Recent developments in prophet inequalities

Correa, José R.; Foncea, Patricio; Hoeksma, Ruben; Oosterwijk, Tim; Vredeveld, Tjark

doi:10.1145/3331033.3331039

Cited by 45 publications

(30 citation statements)

References 24 publications

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“…In the more general settings of the prophet inequality, the assumption that the random variables are identically distributed is discarded, and each random variable is allowed to be drawn from a different known distribution (for a recent survey see [6]). The single sample prophet inequality (see [1] for example) relaxes the assumption that the distributions are known.…”

Section: Further Related Workmentioning

confidence: 99%

Competitive Analysis with a Sample and the Secretary Problem

Kaplan¹,

Naori²,

Raz³

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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We extend the standard online worst-case model to accommodate past experience which is available to the online player in many practical scenarios. We do this by revealing a random sample of the adversarial input to the online player ahead of time. The online player competes with the expected optimal value on the part of the input that arrives online. Our model bridges between existing online stochastic models (e.g., items are drawn i.i.d. from a distribution) and the online worst-case model. We also extend in a similar manner (by revealing a sample) the online random-order model.We study the classical secretary problem in our new models. In the worst-case model we present a simple online algorithm with optimal competitive-ratio for any sample size. In the random-order model, we also give a simple online algorithm with an almost tight competitiveratio for small sample sizes. Interestingly, we prove that for a large enough sample, no algorithm can be simultaneously optimal both in the worst-cast and random-order models.

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Section: Further Related Workmentioning

confidence: 99%

Competitive Analysis with a Sample and the Secretary Problem

Kaplan¹,

Naori²,

Raz³

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…Therefore, it is enough to prove that a(r * ) − 1+r * 8 ≥ a(r * )/2, which is equivalent to 4a(r * ) ≥ 1 + r * . If we replace a(r * ) with the explicit formula in the right hand of Equation (7) and rearrange terms, we obtain the inequality 3 − 4 ln(2) ≥ r * (5 − 6 ln (2)). Note that this is satisfied with equality by r * .…”

Section: P(algmentioning

confidence: 99%

The Two-Sided Game of Googol and Sample-Based Prophet Inequalities

Correa¹,

Cristi²,

Epstein³

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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The secretary problem or the game of Googol are classic models for online selection problems that have received significant attention in the last five decades. In this paper we consider a variant of the problem and explore its connections to data-driven online selection. Specifically, we are given n cards with arbitrary non-negative numbers written on both sides. The cards are randomly placed on n consecutive positions on a table, and for each card, the visible side is also selected at random. The player sees the visible side of all cards and wants to select the card with the maximum hidden value. To this end, the player flips the first card, sees its hidden value and decides whether to pick it or drop it and continue with the next card.We study algorithms for two natural objectives. In the first one, similar to the secretary problem, the player wants to maximize the probability of selecting the maximum hidden value. We show that this can be done with probability at least 0.45292. In the second objective, similar to the prophet inequality, the player wants to maximize the expectation of the selected hidden value. Here we show a guarantee of at least 0.63518 with respect to the expected maximum hidden value.Our algorithms result from combining three basic strategies. One is to stop whenever we see a value larger than the initial n visible numbers. The second one is to stop the first time the last flipped card's value is the largest of the currently n visible numbers in the table. And the third one is similar to the latter but to stop it additionally requires that the last flipped value is larger than the value on the other side of its card.We apply our results to the prophet secretary problem with unknown distributions, but with access to a single sample from each distribution. In particular, our guarantee improves upon 1 − 1/e for this problem, which is the currently best known guarantee and only works for the i.i.d. prophet inequality with samples.

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“…We focus on reviewing papers that either study the prophet secretary problem or derive prophet inequalities based on the number of units k, and summarize the results most related to ours in Table 1. For a more expansive survey of prophet inequalities, we refer to Correa et al (2018).…”

Section: Literature Review and Comparison Of Techniquesmentioning

confidence: 99%

Tight Guarantees for Static Threshold Policies in the Prophet Secretary Problem

Arnosti¹,

Ma²

2021

Preprint

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In the prophet secretary problem, n values are drawn independently from known distributions, and presented in random order. A decision-maker must accept or reject each value when it is presented, and may accept at most k values in total. The objective is to maximize the expected sum of accepted values.We study the performance of static threshold policies, which accept the first k values exceeding a fixed threshold (or all such values, if fewer than k exist). We show that using an optimal threshold guarantees afraction of the offline optimal solution, and provide an example demonstrating that this guarantee is tight. We also provide simple algorithms that achieve this guarantee. The first sets a threshold such that the expected number of values exceeding the threshold is equal to k, and offers a guarantee of γ k if k ≥ 5. The second sets a threshold so that k • γ k values are accepted in expectation, and offers a guarantee of γ k for all k. To establish these guarantees, we prove a result about sums of independent Bernoulli random variables, which extends a classical result of Hoeffding (1956) and is of general interest.Finally, we note that our algorithms can be implemented in settings with restricted information about agents' values. This makes them practical in settings such as the allocation of COVID-19 vaccines.

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Recent developments in prophet inequalities

Cited by 45 publications

References 24 publications

Competitive Analysis with a Sample and the Secretary Problem

Competitive Analysis with a Sample and the Secretary Problem

The Two-Sided Game of Googol and Sample-Based Prophet Inequalities

Tight Guarantees for Static Threshold Policies in the Prophet Secretary Problem

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