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 a uniformly 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 analyze 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 an appropriate threshold guarantees γ_k =1 - e ⁻^kk^k /k! times the value of the offline optimal solution. Note that γ₁ = 1 - 1/e, and by Stirling’s approximation, (Formular Presented). This represents the best-known guarantee for the prophet secretary problem for all k > 1 and is tight for all k for the class of static threshold policies. We provide two simple methods for setting the threshold. Our first method sets a threshold such that k · γ_k values are accepted in expectation, and offers an optimal guarantee for all k. Our second sets a threshold such that the expected number of values exceeding the threshold is equal to k. This approach gives an optimal guarantee if k > 4 but gives suboptimal guarantees for k ≤ 4. Our proofs use a new result for optimizing sums of independent Bernoulli random variables, which extends a result of Hoeffding from 1956 and could be of independent interest.