Score-Based Secretary Problem

Abstract

In the celebrated “Secretary Problem,” involving n candidates who have applied for a single vacant secretarial position, the employer interviews them one by one in random order and learns their relative ranks. As soon as each interview is over, the employer must either hire the candidate (and stop the process) or reject her (never to be recalled). We consider a variation of this problem where the employer also learns the scores of the already interviewed candidates, which are assumed to be independent and drawn from a known continuous probability distribution. Endowed with this additional information, what strategy should the employer follow in order to maximize his chance of hiring the candidate with the highest score among all n candidates? What is the maximum probability of hiring the best candidate?

Appendix

Proof of Theorem 2. Suppose that the employer hires Candidate k, where $k=1,2,\ldots ,n$. Recall that the employer hires Candidate 1 iff $X_1>\xi _n$. Otherwise, he hires Candidate $k\ge 2$ provided he has not already hired anyone earlier and $X_k=s$ exceeds both the previous maximum $r=\max \{X_1,\ldots , X_{k-1}\}$ and the threshold value $\xi _{n+1-k}$ for Candidate k. In order to express the probability density function of r, subject to the fact that no candidate among Candidates $1, 2, \ldots , k-1$ has been hired, we must account for which candidate possibly achieved the record score r among Candidates $1, 2, \ldots , k-1$. Therefore, we split the range $(0,\xi _n)$ of r as

$$(\xi _{n-1},\xi _n), (\xi _{n-2},\xi _{n-1}), \ldots , (\xi _{n+1-k},\xi _{n+2-k}), (0,\xi _{n+1-k})\;;$$

and note that the number of possible candidates who might have achieved r is $1, 2, \ldots , k-1, k-1$, respectively, in these intervals. See Fig. 6.

Fig. 6

If Candidate $k\ge 2$ with score $X_k=s$ is hired, then what is the likely score $r=\max \{X_1,\ldots ,X_{k-1}\}$ of the previous leading candidate, and which candidate is she?

Having hired Candidate k, the employer wins (or hires the best candidate) only if all future Candidates also score below Candidate k. Adding up the joint probability that the employer hires Candidate k and he wins, we obtain

$$\begin{aligned}&P_n(\mathrm{Win}) = \sum \limits _{k=1}^n P_n(\mathrm{Hire }~k~ \mathrm{and~ Win})\\= & {} \int _{\xi _n}^1 s^{n-1}\,ds + \sum \limits _{k=2}^{n-1} \left\{ \sum \limits _{j=1}^{k-1} \int _{\xi _{n-j}}^{\xi _{n+1-j}} j\,r^{k-2} \int _r^1 s^{n-k}\,ds \; dr \right. \\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \left. + \int _{0}^{\xi _{n+1-k}} (k-1)\,r^{k-2} \int _{\xi _{n+1-k}}^1 s^{n-k}\,ds \; dr \right\} \\&~~~~~~~~~~~~~~~~~~~~ + \sum \limits _{j=1}^{n-1} \int _{\xi _{n-j}}^{\xi _{n+1-j}} j\,r^{n-2} \int _r^1 s^{0}\,ds \; dr \\= & {} \frac{1-\xi _n^n}{n} + \sum \limits _{k=2}^{n-1} \left\{ \sum \limits _{j=1}^{k-1} \int _{\xi _{n-j}}^{\xi _{n+1-j}} j\, \frac{r^{k-2}-r^{n-1}}{n+1-k}\; dr \right. \\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \left. + \int _{0}^{\xi _{n+1-k}} (k-1)\,r^{k-2} \frac{1-\xi _{n+1-k}^{n+1-k}}{n+1-k} \; dr \right\} \\&~~~~~~~~~~~~~~~~~~~~ + \sum \limits _{j=1}^{n-1} \int _{\xi _{n-j}}^{\xi _{n+1-j}} j\,(r^{n-2}-r^{n-1})\; dr \\= & {} \frac{1-\xi _n^n}{n} + \sum \limits _{k=2}^{n} \frac{1}{n+1-k}\;\{ \sum \limits _{j=1}^{k-1} j\, \left( \frac{\xi _{n-j}^{k-1}-\xi _{n+1-j}^{k-1}}{k-1} - \frac{\xi _{n-j}^{n}-\xi _{n+1-j}^{n}}{n}\right) \\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \left. + \left( \xi _{n+1-k}^{k-1} - \xi _{n+1-k}^n \right) \right\} \\= & {} \frac{1}{n}\, [1-\xi _n^n-\xi _{n-1}^n-\cdots -\xi _2^n-\xi _1^n] \\&~~~ +\frac{1}{n-1}\left( \xi _n-\frac{\xi _n^n}{n} \right) +\frac{1}{n-2} \left[ \;\left( \frac{\xi _n^2}{2}-\frac{\xi _n^n}{n} \right) +\left( \frac{\xi _{n-1}^2}{2}-\frac{\xi _{n-1}^n}{n} \right) \right] \\&~~~ + \frac{1}{n-3} \left[ \;\left( \frac{\xi _n^3}{3}-\frac{\xi _n^n}{n} \right) +\left( \frac{\xi _{n-1}^3}{3}-\frac{\xi _{n-1}^n}{n} \right) +\left( \frac{\xi _{n-2}^3}{3}-\frac{\xi _{n-2}^n}{n} \right) \right] \\&~~~ + \cdots \\&~~~ + \frac{1}{1} \left[ \;\left( \frac{\xi _n^{n-1}}{n-1}-\frac{\xi _n^n}{n} \right) +\left( \frac{\xi _{n-1}^{n-1}}{n-1}-\frac{\xi _{n-1}^n}{n} \right) +\cdots +\left( \frac{\xi _{2}^{n-1}}{n-1}-\frac{\xi _{2}^n}{n} \right) \right] \\= & {} \frac{1}{n}\, [1-\xi _n^n-\xi _{n-1}^n-\cdots -\xi _2^n] \\&~~~ +\left[ \frac{\xi _n}{1\cdot (n-1)}+ \frac{\xi _n^2+\xi _{n-1}^2}{2\cdot (n-2)}+\cdots +\frac{\xi _n^{n-1}+\xi _{n-1}^{n-1}+\cdots +\xi _2^{n-1}}{(n-1)\cdot 1)} \right] \\&~~~~ - \frac{\xi _n^n}{n}H_{n-1} - \frac{\xi _{n-1}^n}{n}H_{n-2}-\cdots - \frac{\xi _{2}^n}{n}H_{1}\;, \end{aligned}$$

whence the theorem follows. Q.E.D.