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Prophet Inequalities Made Easy: Stochastic Optimization by Pricing Nonstochastic Inputs

Authors

Paul Dütting, Michal Feldman, Thomas Keßelheim, Brendan Lucier
Type: Article
Publication Date: 2020-01-01
Citations: 58
DOI: https://doi.org/10.1137/20m1323850

Abstract

We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold algorithms for settings beyond binary selection. Our analysis takes the form of an extension theorem: we derive sufficient conditions on prices when all weights are known in advance, then prove that the resulting approximation guarantees extend directly to stochastic settings. Our framework unifies and simplifies much of the existing literature on prophet inequalities and posted price mechanisms and is used to derive new and improved results for combinatorial markets (with and without complements), multidimensional matroids, and sparse packing problems. Finally, we highlight a surprising connection between the smoothness framework for bounding the price of anarchy of mechanisms and our framework, and show that many smooth mechanisms can be recast as posted price mechanisms with comparable performance guarantees.

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  • SIAM Journal on Computing
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  • arXiv (Cornell University)
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  • London School of Economics and Political Science Research Online (London School of Economics and Political Science)
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Prophet Inequalities Made Easy: Stochastic Optimization by Pricing Non-Stochastic Inputs

2016-12-09
Paul Dütting , Michal Feldman , Thomas Keßelheim +1 more
We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold … We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold algorithms for settings beyond binary selection. Our analysis takes the form of an extension theorem: we derive sufficient conditions on prices when all weights are known in advance, then prove that the resulting approximation guarantees extend directly to stochastic settings. Our framework unifies and simplifies much of the existing literature on prophet inequalities and posted price mechanisms, and is used to derive new and improved results for combinatorial markets (with and without complements), multi-dimensional matroids, and sparse packing problems. Finally, we highlight a surprising connection between the smoothness framework for bounding the price of anarchy of mechanisms and our framework, and show that many smooth mechanisms can be recast as posted price mechanisms with comparable performance guarantees.

Prophet Inequalities Made Easy: Stochastic Optimization by Pricing Non-Stochastic Inputs

2017-10-01
Paul Dütting , Michal Feldman , Thomas Keßelheim +1 more
We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold … We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold algorithms for settings beyond binary selection. Our analysis takes the form of an extension theorem: we derive sufficient conditions on prices when all weights are known in advance, then prove that the resulting approximation guarantees extend directly to stochastic settings. Our framework unifies and simplifies much of the existing literature on prophet inequalities and posted price mechanisms, and is used to derive new and improved results for combinatorial markets (with and without complements), multi-dimensional matroids, and sparse packing problems. Finally, we highlight a surprising connection between the smoothness framework for bounding the price of anarchy of mechanisms and our framework, and show that many smooth mechanisms can be recast as posted price mechanisms with comparable performance guarantees.
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Prophet Inequalities Made Easy: Stochastic Optimization by Pricing Non-Stochastic Inputs

2016-01-01
Paul Dütting , Michal Feldman , Thomas Keßelheim +1 more
We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold … We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold algorithms for settings beyond binary selection. Our analysis takes the form of an extension theorem: we derive sufficient conditions on prices when all weights are known in advance, then prove that the resulting approximation guarantees extend directly to stochastic settings. Our framework unifies and simplifies much of the existing literature on prophet inequalities and posted price mechanisms, and is used to derive new and improved results for combinatorial markets (with and without complements), multi-dimensional matroids, and sparse packing problems. Finally, we highlight a surprising connection between the smoothness framework for bounding the price of anarchy of mechanisms and our framework, and show that many smooth mechanisms can be recast as posted price mechanisms with comparable performance guarantees.
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The IID Prophet Inequality with Limited Flexibility

