Query Efficient Prophet Inequality with Unknown I.I.D. Distributions

We study the single-choice prophet inequality problem, where a gambler faces a sequence of $n$ online i.i.d. random variables drawn from an unknown distribution. When a variable reveals its value, the gambler needs to decide irrevocably whether or not to accept it. The goal is to maximize the competitive ratio between the expected gain of the gambler and that of the maximum variable. It is shown by Correa et al. that when the distribution is unknown or only $o(n)$ uniform samples from the distribution are given, the best an algorithm can do is $1/e$-competitive. In contrast, when the distribution is known or $\Omega(n)$ uniform samples are given, the optimal competitive ratio 0.7451 can be achieved. In this paper, we propose a new model in which the algorithm has access to an oracle that answers quantile queries about the distribution, and study the extent to which we can use a small number of queries to achieve good competitive ratios. Naturally, by making queries to the oracle, one can implement the threshold-based blind strategies that use the answers from the queries as thresholds to accept variables. Our first contribution is to prove that the competitive ratio improves gracefully with the number of thresholds. Particularly with two thresholds our algorithm achieves a competitive ratio of 0.6786. Our second contribution, surprisingly, shows that with a single query, we can do strictly better than with a single threshold. The algorithm sets a threshold in the first phase by making a single query and uses the maximum realization from the first phase as the threshold for the second phase. It can be viewed as a natural combination of the single-threshold algorithm and the algorithm for the secretary problem. By properly choosing the quantile to query and the break-point between the two phases, we achieve a competitive ratio of 0.6718.