On the Significance of Knowing the Arrival Order in Prophet Inequality

In a prophet inequality problem, $n$ boxes arrive online, each containing some value that is drawn independently from a known distribution. Upon the arrival of a box, its value is realized, and an online algorithm decides, immediately and irrevocably, whether to accept it or proceed to the next box. Clearly, an online algorithm that knows the arrival order may be more powerful than an online algorithm that is unaware of the order. Despite the growing interest in the role of the arrival order on the performance of online algorithms, the effect of knowledge of the order has been overlooked thus far. Our goal in this paper is to quantify the loss due to unknown order. We define the order competitive ratio as the worst-case ratio between the performance of the best order-unaware and the best order-aware algorithms. We study the order competitive ratio for two objective functions, namely (i) max-expectation: maximizing the expected accepted value, and (ii) max-probability: maximizing the probability of accepting the box with the largest value. For the max-expectation objective, we're golden: we give a deterministic order-unaware algorithm that achieves an order competitive ratio of the inverse of the golden ratio (i.e., $1/\phi \approx 0.618$). For the max-probability objective, we give a deterministic order-unaware algorithm that achieves an order competitive ratio of $\ln \frac{1}{\lambda} \approx 0.806$ (where $\lambda$ is the unique solution to $\frac{x}{1-x}= \ln \frac{1}{x}$). Both results are tight. Our algorithms are inevitably adaptive and go beyond single-threshold algorithms.