2022-01-01
Sebastian Perez-Salazar , Mohit Singh , Alejandro Toriello
In online sales, sellers usually offer each potential buyer a posted price in a take-it-or-leave fashion. Buyers can sometimes see posted prices faced by other buyers, and changing the price … In online sales, sellers usually offer each potential buyer a posted price in a take-it-or-leave fashion. Buyers can sometimes see posted prices faced by other buyers, and changing the price frequently could be considered unfair. The literature on posted price mechanisms and prophet inequality problems has studied the two extremes of pricing policies, the fixed price policy and fully dynamic pricing. The former is suboptimal in revenue but is perceived as fairer than the latter. This work examines the middle situation, where there are at most $k$ distinct prices over the selling horizon. Using the framework of prophet inequalities with independent and identically distributed random variables, we propose a new prophet inequality for strategies that use at most $k$ thresholds. We present asymptotic results in $k$ and results for small values of $k$. For $k=2$ prices, we show an improvement of at least $11\%$ over the best fixed-price solution. Moreover, $k=5$ prices suffice to guarantee almost $99\%$ of the approximation factor obtained by a fully dynamic policy that uses an arbitrary number of prices. From a technical standpoint, we use an infinite-dimensional linear program in our analysis; this formulation could be of independent interest to other online selection problems.

An $O(\log \log m)$ Prophet Inequality for Subadditive Combinatorial Auctions

2022-05-19
Paul Dütting , Thomas Keßelheim , Brendan Lucier
Prophet inequalities compare the expected performance of an online algorithm for a stochastic optimization problem to the expected optimal solution in hindsight. They are a major alternative to classic worst-case … Prophet inequalities compare the expected performance of an online algorithm for a stochastic optimization problem to the expected optimal solution in hindsight. They are a major alternative to classic worst-case competitive analysis, of particular importance in the design and analysis of simple (posted-price) incentive compatible mechanisms with provable approximation guarantees. A central open problem in this area concerns subadditive combinatorial auctions. Here $n$ agents with subadditive valuation functions compete for the assignment of $m$ items. The goal is to find an allocation of the items that maximizes the total value of the assignment. The question is whether there exists a prophet inequality for this problem that significantly beats the best known approximation factor of $O(\log m)$. We make major progress on this question by providing an $O(\log \log m)$ prophet inequality. Our proof goes through a novel primal-dual approach. It is also constructive, resulting in an online policy that takes the form of static and anonymous item prices that can be computed in polynomial time given appropriate query access to the valuations. As an application of our approach, we construct a simple and incentive compatible mechanism based on posted prices that achieves an $O(\log \log m)$ approximation to the optimal revenue for subadditive valuations under an item-independence assumption.
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An O(log log m) Prophet Inequality for Subadditive Combinatorial Auctions

2020-01-01
Paul Dütting , Thomas Keßelheim , Brendan Lucier
Prophet inequalities compare the expected performance of an online algorithm for a stochastic optimization problem to the expected optimal solution in hindsight. They are a major alternative to classic worst-case … Prophet inequalities compare the expected performance of an online algorithm for a stochastic optimization problem to the expected optimal solution in hindsight. They are a major alternative to classic worst-case competitive analysis, of particular importance in the design and analysis of simple (posted-price) incentive compatible mechanisms with provable approximation guarantees. A central open problem in this area concerns subadditive combinatorial auctions. Here $n$ agents with subadditive valuation functions compete for the assignment of $m$ items. The goal is to find an allocation of the items that maximizes the total value of the assignment. The question is whether there exists a prophet inequality for this problem that significantly beats the best known approximation factor of $O(\log m)$. We make major progress on this question by providing an $O(\log \log m)$ prophet inequality. Our proof goes through a novel primal-dual approach. It is also constructive, resulting in an online policy that takes the form of static and anonymous item prices that can be computed in polynomial time given appropriate query access to the valuations. As an application of our approach, we construct a simple and incentive compatible mechanism based on posted prices that achieves an $O(\log \log m)$ approximation to the optimal revenue for subadditive valuations under an item-independence assumption.
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Prophet Inequalities with Limited Information

