IID prophet inequality with a single data point

In this work, we study the single-choice prophet inequality problem, where a seller encounters a sequence of n online bids. These bids are modeled as independent and identically distributed (i.i.d.) random variables drawn from an unknown distribution. Upon the revelation of each bid's value, the seller must make an immediate and irrevocable decision on whether to accept the bid and sell the item to the bidder. The objective is to maximize the competitive ratio between the expected gain of the seller and that of the maximum bid. It is shown by Correa et al. [1] that when the distribution is unknown or only $o\left(n\right)$ uniform samples from the distribution are given, the best an algorithm can do is $1/e$-competitive. In contrast, when the distribution is known [2], or when $\Omega \left(n\right)$ uniform samples are given [3], the optimal competitive ratio of 0.7451 can be achieved. In this paper, we study the setting when the seller has access to a single point in the cumulative density function of the distribution, which can be learned from historical sales data. We investigate how effectively this data point can be used to design competitive algorithms. Motivated by the algorithm for the secretary problem, we propose the observe-and-accept algorithm that sets a threshold in the first phase using the data point and adopts the highest bid from the first phase as the threshold for the second phase. It can be viewed as a natural combination of the single-threshold algorithm for prophet inequality and the secretary problem algorithm. We show that our algorithm achieves a good competitive ratio for a wide range of data points, reaching up to 0.6785-competitive as $n\to \infty$ for certain data points. Additionally, we study an extension of the algorithm that utilizes more than two phases and show that the competitive ratio can be further improved to at least 0.6862.