The Infinite Secretary Problem with Recall

Abstract

The infinite secretary problem, in which an infinite number of rankable items arrive at times which are i.i.d., uniform on (0, 1), is modified to allow for a fixed period of recall of length $\alpha, 0 \leq \alpha \leq 1$. The goal is to find the maximum probability of best choice, $v = v(\alpha)$, as well as an optimal stopping time $\tau^\ast = \tau^\ast(\alpha)$. A differential-delay equation is derived, the solution of which yields $v(\alpha)$ and $\tau^\ast(\alpha)$, the latter given in terms of a constant $t^\ast \lbrack = t^\ast(\alpha)\rbrack$. For $\alpha \geq 1/2$, the complete solution to the problem is obtained. For $0 < \alpha < 1/2, v(\alpha)$ cannot be put in closed form, so upper and lower bounds for $v(\alpha)$ and $t^\ast(\alpha)$ are obtained and are investigated for $\alpha$ near 0 and near 1/2, where the solutions are known. We also find asymptotic expansions of $v(\alpha)$ and $t^\ast(\alpha)$ about $\alpha = 0$ and $\alpha = 1/2$. Finally, the solution to the finite, $n$-item length-$m$ recall problem introduced by Smith and Deely is shown to converge to the solution of the infinite problem when $m/n \rightarrow \alpha$.