The satisficing secretary problem: when closed-form solutions meet simulated annealing

The secretary problem has been a focus of extensive study with a variety of extensions that offer useful insights into the theory of optimal stopping. The original solution is to set one stopping threshold that gives rise to an immediately rejected sample $r$ out of the candidate pool of size $n$ and to accept the first candidate that is subsequently interviewed and bests the prior $r$ rejected. In reality, it is not uncommon to draw a line between job candidates to distinguish those above the line vs those below the line. Here we consider such satisficing sectary problem that views suboptimal choices (finding any of the top $d$ candidates) also as hiring success. We use a multiple stopping criteria $(r_1, r_2, \cdots, r_d)$ that sequentially lowers the expectation when the prior selection criteria yields no choice. We calculate the probability of securing the top $d$ candidate in closed form solutions which are in excellent agreement with computer simulations. The exhaustive search for optimal stopping has to deal with large parameter space and thus can quickly incur astronomically large computational times. In light of this, we apply the effective computational method of simulated annealing to find optimal $r_i$ values which performs reasonably well as compared with the exhaustive search despite having significantly less computing time. Our work sheds light on maximizing the likelihood of securing satisficing (suboptimal) outcomes using adaptive search strategy in combination with computation methods.