Stable Secretaries

Abstract

In the classical secretary problem, multiple secretaries arrive one at a time to compete for a single position, and the goal is to choose the best secretary to the job while knowing the candidate’s quality only with respect to the preceding candidates. In this paper we define and study a new variant of the secretary problem, in which there are multiple jobs. The applicants are ranked relatively upon arrival as usual, and, in addition, we assume that the jobs are also ranked. The main conceptual novelty in our model is that we evaluate a matching using the notion of blocking pairs from Gale and Shapley’s stable matching theory. Specifically, our goal is to maximize the number of matched jobs (or applicants) that do not take part in a blocking pair. We study the cases where applicants arrive randomly or in adversarial order, and provide upper and lower bounds on the quality of the possible assignment assuming all jobs and applicants are totally ordered. Among other results, we show that when arrival is uniformly random, a constant fraction of the jobs can be satisfied in expectation, or a constant fraction of the applicants, but not a constant fraction of the matched pairs.

Appendices

Appendix

Additional Proofs

Proof of Observation 2.6

Fix $q = \lceil k/\ell \rceil$ and observe that for every $r \le \ell$, we have

$$\begin{aligned} \Pr (R > r) \, = \, \frac{k - q}{k} \cdot \frac{k - q - 1}{k - 1} \cdots \frac{k - q - (r - 1)}{k - (r - 1)} \, . \end{aligned}$$

It follows that

$$\begin{aligned} \Pr (R > r) \, \le \, \frac{k - k/\ell }{k} \cdot \frac{k - k/\ell - 1}{k - 1} \cdots \frac{k - k/ \ell - (r - 1)}{k - (r - 1)} \, \le \, \left( 1 - \frac{1}{\ell } \right) ^{r} \, < \, e^{- r/\ell } \end{aligned}$$

and

$$\begin{aligned} \Pr (R> r) \, \ge \, \frac{k - k/\ell - 1}{k} \cdot \frac{k - k/ \ell - 2}{k - 1} \cdots \frac{k - k/\ell - r}{k - (r - 1)} \, > \, \left( 1 - \frac{k/\ell + 1}{k - r} \right) ^{r} \, . \end{aligned}$$

By taking $c \ge 3$ so that $r \le \ell \le k/3$, we ensure that

$$\begin{aligned} \ell \le k - 2 r \, \Longleftrightarrow \, k + \ell \le 2 k - 2 r \, \Longleftrightarrow \, \frac{k/\ell + 1}{k - r} \le \frac{2}{\ell } \, , \end{aligned}$$

thus

$$\begin{aligned} \Pr (R> r) \,> \, \left( 1 - \frac{2}{\ell } \right) ^{r} \, > \, e^{-4 r/\ell } \, , \end{aligned}$$

where the second transition follows by taking $c \ge 2.54$ so that $2/\ell \le 0.79$. Therefore,

$$\begin{aligned} \Pr (\ell /5 < R \le \ell ) \, = \, \Pr (R> \ell /5) - \Pr (R> \ell ) \, > \, e^{-4/5} - e^{-1} \end{aligned}$$

which establishes the assertion as $e^{-4/5} - e^{-1} \approx 1/12.28$. $\square$

Proof of Observation 3.4

Recall the classical secretary problem in which $\mathtt {DM}$ has to stop upon the arrival of some x and the objective is to maximize the probability that x is the most preferred boy. The optimal strategy in the secretary problem is to wait until time $k\approx \tfrac{1}{e} n$, and then stop upon the first arrival of a boy who is more preferred than all of the previous boys. The probability of success converges to $\tfrac{1}{e}$, as n grows.

From the solution to the secretary problem we device a matching strategy as follows: in the first $k=\lfloor \tfrac{1}{e} n\rfloor$ steps, match the boys with arbitrary girls who are neither the most preferred nor the least preferred girl. Continue in the same manner while reserving the most and least preferred girls for the first arrivals of boys who are either more preferred or less preferred than all previous boys. Upon the first arrival of a boy x who is more preferred than all previous boys, match x with the most preferred girl. Similarly, match the first boy who is less preferred than all previous boys with the least preferred girl. At times $n-1$ and n match the arriving boys arbitrarily.

In any matching in which the most (resp. least) preferred boy and girl are matched together, they form a satisfied pair; therefore, by the guarantee of the secretary problem solution and the additivity of expectation, the proposed algorithm guarantees an expected number of $\frac{2}{e} -o(1)$ satisfied pairs. $\square$