Strategic Interview Positioning under Uncertain Self-Rank

We study interview positioning in the secretary problem when an applicant is uncertain about their own absolute rank. The employer uses a given rejection threshold, and the applicant’s self-rank distribution is truncated geometric. We show that the applicant need only compare the first eligible position with the final position and that, for finite $n$, a unique critical value exists precisely when $r(r+1)>n-1$. If $r_n$ is the employer’s rejection threshold and $r_n\mathrm{/}n\to \alpha \in (0,1)$ as $n\to \mathrm{\infty }$, then $$\underset{n\to \mathrm{\infty }}{\lim }{q}_c(n,r_n)=q_c(\alpha )=\frac{1-\alpha }{1-\alpha +{\alpha }^2}\mathrm{.}$$ Thus uncertainty about absolute rank produces an explicit asymptotic phase transition in optimal interview positioning.