. But once again, this theorem doesn’t tells us what happens to the agents that are not matched, to imply this theorem holds for the unmatched agents we must need to prove it.
Actually, the “Lone Wolf Theorem” is the answer to my question, and it is a classic result in matching theory. It states that: “Any agent who is unmatched at some stable marriage matching is unmatched in all stable marriage matchings”.
The Lone Wolf Theorem and its generalizations are central to the now-standard strategy for
deriving (one-sided) strategy-proofness results for deferred acceptance (see Hatfield and Milgrom
(2005) and Jagadeesan et al. (2016b)).
This theorem has now 2 proofs.
The first one is know as The standard proof of the Lone Wolf Theorem uses the lattice structure of stable matchings by (Knuth (1976); Roth and Sotomayor (1990)).
And now there is an Elementary proof that does not use any structural properties. I highly recommend the paper “An Elementary proof of the Lone Wolf Theorem” written by Ciupan, Hatfield and Kominers. 2016. “http://www.scottkom.com/articles/Ciupan_Hatfield_Kominers_Elementary_Proof_of_the_Lone_Wolf_Theorem.pdf”
So now you know, if any agent i is unmatched it stays unmatched!