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Prophet Inequality from Samples: Is the More the Merrier?

Tomer Ezra

Abstract

We study a variant of the single-choice prophet inequality problem where the decision-maker does not know the underlying distribution and has only access to a set of samples from the distributions. Rubinstein et al. [2020] showed that the optimal competitive-ratio of $\frac{1}{2}$ can surprisingly be obtained by observing a set of $n$ samples, one from each of the distributions. In this paper, we prove that this competitive-ratio of $\frac{1}{2}$ becomes unattainable when the decision-maker is provided with a set of more samples. We then examine the natural class of ordinal static threshold algorithms, where the algorithm selects the $i$-th highest ranked sample, sets this sample as a static threshold, and then chooses the first value that exceeds this threshold. We show that the best possible algorithm within this class achieves a competitive-ratio of $0.433$. Along the way, we utilize the tools developed in the paper and provide an alternative proof of the main result of Rubinstein et al. [2020].

Prophet Inequality from Samples: Is the More the Merrier?

Abstract

We study a variant of the single-choice prophet inequality problem where the decision-maker does not know the underlying distribution and has only access to a set of samples from the distributions. Rubinstein et al. [2020] showed that the optimal competitive-ratio of can surprisingly be obtained by observing a set of samples, one from each of the distributions. In this paper, we prove that this competitive-ratio of becomes unattainable when the decision-maker is provided with a set of more samples. We then examine the natural class of ordinal static threshold algorithms, where the algorithm selects the -th highest ranked sample, sets this sample as a static threshold, and then chooses the first value that exceeds this threshold. We show that the best possible algorithm within this class achieves a competitive-ratio of . Along the way, we utilize the tools developed in the paper and provide an alternative proof of the main result of Rubinstein et al. [2020].
Paper Structure (18 sections, 11 theorems, 54 equations)

This paper contains 18 sections, 11 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Our Results and Techniques
  3. Main Result 1 (Section \ref{['sec:main']}):
  4. Main Result 2 (Section \ref{['sec:ordinal']}):
  5. Simplified Analysis for the Single-Sample Case.
  6. Related Work
  7. Preliminaries
  8. Probability definitions and tools.
  9. Hardness of Approximation
  10. Case 1: The algorithm over-select for some $\ell\in[0,2\epsilon]$.
  11. Case 2: The algorithm does not over-select for any $\ell\in[0,2\epsilon]$.
  12. Ordinal Static Threshold Algorithms
  13. Proof of Claim \ref{['item:1']}.
  14. Proof of Claim \ref{['item:2']}.
  15. Proof of Claim \ref{['item:3']}.
  16. ...and 3 more sections

Key Result

Theorem 2.1

Fixing $p\in(0,1)$, and taking $n$ goes to infinity, the total variation distance between binomial with parameters $n,p$, and a normal distribution with the same mean and variance goes to $0$. Formally, where

Theorems & Definitions (21)

  • Definition 2.1: Total Variation Distance
  • Theorem 2.1: gavalakis2024entropy
  • Theorem 2.2: devroye2018total
  • Theorem 2.3: Chernoff Bound
  • Definition 2.2: Stochastic Domination
  • Theorem 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • ...and 11 more