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Oracle-Augmented Prophet Inequalities

Sariel Har-Peled, Elfarouk Harb, Vasilis Livanos

TL;DR

The paper extends prophet inequalities by introducing an oracle that answers whether the current observation exceeds the suffix maximum, enabling up to $m$ oracle calls. It shows that for the PbM objective, the oracle model with $m$ calls is equivalent to the Top-$1$-of-$(m{+}1)$ model, while for RoE the equivalence fails, but oracle-based guarantees still translate to Top-$1$-of-$(m{+}1)$ bounds in one direction. A tight single-threshold algorithm yields a competitive ratio of $1 - e^{-\xi_m}$, with $\xi_m$ the unique root of $1 - e^{-\xi_m} = \Gamma(m{+}1, \xi_m)/m!$, and matching upper bounds are provided via carefully constructed inputs. The analysis leverages sharding and Poissonization, develops an exponent sequence with rich asymptotics, and extends the results to both non-IID and IID settings, delivering substantial improvements over prior Top-$1$-of-$m$ bounds and offering practical near-optimal strategies when multiple oracle calls are available.

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.

Oracle-Augmented Prophet Inequalities

TL;DR

The paper extends prophet inequalities by introducing an oracle that answers whether the current observation exceeds the suffix maximum, enabling up to oracle calls. It shows that for the PbM objective, the oracle model with calls is equivalent to the Top--of- model, while for RoE the equivalence fails, but oracle-based guarantees still translate to Top--of- bounds in one direction. A tight single-threshold algorithm yields a competitive ratio of , with the unique root of , and matching upper bounds are provided via carefully constructed inputs. The analysis leverages sharding and Poissonization, develops an exponent sequence with rich asymptotics, and extends the results to both non-IID and IID settings, delivering substantial improvements over prior Top--of- bounds and offering practical near-optimal strategies when multiple oracle calls are available.

Abstract

In the classical prophet inequality settings, a gambler is given a sequence of random variables , 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 . 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 oracle calls is equivalent to the \textsc{Top--of-} 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--of-} model. We resolve the oracle model for any , 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--of-} model.
Paper Structure (34 sections, 20 theorems, 53 equations, 2 figures, 1 table)

This paper contains 34 sections, 20 theorems, 53 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Type of Adversary.
  3. Our contributions
  4. Our results in detail
  5. Equivalence of models as far as $\mathbb{P}_{\max}$
  6. Difference of the models as far as RoE
  7. Bounding the performance of the oracle model
  8. Matching upper bound.
  9. The benefit of several oracle calls
  10. Additional related work
  11. Reductions
  12. The $\mathbb{P}_{\max}$ objective
  13. For the RoE objective oracle diff prophet
  14. The non-IID settings
  15. The exponent sequence
  16. ...and 19 more sections

Key Result

Lemma 2.1

Fix an instance of the prophet problem. Let $\mathcal{A}$ be an algorithm for this instance in $\mathcal{M}( { def:prop:h:o:m } \xspace_m, y, {{ { def:PbM } }} \xspace)$, where $y = { def:toiidornot } \xspace$ or def:toiidornot . Then, there exists an algorithm $\m

Figures (2)

  • Figure 1.1: The value of $1-e^{-\xi_m}$ for $m=1,\dots, 15$.
  • Figure 1.2: State of the art.

Theorems & Definitions (33)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 23 more