[Paper Review] Prophet Secretary: Surpassing the $1-1/e$ Barrier

Yossi Azar, Ashish Chiplunkar+1 more|arXiv (Cornell University)|Nov 6, 2017

Optimization and Search Problems8 references3 citations

TL;DR

This paper presents a novel algorithm for the Prophet Secretary problem that surpasses the long-standing $1 - 1/e$ competitive ratio barrier, achieving a competitive ratio of $1 - 1/e + 1/400$. By introducing new analytical techniques beyond prior work, the authors design a threshold-based algorithm that adaptively leverages the random order of arrivals and distribution knowledge to outperform existing methods in stochastic online selection under known i.i.d. arrival order.

ABSTRACT

In the Prophet Secretary problem, samples from a known set of probability distributions arrive one by one in a uniformly random order, and an algorithm must irrevocably pick one of the samples as soon as it arrives. The goal is to maximize the expected value of the sample picked relative to the expected maximum of the distributions. This is one of the most simple and fundamental problems in online decision making that models the process selling one item to a sequence of costumers. For a closely related problem called the Prophet Inequality where the order of the random variables is adversarial, it is known that one can achieve in expectation $1/2$ of the expected maximum, and no better ratio is possible. For the Prophet Secretary problem, that is, when the variables arrive in a random order, Esfandiari et al.\ (ESA 2015) showed that one can actually get $1-1/e$ of the maximum. The $1-1/e$ bound was recently extended to more general settings (Ehsani et al., 2017). Given these results, one might be tempted to believe that $1-1/e$ is the correct bound. We show that this is not the case by providing an algorithm for the Prophet Secretary problem that beats the $1-1/e$ bound and achieves $1-1/e+1/400$ of the optimum value. We also prove a hardness result on the performance of algorithms under a natural restriction which we call deterministic distribution-insensitivity.

Motivation & Objective

• To overcome the perceived tightness of the $1 - 1/e$ competitive ratio in the Prophet Secretary problem, which had been considered optimal since Esfandiari et al.'s 2017 result.
• To develop a new algorithmic framework that improves upon the $1 - 1/e$ bound in the random-order arrival model, where distributions are known in advance but samples arrive in uniformly random order.
• To establish a provably better competitive ratio than $1 - 1/e$ under the standard benchmark of $\mathbb{E}[\max_i X_i]$, using techniques that go beyond simple thresholding.
• To analyze the limitations of deterministic distribution-insensitive algorithms by proving an upper bound of $11/15$ on their competitive ratio.

Proposed method

• Designing a threshold-based algorithm that sets dynamic thresholds $\alpha_k$ for each position $k$ in the random arrival sequence, dependent on $n$ and $\mathbb{E}[\max_i X_i]$.
• Introducing new analytical techniques to bound the performance of the algorithm across adversarial instances, particularly by constructing worst-case distribution families to test algorithmic robustness.
• Using probabilistic analysis and asymptotic limits ($\varepsilon \to 0$) to derive tight upper bounds on expected profit in critical scenarios involving rare high-value outcomes.
• Applying lemmas to bound the competitive ratio under various threshold ordering constraints ($\alpha_1 > \alpha_2$, $\alpha_1 \leq \alpha_2$, etc.) and combining them via weighted averaging to derive the overall upper bound.
• Constructing four distinct adversarial instances (Instance 1–4) to test algorithm behavior under different distributional assumptions and arrival order probabilities.
• Proving that the competitive ratio is bounded above by $11/15 \approx 0.733$ for any deterministic distribution-insensitive algorithm, under a natural restriction.

Experimental results

Research questions

• RQ1 Can the $1 - 1/e$ competitive ratio for the Prophet Secretary problem be surpassed using a randomized arrival order and known distributions?
• RQ2 What is the true optimal competitive ratio for the Prophet Secretary problem, and is $1 - 1/e$ tight?
• RQ3 Are there algorithmic techniques beyond simple thresholding that can yield better performance in the Prophet Secretary setting?
• RQ4 What are the fundamental limitations of deterministic distribution-insensitive algorithms in this problem class?
• RQ5 Can tighter upper bounds be established on the competitive ratio under natural algorithmic restrictions?

Key findings

• The paper presents an algorithm that achieves a competitive ratio of $1 - 1/e + 1/400 \approx 0.6321$, exceeding the prior $1 - 1/e \approx 0.6321$ bound.
• The improvement is achieved through a novel thresholding strategy that adapts to the random order and known distributions, going beyond the methods of Esfandiari et al.
• The authors prove that no deterministic distribution-insensitive algorithm can achieve a competitive ratio better than $11/15 \approx 0.7333$, establishing a new upper bound under this restriction.
• Through four carefully constructed adversarial instances, the paper derives tight upper bounds on the performance of threshold-based algorithms under different threshold orderings and distribution types.
• The analysis shows that the competitive ratio is bounded above by $\min(1+2\alpha_1, 2 - \alpha_1 + 2\alpha_2, 3 - 2\alpha_2)/3$ under certain threshold constraints, leading to the $11/15$ upper bound.
• The results refute the long-standing belief that $1 - 1/e$ is the optimal competitive ratio for the Prophet Secretary problem, demonstrating that better performance is achievable.