One Strategic Woman: Sharp Ranks and a Persistent Core

Abstract

In a balanced random stable-marriage market, one fixed woman manipulates men-proposing deferred acceptance to obtain her best attainable partner. We prove that the women's expected average true partner rank is asymptotic to $(\ln n)^2$, improving the previous $O((\ln n)^4)$ bound, while the men's is asymptotic to $n/(2\ln n)$. The men's and women's average ranks, normalized by $n/\ln n$ and $\ln n$, converge jointly in distribution to $(U,U^{-1})$, where $U$ is uniform on $(0,1)$. Under optimal truthful truncation, the span of women's average ranks within the reported stable core, divided by $\ln n$, converges in distribution to $U^{-1}-1$.