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Tight Guarantees for Static Threshold Policies in the Prophet Secretary Problem
arXiv - CS - Computer Science and Game Theory Pub Date : 2021-08-29 , DOI: arxiv-2108.12893
Nick Arnosti, Will Ma

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 a $\gamma_k = 1 - e^{-k}k^k/k!$ fraction 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 $\gamma_k$ if $k \geq 5$. The second sets a threshold so that $k \cdot \gamma_k$ values are accepted in expectation, and offers a guarantee of $\gamma_k$ 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.

中文翻译:

先知秘书问题中静态阈值策略的严格保证

在先知秘书问题中,$n$ 值独立于已知分布抽取,并以随机顺序呈现。当每个值出现时,决策者必须接受或拒绝,并且最多可以接受总计 $k$ 值。目标是最大化接受值的预期总和。我们研究静态阈值策略的性能,它接受超过固定阈值的第一个 $k$ 值(或所有这些值,如果存在的值少于 $k$)。我们表明使用最佳阈值保证 $\gamma_k = 1 - e^{-k}k^k/k!$ 离线最佳解决方案的分数,并提供一个例子证明这种保证是严格的。我们还提供了实现这一保证的简单算法。第一个设置阈值,使得超过阈值的预期值数量等于 $k$,如果 $k \geq 5$,则提供 $\gamma_k$ 的保证。第二个设置一个阈值,以便 $k \cdot \gamma_k$ 值在预期中被接受,并为所有 $k$ 提供 $\gamma_k$ 的保证。为了建立这些保证,我们证明了一个关于独立伯努利随机变量之和的结果,它扩展了 Hoeffding (1956) 的经典结果并且具有普遍意义。最后,我们注意到我们的算法可以在关于代理值的信息受限的环境中实现。这使它们在分配 COVID-19 疫苗等环境中变得实用。它扩展了 Hoeffding (1956) 的经典结果并引起普遍关注。最后,我们注意到我们的算法可以在关于代理值的信息受限的环境中实现。这使它们在分配 COVID-19 疫苗等环境中变得实用。它扩展了 Hoeffding (1956) 的经典结果并引起普遍关注。最后,我们注意到我们的算法可以在关于代理值的信息受限的环境中实现。这使它们在分配 COVID-19 疫苗等环境中变得实用。
更新日期:2021-08-31
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