Tight tail probability bounds for distribution-free decision making

Roos, Ernst; Brekelmans, Ruud; Eekelen, Wouter van; Hertog, Dick den; Leeuwaarden, Johan van

doi:10.1016/j.ejor.2021.12.010

Cited by 5 publications

(5 citation statements)

References 26 publications

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“…The absolute revenue maximization problem is studied by Carrasco et al [11]. For further directions in distributionally robust mechanism design (some of which considering mean-dispersion information), see, e.g., [8,2,30,12,25,48,50,39,1,3] and references therein.…”

Section: Related Work and Further Discussionmentioning

confidence: 99%

“…For P(µ, σ 2 , L) this rederives a result in [14], and for P(µ, d, L) this gives a similar result but then for MAD instead of variance. Related results for mean-MAD ambiguity in the context of Chebyshev-like inequalities can be found [39].…”

Section: Primal-dual Methods By Examplementioning

confidence: 98%

“…In order to prove Theorem 1.4, we rely on the following lemma, which can be seen as a mean-MAD-support analogue of the result of Cox [14]. A related result for mean-MAD ambiguity in the context of Chebyshev-like inequalities is given in [39]. The latter condition is always satisfied in order to ensure that P(µ, d, L) is nonempty.…”

Section: Proof Of Theorem 14mentioning

confidence: 99%

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Optimal Stopping Theory for a Distributionally Robust Seller

Kleer¹,

Leeuwaarden²

2022

Preprint

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Sellers in online markets face the challenge of determining the right time to sell in view of uncertain future offers. Classical stopping theory assumes that sellers have full knowledge of the value distributions, and leverage this knowledge to determine stopping rules that maximize expected welfare. In practice, however, stopping rules must often be determined under partial information, based on scarce data or expert predictions. Consider a seller that has one item for sale and receives successive offers drawn from some value distributions. The decision on whether or not to accept an offer is irrevocable, and the value distributions are only partially known. We therefore let the seller adopt a robust maximin strategy, assuming that value distributions are chosen adversarially by nature to minimize the value of the accepted offer. We provide a general maximin solution to this stopping problem that identifies the optimal (threshold-based) stopping rule for the seller for all possible statistical information structures. We then perform a detailed analysis for various ambiguity sets relying on knowledge about the common mean, dispersion (variance or mean absolute deviation) and support of the distributions. We show for these information structures that the seller's stopping rule consists of decreasing thresholds converging to the common mean, and that nature's adversarial response, in the long run, is to always create an all-or-nothing scenario. The maximin solutions also reveal what happens as dispersion or the number of offers grows large.

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Section: Related Work and Further Discussionmentioning

confidence: 99%

Section: Primal-dual Methods By Examplementioning

confidence: 98%

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Optimal Stopping Theory for a Distributionally Robust Seller

Kleer¹,

Leeuwaarden²

2022

Preprint

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“…Subsequently, He et al (2010) extend this result to the problem with first-, second-, and fourth-order moments. Recently, Roos et al (2021) provide alternative tight lower and upper bounds on the tail probability under a bounded support, given the mean and mean absolute deviation of the random variable.…”

Section: Related Literaturementioning

confidence: 99%

“…Following this approach, a large volume of literature has discovered more closed-form decisions for some variants of Scarf's model by considering the asymmetry of demands (Natarajan et al 2018), heavy-tailed distributed demands (Das et al 2021), risk-averse objectives (Han et al 2014), etc. The study of closed-form solutions is also prevalent in probability theory (Bertsimas and Popescu 2005, He et al 2010, Roos et al 2021 and portfolio selection (Ghaoui et al 2003, Zuluaga et al 2009, Chen et al 2011, Li 2018. However, the techniques that have been used in the literature to develop these closed forms are mathematically different and are only tailored for specific problems.…”

Section: Introductionmentioning

confidence: 99%

A Unified Framework for Generalized Moment Problems: a Novel Primal-Dual Approach

Guo¹,

He²,

Jiang³

et al. 2022

Preprint

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Generalized moment problems optimize functional expectation over a class of distributions with generalized moment constraints, i.e., the function in the moment can be any measurable function. These problems have recently attracted growing interest due to their great flexibility in representing nonstandard moment constraints, such as geometry-mean constraints, entropy constraints, and exponential-type moment constraints. Despite the increasing research interest, analytical solutions are mostly missing for these problems, and researchers have to settle for nontight bounds or numerical approaches that are either suboptimal or only applicable to some special cases. In addition, the techniques used to develop closed-form solutions to the standard moment problems are tailored for specific problem structures. In this paper, we propose a framework that provides a unified treatment for any moment problem. The key ingredient of the framework is a novel primal-dual optimality condition. This optimality condition enables us to reduce the original infinite dimensional problem to a nonlinear equation system with a finite number of variables. In solving three specific moment problems, the framework demonstrates a clear path for identifying the analytical solution if one is available, otherwise, it produces semi-analytical solutions that lead to efficient numerical algorithms. Finally, through numerical experiments, we provide further evidence regarding the performance of the resulting algorithms by solving a moment problem and a distributionally robust newsvendor problem.

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Chance-Constrained Generic Energy Storage Operations Under Decision-Dependent Uncertainty

Almassalkhi

Cheng

et al. 2023

IEEE Trans. Sustain. Energy

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Tight tail probability bounds for distribution-free decision making

Cited by 5 publications

References 26 publications

Optimal Stopping Theory for a Distributionally Robust Seller

Optimal Stopping Theory for a Distributionally Robust Seller

A Unified Framework for Generalized Moment Problems: a Novel Primal-Dual Approach

Chance-Constrained Generic Energy Storage Operations Under Decision-Dependent Uncertainty

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