Tight Guarantees for Multi-unit Prophet Inequalities and Online Stochastic Knapsack

Jiashuo, Jiang,; Ma, Will; Zhang, Jiawei

doi:10.1137/1.9781611977073.51

Cited by 14 publications

(19 citation statements)

References 15 publications

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“…In a seminal work, Alaei (2014) obtains a 1 − 1 √ k+3 -guarantee when there are k slots in total. Jiang et al (2022) improves the guarantee for every k > 1 and shows that their guarantee is tight with respect to the Ex-Ante benchmark. Our work further improves the existing results by formulating an LP to compute the ratio of a policy for every given value distribution densities.…”

Section: Related Work and Comparisons Of Resultsmentioning

confidence: 89%

“…The goal is to achieve the maximum θ. It has been shown in Jiang et al (2022) that the optimal guarantees for the k-unit Magician/OCRS problems are actually the same, leading to tight Ex-Ante prophet inequalities.…”

Section: Dp(i) Proph(i)mentioning

confidence: 99%

“…The benefits of the Ex-Ante benchmark are twofold: (i) it is generally easier to analyze (Alaei 2014), and possesses special structures such that the optimal policy and the tight guarantees can be derived even when k > 1 (Jiang et al 2022); (ii) it enables a decomposition technique where policies and guarantees for a single resource can easily extend to online allocations problems with multiple resources (i.e. multiple types of slots) (Alaei et al 2012), even allowing for very general allocation procedures involving customer choice (Gallego et al 2015).…”

Section: Introductionmentioning

confidence: 99%

“…Various online policies have been developed in the literature in order to achieve a good performance guarantee relative to the benchmarks. Some of the tight performance guarantees are derived from optimal dynamic policies, where the decision depends both on the realization of the current agent's value and the number of currently remaining slots; see, e.g., Correa et al (2017) and Jiang et al (2022). Simpler but more restrictive static threshold policies also enjoy strong performance guarantees (Hajiaghayi et al 2007), where the decision is given by comparing the realized value of the agent against a threshold and the threshold is fixed for each agent, regardless of the remaining number of empty slots.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Tightness without Counterexamples: A New Approach and New Results for Prophet Inequalities

Jiashuo¹,

Ma²,

Zhang³

2022

Preprint

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Prophet inequalities consist of many beautiful statements that establish tight performance ratios between online and offline allocation algorithms. Typically, tightness is established by constructing an algorithmic guarantee and a worst-case instance separately, whose bounds match as a result of some "ingenuity". In this paper, we instead formulate the construction of the worst-case instance as an optimization problem, which directly finds the tight ratio without needing to construct two bounds separately.Our analysis of this complex optimization problem involves identifying structure in a new "Type Coverage" dual problem. It can be seen as akin to the celebrated Magician and OCRS (Online Contention Resolution Scheme) problems, except more general in that it can also provide tight ratios relative to the optimal offline allocation, whereas the earlier problems only establish tight ratios relative to the ex-ante relaxation of the offline problem.Through this analysis, our paper provides a unified framework that derives new results and recovers many existing ones. First, we show that the "oblivious" method of setting a static threshold due to Chawla et al.(2020) is, surprisingly, best-possible among all static threshold algorithms, for any number k of starting units.We emphasize that this result is derived without needing to explicitly find any counterexample instances. We establish similar "no separation" results for static thresholds in the IID setting, which although previously known, required the construction of complicated counterexamples. Finally, our framework and in particular our Type Coverage problem yields a simplified derivation of the tight 0.745 ratio when k = 1 in the IID setting.

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Section: Related Work and Comparisons Of Resultsmentioning

confidence: 89%

Section: Dp(i) Proph(i)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Tightness without Counterexamples: A New Approach and New Results for Prophet Inequalities

Jiashuo¹,

Ma²,

Zhang³

2022

Preprint

Self Cite

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show abstract

“…Also, establishing connections to the price of anarchy Dutting et al [2020] and online contention resolution schemes Feldman et al [2016] have been of particular interest in this literature. Generalizations of the simple prophet inequality problem to combinatorial settings have also been studied, where the examples are matroids Krengel and Sucheston [1978], Ehsani et al [2018] and knapsack Feldman et al [2016], Jiang et al [2022]. For a survey of recent developments in prophet inequality see Correa et al [2019] and for connections to mechanism design see Lucier [2017].…”

Section: Prophet Inequalitymentioning

confidence: 99%

Descending Price Auctions with Bounded Number of Price Levels and Batched Prophet Inequality

Alaei¹,

Makhdoumi²,

Malekian³

et al. 2022

Preprint

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We consider descending price auctions for selling m units of a good to unit demand i.i.d. buyers where there is an exogenous bound of k on the number of price levels the auction clock can take. The auctioneer's problem is to choose price levels p 1 > p 2 > • • • > p k for the auction clock such that auction expected revenue is maximized. The prices levels are announced prior to the auction. We reduce this problem to a new variant of prophet inequality, which we call batched prophet inequality, where a decision-maker chooses k (decreasing) thresholds and then sequentially collects rewards (up to m) that are above the thresholds with ties broken uniformly at random. For the special case of m = 1 (i.e., selling a single item), we show that the resulting descending auction with k price levels achieves 1 − 1/e k of the unrestricted (without the bound of k) optimal revenue. That means a descending auction with just 4 price levels can achieve more than 98% of the optimal revenue. We then extend our results for m > 1 and provide a closed-form bound on the competitive ratio of our auction as a function of the number of units m and the number of price levels k.

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Prophet Inequalities via the Expected Competitive Ratio

Ezra,

Leonardi,

Reiffenhäuser

et al. 2023

Lecture Notes in Computer Science

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Tight Guarantees for Multi-unit Prophet Inequalities and Online Stochastic Knapsack

Cited by 14 publications

References 15 publications

Tightness without Counterexamples: A New Approach and New Results for Prophet Inequalities

Tightness without Counterexamples: A New Approach and New Results for Prophet Inequalities

Descending Price Auctions with Bounded Number of Price Levels and Batched Prophet Inequality

Prophet Inequalities via the Expected Competitive Ratio

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