A survey of prophet inequalities in optimal stopping theory

Hill, Theodore P.; Kertz, Robert P.

doi:10.1090/conm/125/1160620

Cited by 77 publications

(34 citation statements)

References 22 publications

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“…There is an extensive literature on prophet inequalities, part of which is surveyed in Hill and Kertz (1992). The more recent survey of Hartline (2012) studies specifically prophet inequalities obtained by simple algorithms like ours.…”

Section: Related Workmentioning

confidence: 99%

Matroid prophet inequalities and applications to multi-dimensional mechanism design

Kleinberg

Weinberg

2019

Games and Economic Behavior

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Section: Related Workmentioning

confidence: 99%

Matroid prophet inequalities and applications to multi-dimensional mechanism design

Kleinberg

Weinberg

2019

Games and Economic Behavior

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“…1 Note that some notion of independence assumption is necessary as even for the single choice problem, if values are arbitrarily correlated then every online algorithm is Ω(n) competitive [35]. 2 In fact, for general downward-closed constraint, even with additive valuations one should not expect efficient algorithms with membership queries [45].…”

Section: Theorem 13 (Monotone Subadditivementioning

confidence: 99%

Combinatorial Prophet Inequalities

Rubinstein¹,

Singla²

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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We introduce a novel framework of Prophet Inequalities for combinatorial valuation functions. For a (non-monotone) submodular objective function over an arbitrary matroid feasibility constraint, we give an O(1)-competitive algorithm. For a monotone subadditive objective function over an arbitrary downwardclosed feasibility constraint, we give an O(log n log 2 r)-competitive algorithm (where r is the cardinality of the largest feasible subset).Inspired by the proof of our subadditive prophet inequality, we also obtain an O(log n · log 2 r)-competitive algorithm for the Secretary Problem with a monotone subadditive objective function subject to an arbitrary downward-closed feasibility constraint. Even for the special case of a cardinality feasibility constraint, our algorithm circumvents an Ω( √ n) lower bound by Bateni, Hajiaghayi, and Zadimoghaddam [10] in a restricted query model.En route to our submodular prophet inequality, we prove a technical result of independent interest: we show a variant of the Correlation Gap Lemma [14, 1] for nonmonotone submodular functions.

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“…In a classical prophet problem, the expected return of a prophet, who has complete foresight into the future, is compared with paper by Hill and Kertz [9]. A more recent overview is the book by Harten et al [6], along with its revised and translated version by Schmitz [15].…”

Section: Introductionmentioning

confidence: 99%

An application of prophet regions to optimal stopping with a random number of observations

Allaart

2004

Optimization

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Let X 1 , X 2 , . . . be any sequence of nonnegative integrable random variables, and let N ∈ {1, 2, . . . } be a random variable with known distribution, independent of X 1 , X 2 , . . . . The optimal stopping value sup t E(X t I(N ≥ t)) is considered for two players: one who has advance knowledge of the value of N , and another who does not. Sharp ratio and difference inequalities relating the two players' optimal values are given in a number of settings. The key to the proofs is an application of a prophet region for arbitrarily dependent random variables by Hill and Kertz (Trans. Amer. Math. Soc. 278, 197-207 (1983)).AMS 2000 subject classification: 60G40, 62L15.

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A survey of prophet inequalities in optimal stopping theory

Cited by 77 publications

References 22 publications

Matroid prophet inequalities and applications to multi-dimensional mechanism design

Matroid prophet inequalities and applications to multi-dimensional mechanism design

Combinatorial Prophet Inequalities

An application of prophet regions to optimal stopping with a random number of observations

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