A direct proof of the short-side advantage in random matching markets

We study the stable matching problem under the random matching model where the preferences of the doctors and hospitals are sampled uniformly and independently at random. In a balanced market with n doctors and n hospitals, the doctor-proposing deferred-acceptance algorithm gives doctors an expected rank of order $\log n$ for their partners and hospitals an expected rank of order $\frac{n}{\log n}$ for their partners (Pittel, 1989; Wilson, 1972). This situation is reversed in an unbalanced market with $n+1$ doctors and n hospitals (Ashlagi et al., 2017), a phenomenon known as the short-side advantage. The current proofs (Ashlagi et al., 2017; Cai and Thomas, 2022) of this fact are indirect, counter-intuitively being based upon analyzing the hospital-proposing deferred-acceptance algorithm. In this paper we provide a direct proof of the short-side advantage, explicitly analyzing the doctor-proposing deferred-acceptance algorithm. Our proof sheds light on how and why the phenomenon arises.