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.

翻译：在预言秘书问题中, 美元值独立于已知的发行量之外, 并按随机顺序排列。 决策者必须接受或拒绝每个值, 并在提出时接受每个值的一小部分, 并且可以接受最多为美元的总价值。 目标是最大限度地实现所接受值的预期总和。 我们研究静态阈值政策的表现, 接受第一个超过固定阈值的K美元值( 或所有此类值, 如果存在美元, 如果存在美元, 则低于美元 5 美元 ) 。 我们显示, 使用最佳阈值保证 $\ gamma_k k kk/k! 最佳设置时必须接受或拒绝每个值的每个值, 并提供一个示例, 证明这一保证非常严格。 我们还提供了实现这一保证的简单算法。 第一个阈值的预期值数等于 $k$, 如果存在 $k egeq 5, 则提供1 $gam_k 的保证。 第二个设定了一个阈值, 美元的实际值值=cdcot k coal 值被接受, exmalal expralalal exalfalfalformaxes, 我们可以证明 $xal lifal $56 。