Optimal selection based on relative rank (the “secretary problem”)

Abstract

n rankable persons appear sequentially in random order. At theith stage we observe the relative ranks of the firsti persons to appear, and must either select theith person, in which case the process stops, or pass on to the next stage. For that stopping rule which minimizes the expectation of the absolute rank of the person selected, it is shown that asn → ∞ this tends to the value

$$\prod\limits_{j = 1}^\infty {(\tfrac{{j + 2}}{j})^{1/j + 1} } \cong 3.8695$$

.