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Oracle-Augmented Prophet Inequalities

Authors Sariel Har-Peled , Elfarouk Harb , Vasilis Livanos

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Author Details

Sariel Har-Peled
  • Department of Computer Science, University of Illinois, Urbana, IL, USA
Elfarouk Harb
  • Department of Computer Science, University of Illinois, Urbana, IL, USA
Vasilis Livanos
  • Department of Industrial Engineering, University of Chile, Santiago, Chile

Funding

  • Har-Peled, Sariel: Work on this paper was partially supported by NSF AF award CCF-2317241.
  • Harb, Elfarouk: Work on this paper was partially supported by NSF award CCF-1910149.
  • Livanos, Vasilis: Work on this paper was partially supported by NSF award CCF-1750436.

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Sariel Har-Peled, Elfarouk Harb, and Vasilis Livanos. Oracle-Augmented Prophet Inequalities. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 81:1-81:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ICALP.2024.81

Abstract

In the classical prophet inequality setting, a gambler is given a sequence of n random variables X₁, … , X_n, taken from known distributions, observes their values in adversarial order and selects one of them, immediately after it is being observed, aiming to select a value that is as high as possible. The classical prophet inequality shows a strategy that guarantees a value at least half of the value of an omniscience prophet that always 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 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 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 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 Top-1-of-m model.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algorithmic mechanism design
Keywords
  • prophet inequalities
  • predictions
  • top-1-of-k model

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