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New Prophet Inequalities via Poissonization and Sharding

Elfarouk Harb

TL;DR

This work addresses prophet inequalities by introducing a unified framework built on Poissonization and sharding. The approach analyzes complex sequential decision problems via Poissonized shard counts and stochastic dominance, yielding tighter bounds across many variants (Top-$1$-of-$k$, Prophet Secretary, Semi-Online, SOLM) and simplifying proofs. Key contributions include improved competitive ratios for non-IID and IID settings, near-tight results for small $k$, adaptive threshold strategies, and an $O(\log^{*} n)$ load algorithm for SOLM, along with algorithmic templates that extend to order-selection and oracle-augmented models. The framework provides both practical improvements in constants and conceptual clarity, with potential to unlock further results in related prophet-inequality variants and dynamic settings.

Abstract

This work introduces \emph{sharding} and \emph{Poissonization} as a unified framework for analyzing prophet inequalities. Sharding involves splitting a random variable into several independent random variables, shards, that collectively mimic the original variable's behavior. We combine this with Poissonization, where these shards are modeled using a Poisson distribution. Despite the simplicity of our framework, we improve the competitive ratio analysis of a dozen well studied prophet inequalities in the literature, some of which have been studied for decades. This includes the \textsc{Top-$1$-of-$k$} prophet inequality, prophet secretary inequality, and semi-online prophet inequality, among others. This approach not only refines the constants but also offers a more intuitive and streamlined analysis for many prophet inequalities in the literature. Furthermore, it simplifies proofs of several known results and may be of independent interest for other variants of the prophet inequality, such as order-selection.

New Prophet Inequalities via Poissonization and Sharding

TL;DR

This work addresses prophet inequalities by introducing a unified framework built on Poissonization and sharding. The approach analyzes complex sequential decision problems via Poissonized shard counts and stochastic dominance, yielding tighter bounds across many variants (Top--of-, Prophet Secretary, Semi-Online, SOLM) and simplifying proofs. Key contributions include improved competitive ratios for non-IID and IID settings, near-tight results for small , adaptive threshold strategies, and an load algorithm for SOLM, along with algorithmic templates that extend to order-selection and oracle-augmented models. The framework provides both practical improvements in constants and conceptual clarity, with potential to unlock further results in related prophet-inequality variants and dynamic settings.

Abstract

This work introduces \emph{sharding} and \emph{Poissonization} as a unified framework for analyzing prophet inequalities. Sharding involves splitting a random variable into several independent random variables, shards, that collectively mimic the original variable's behavior. We combine this with Poissonization, where these shards are modeled using a Poisson distribution. Despite the simplicity of our framework, we improve the competitive ratio analysis of a dozen well studied prophet inequalities in the literature, some of which have been studied for decades. This includes the \textsc{Top--of-} prophet inequality, prophet secretary inequality, and semi-online prophet inequality, among others. This approach not only refines the constants but also offers a more intuitive and streamlined analysis for many prophet inequalities in the literature. Furthermore, it simplifies proofs of several known results and may be of independent interest for other variants of the prophet inequality, such as order-selection.
Paper Structure (74 sections, 28 theorems, 118 equations, 10 figures, 1 table, 7 algorithms)

This paper contains 74 sections, 28 theorems, 118 equations, 10 figures, 1 table, 7 algorithms.

Table of Contents

  1. Introduction, Related Work, and Contributions.
  2. Prophet Secretary
  3. Top-$1$-of-$k$
  4. Order-selection
  5. Semi-Online
  6. Poissonization
  7. Sharding
  8. Simple Application.
  9. Case 1: $\ell \in [0, \tau]$.
  10. Case 2: $\ell \in (\tau, +\infty)$.
  11. New results exceeds parameter optimization.
  12. Organization
  13. Notation, assumptions, and existing results.
  14. Notation
  15. Continuity Assumptions
  16. ...and 59 more sections

Key Result

Theorem 1.1

(Proof in sec:best1ofk) There exists an algorithm for the TOP-$1$-of-$2$ problem with a competitive ratio of $0.781$. No algorithm can achieve a competitive ratio higher than $0.794$. For any $k$, there exists an algorithm for TOP-$1$-of-$k$ with a competitive ratio of at least $1-e^{-kW(\frac{\sqrt

Figures (10)

  • Figure 1: Level $7$ canonical boxes of $\tau_7 = \Xi(q)$
  • Figure 2: The two cases of \ref{['lemma:iid:monstrosity']}. Curves are labeled either as $C$ or $\tau_C$.
  • Figure 3: The function $C^\ast(x)$ for $c=0.745$ solved using numerical methods, truncated at $x=0.99$.
  • Figure 4:
  • Figure 5: The 3 cases of the analysis for Top-$1$-of-$2$ algorithm from left to right.
  • ...and 5 more figures

Theorems & Definitions (36)

  • Remark 1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 2
  • Lemma 1.1
  • Lemma 2.1
  • Lemma 2.2
  • ...and 26 more