Tight Guarantees for Static Threshold Policies in the Prophet Secretary Problem

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.