In a balanced matching market — n doctors and n hospitals, random preferences — the doctor-proposing deferred acceptance algorithm gives doctors an expected rank of order log n. Each doctor gets approximately their log(n)-th choice. Hospitals fare worse: they get approximately their n/log(n)-th choice. The proposing side has a structural advantage.
Add one doctor. Now n+1 doctors compete for n positions. The advantage flips. Hospitals — now the short side — gain dramatically. The extra competitor doesn't just slightly shift outcomes; it transfers the entire structural advantage to the other side.
The paper provides a direct proof of why. The mechanism is competition at the margin. In a balanced market, each doctor's proposal chain terminates quickly — there are enough positions that rejection cascades are short. Adding one doctor extends the cascades: the extra competitor creates a chain of displacements where each rejected doctor proposes to their next choice, potentially displacing an already-matched doctor, who in turn proposes elsewhere. The cascade length increases from O(log n) to much larger, and the hospitals benefit from seeing more applicants.
The short-side advantage is not a small effect. It is the difference between getting your top choices and getting your median choice. One extra participant on the other side is sufficient to generate it.
The structural point: market power in matching is determined by the side with fewer participants, not by preference structure, information, or strategy. The short side wins because it creates competition on the long side. The advantage is arithmetical — it follows from the count, not the content.