Uniformly Bounded Regret in the Multisecretary Problem

Abstract

In the secretary problem of Cayley (1875) and Moser (1956), n non-negative, independent, random variables with common distribution are sequentially presented to a decision maker who decides when to stop and collect the most recent realization. The goal is to maximize the expected value of the collected element.In the k-choice variant, the decision maker is allowed to make k ď n selections to maximize the expected total value of the selected elements. Assuming that the values are drawn from a known distribution with finite support, we prove that the best regret-the expected gap between the optimal online policy and its offline counterpart in which all n values are made visible at time 0-is uniformly bounded in the number of candidates n and the budget k. Our proof is constructive: we develop an adaptive Budget-Ratio policy that achieves this performance. The policy selects or skips values depending on where the ratio of the residual budget to the remaining time stands relative to multiple thresholds that correspond to middle points of the distribution. We also prove that being adaptive is crucial: in general, the minimal regret among non-adaptive policies grows like the square root of n. The difference is the value of adaptiveness.

Does the Multisecretary Problem Always Have Bounded Regret?

“…I show that this myopic regret is extremely easy to work with: it is simply a quadratic function of k when secretary valuations are uniformly distributed, and it is bounded by the probability of a binomial random variable being far out in its tail when secretary valuations have finite support. This latter fact yields a shorter and more direct proof of Arlotto and Gurvich's (2019) original result.…”

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Arlotto and Gurvich (2019) showed that regret in the multisecretary problem is bounded, both in the number of job openings, n, and the number of applicants, k, provided that the applicant valuations have finite support. I show that this result does not hold when applicant valuations are drawn from a standard uniform distribution.

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“…Thus, you face an optimal stopping problem, with the objective being to maximize the expected capability of the man you hire or, equivalently, minimize your regret, which is defined as the expected capability difference between the most capable man and the man you hire. Arlotto and Gurvich (2019) study the multisecretary problem, which is the same as above except with k posts to fill, for 1 ≤ k ≤ n − 1. In this version of the problem, your regret is the expected capability difference between the k most capable men and the k men you hire.…”

“…To answer this question, I show that Arlotto and Gurvich's (2019) uniform bound does not hold when secretary capabilities are standard uniform random variables. Specifically, I show that the regret is between log(n)/16 − 1/4 and log(n + 1)/8 in this case, when k = n/2 and n ≥ 16.…”

“…Why does extending the range of secretary valuations from finite set {v 1 , · · · , v ℓ } to infinite set [0, 1] break Arlotto and Gurvich's (2019) result? Well, in the former case, secretaries are interchangeable parts, which means that your hiring errors can be perfectly "undone" in the future.…”