Tight Guarantees for Static Threshold Policies in the Prophet Secretary Problem

Abstract

In the prophet secretary problem, n values are drawn independently from known distributions, and presented in random order. A decision-maker must accept or reject each value when it is presented, and may accept at most k values in total. The objective is to maximize the expected sum of accepted values.We study the performance of static threshold policies, which accept the first k values exceeding a fixed threshold (or all such values, if fewer than k exist). We show that using an optimal threshold guarantees afraction of the offline optimal solution, and provide an example demonstrating that this guarantee is tight. We also provide simple algorithms that achieve this guarantee. The first sets a threshold such that the expected number of values exceeding the threshold is equal to k, and offers a guarantee of γ k if k ≥ 5. The second sets a threshold so that k • γ k values are accepted in expectation, and offers a guarantee of γ k for all k. To establish these guarantees, we prove a result about sums of independent Bernoulli random variables, which extends a classical result of Hoeffding (1956) and is of general interest.Finally, we note that our algorithms can be implemented in settings with restricted information about agents' values. This makes them practical in settings such as the allocation of COVID-19 vaccines.

Descending Price Auctions with Bounded Number of Price Levels and Batched Prophet Inequality

“…We conclude this section by noting that Theorem 3 covers the result of Arnosti and Ma [2021] that establishes 1 − e −m m m m! for a single threshold and a more involved proof technique than our analysis.…”

“…rewards (or the homogeneous prophet secretary problem) with a static policy when the decision-maker can collect m rewards. This problem has been studied in Yan [2011] and Arnosti and Ma [2021], where the authors establish the existence of a distribution for rewards such that no single threshold policy can achieve better than 1 − e −m m m m! of the optimal expected reward, and show this approximation factor can be achieved by a single threshold.…”

“…of the optimal expected reward, and show this approximation factor can be achieved by a single threshold. As a side note, our analysis for the setting with multiple items can be viewed as an extension of the results of Arnosti and Ma [2021] to any k > 1 by using a different, and arguably simpler, analysis.…”

“…[2017], Chawla et al [2020], Arnosti and Ma [2021] (also see Correa et al [2021] that introduce the blind strategies, which is a generalization of the static policy to a multi-threshold setting), i.i.d. and random order Abolhassani et al [2017], , and ordered prophets (a.k.a.…”

Tightness without Counterexamples: A New Approach and New Results for Prophet Inequalities

“…Proph(I) can also be constructed (see Arnosti and Ma (2021)). However, these upper bounds for static threshold policies relative to the weaker prophet benchmark require a complicated family of examples with messy calculations, whereas our framework elegantly establishes the tightness of the ratio E[min{Bin(n,k/n),k}] k without needing to construct any counterexamples.…”