2013-12-18
Pablo Azar , Robert Kleinberg , S. Matthew Weinberg
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2014 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Prophet Inequalities with Limited InformationPablo D. Azar, Robert Kleinberg, and S. Matthew WeinbergPablo D. … Previous chapter Next chapter Full AccessProceedings Proceedings of the 2014 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Prophet Inequalities with Limited InformationPablo D. Azar, Robert Kleinberg, and S. Matthew WeinbergPablo D. Azar, Robert Kleinberg, and S. Matthew Weinbergpp.1358 - 1377Chapter DOI:https://doi.org/10.1137/1.9781611973402.100PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract In the classical prophet inequality, a gambler observes a sequence of stochastic rewards V1, …, Vn and must decide, for each reward Vi, whether to keep it and stop the game or to forfeit the reward forever and reveal the next value Vi. The gambler's goal is to obtain a constant fraction of the expected reward that the optimal offline algorithm would get. Recently, prophet inequalities have been generalized to settings where the gambler can choose k items, and, more generally, where he can choose any independent set in a matroid. However, all the existing algorithms require the gambler to know the distribution from which the rewards V1, …, Vn are drawn. The assumption that the gambler knows the distribution from which V1, …, Vn are drawn is very strong. Instead, we work with the much simpler assumption that the gambler only knows a few samples from this distribution. We construct the first single-sample prophet inequalities for many settings of interest, whose guarantees all match the best possible asymptotically, even with full knowledge of the distribution. Specifically, we provide a novel single-sample algorithm when the gambler can choose any k elements whose analysis is based on random walks with limited correlation. In addition, we provide a black-box method for converting specific types of solutions to the related secretary problem to single-sample prophet inequalities, and apply it to several existing algorithms. Finally, we provide a constant-sample prophet inequality for constant-degree bipartite matchings. In addition, we apply these results to design the first posted-price and multi-dimensional auction mechanisms with limited information in settings with asymmetric bidders. Connections between prophet inequalities and posted-price mechanisms are already known, but applying the existing framework requires knowledge of the underlying distributions, as well as the so-called "virtual values" even when the underlying prophet inequalities do not. We therefore provide an extension of this framework that bypasses virtual values altogether, allowing our mechanisms to take full advantage of the limited information required by our new prophet inequalities. Previous chapter Next chapter RelatedDetails Published:2014ISBN:978-1-61197-338-9eISBN:978-1-61197-340-2 https://doi.org/10.1137/1.9781611973402Book Series Name:ProceedingsBook Code:PRDA14Book Pages:viii + 1885
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Prophet Inequalities for Cost Minimization

2022-01-01
Vasilis Livanos , Ruta Mehta
Prophet inequalities for rewards maximization are fundamental to optimal stopping theory with extensive applications to mechanism design and online optimization. We study the \emph{cost minimization} counterpart of the classical prophet … Prophet inequalities for rewards maximization are fundamental to optimal stopping theory with extensive applications to mechanism design and online optimization. We study the \emph{cost minimization} counterpart of the classical prophet inequality: a decision maker is facing a sequence of costs $X_1, X_2, \dots, X_n$ drawn from known distributions in an online manner and \emph{must} ``stop'' at some point and take the last cost seen. The goal is to compete with a ``prophet'' who can see the realizations of all $X_i$'s upfront and always select the minimum, obtaining a cost of $\mathbb{E}[\min_i X_i]$. If the $X_i$'s are not identically distributed, no strategy can achieve a bounded approximation, even for random arrival order and $n = 2$. This leads us to consider the case where the $X_i$'s are independent and identically distributed (I.I.D.). For the I.I.D. case, we show that if the distribution satisfies a mild condition, the optimal stopping strategy achieves a (distribution-dependent) constant-factor approximation to the prophet's cost. Moreover, for MHR distributions, this constant is at most $2$. All our results are tight. We also demonstrate an example distribution that does not satisfy the condition and for which the competitive ratio of any algorithm is infinite. Turning our attention to single-threshold strategies, we design a threshold that achieves a $O\left(polylog{n}\right)$-factor approximation, where the exponent in the logarithmic factor is a distribution-dependent constant, and we show a matching lower bound. Finally, we note that our results can be used to design approximately optimal posted price-style mechanisms for procurement auctions which may be of independent interest. Our techniques utilize the \emph{hazard rate} of the distribution in a novel way, allowing for a fine-grained analysis which could find further applications in prophet inequalities.

