The secretary problem of minimizing the expected rank: a simple suboptimal approach with generalizations

The secretary problem for selecting one item so as to minimize its expected rank, based on observing the relative ranks only, is revisited. A simple suboptimal rule, which performs almost as well as the optimal rule, is given. The rule stops with the smallest i such that R_i ≤ ic/(n + 1 - i) for a given constant c, where R_i is the relative rank of the ith observation and n is the total number of items. This rule has added flexibility. A curtailed version thereof can be used to select an item with a given probability P, P < 1. The rule can be used to select two or more items. The problem of selecting a fixed percentage, a, 0 < α < 1, of n, is also treated. Numerical results are included to illustrate the findings.