Optimal Selection Based on Relative Rank

n rankable persons appear sequentially in random order. At the ith stage we observe the relative ranks of the first i persons to appear, and must either select the ith 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 as n→ ∞ this tends to the value $$\mathop \Pi \limits_{j = 1}^\infty {\left( {\frac{{j + 2}}{j}} \right)^{1/j + 1}} \cong 3.8695.$$