Oracle-Augmented Prophet Inequalities

Abstract

In the classical prophet inequality settings, a gambler is given a sequence of $n$ random variables $X_1, \dots, X_n$, taken from known distributions, observes their values in this (potentially adversarial) order, and select one of them, immediately after it is being observed, so that its value is as high as possible. The classical \emph{prophet inequality} shows a strategy that guarantees a value at least half of that an omniscience prophet that picks the maximum, and this ratio is optimal. Here, we generalize the prophet inequality, allowing the gambler some additional information about the future that is otherwise privy only to the prophet. Specifically, at any point in the process, the gambler is allowed to query an oracle $\mathcal{O}$. The oracle responds with a single bit answer: YES if the current realization is greater than the remaining realizations, and NO otherwise. We show that the oracle model with $m$ oracle calls is equivalent to the \textsc{Top-$1$-of-$(m+1)$} model when the objective is maximizing the probability of selecting the maximum. This equivalence fails to hold when the objective is maximizing the competitive ratio, but we still show that any algorithm for the oracle model implies an equivalent competitive ratio for the \textsc{Top-$1$-of-$(m+1)$} model. We resolve the oracle model for any $m$, giving tight lower and upper bound on the best possible competitive ratio compared to an almighty adversary. As a consequence, we provide new results as well as improvements on known results for the \textsc{Top-$1$-of-$m$} model.