An O(log log m) Prophet Inequality for Subadditive Combinatorial Auctions

2020-11-01
Paul Dütting , Thomas Keßelheim , Brendan Lucier
Prophet inequalities compare the expected performance of an online algorithm for a stochastic optimization problem to the expected optimal solution in hindsight. They are a major alternative to classic worst-case … Prophet inequalities compare the expected performance of an online algorithm for a stochastic optimization problem to the expected optimal solution in hindsight. They are a major alternative to classic worst-case competitive analysis, of particular importance in the design and analysis of simple (posted-price) incentive compatible mechanisms with provable approximation guarantees. A central open problem in this area concerns subadditive combinatorial auctions. Here n agents with subadditive valuation functions compete for the assignment of m items. The goal is to find an allocation of the items that maximizes the total value of the assignment. The question is whether there exists a prophet inequality for this problem that significantly beats the best known approximation factor of O(log m). We make major progress on this question by providing an O(log log m) prophet inequality. Our proof goes through a novel primal-dual approach. It is also constructive, resulting in an online policy that takes the form of static and anonymous item prices that can be computed in polynomial time given appropriate query access to the valuations. As an application of our approach, we construct a simple and incentive compatible mechanism based on posted prices that achieves an O(log log m) approximation to the optimal revenue for subadditive valuations under an item-independence assumption.
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The Competition Complexity of Prophet Inequalities

2024-02-16
Johannes Brustle , José R. Correa , Paul Dütting +3 more
We study the classic single-choice prophet inequality problem through a resource augmentation lens. Our goal is to bound the $(1-\varepsilon)$-competition complexity of different types of online algorithms. This metric asks … We study the classic single-choice prophet inequality problem through a resource augmentation lens. Our goal is to bound the $(1-\varepsilon)$-competition complexity of different types of online algorithms. This metric asks for the smallest $k$ such that the expected value of the online algorithm on $k$ copies of the original instance, is at least a $(1-\varepsilon)$-approximation to the expected offline optimum on a single copy. We show that block threshold algorithms, which set one threshold per copy, are optimal and give a tight bound of $k = \Theta(\log \log 1/\varepsilon)$. This shows that block threshold algorithms approach the offline optimum doubly-exponentially fast. For single threshold algorithms, we give a tight bound of $k = \Theta(\log 1/\varepsilon)$, establishing an exponential gap between block threshold algorithms and single threshold algorithms. Our model and results pave the way for exploring resource-augmented prophet inequalities in combinatorial settings. In line with this, we present preliminary findings for bipartite matching with one-sided vertex arrivals, as well as in XOS combinatorial auctions. Our results have a natural competition complexity interpretation in mechanism design and pricing applications.
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Prophet Secretary for Combinatorial Auctions and Matroids

2017-01-01
Soheil Ehsani , MohammadTaghi Hajiaghayi , Thomas Keßelheim +1 more
The secretary and the prophet inequality problems are central to the field of Stopping Theory. Recently, there has been a lot of work in generalizing these models to multiple items … The secretary and the prophet inequality problems are central to the field of Stopping Theory. Recently, there has been a lot of work in generalizing these models to multiple items because of their applications in mechanism design. The most important of these generalizations are to matroids and to combinatorial auctions (extends bipartite matching). Kleinberg-Weinberg \cite{KW-STOC12} and Feldman et al. \cite{feldman2015combinatorial} show that for adversarial arrival order of random variables the optimal prophet inequalities give a $1/2$-approximation. For many settings, however, it's conceivable that the arrival order is chosen uniformly at random, akin to the secretary problem. For such a random arrival model, we improve upon the $1/2$-approximation and obtain $(1-1/e)$-approximation prophet inequalities for both matroids and combinatorial auctions. This also gives improvements to the results of Yan \cite{yan2011mechanism} and Esfandiari et al. \cite{esfandiari2015prophet} who worked in the special cases where we can fully control the arrival order or when there is only a single item. Our techniques are threshold based. We convert our discrete problem into a continuous setting and then give a generic template on how to dynamically adjust these thresholds to lower bound the expected total welfare.

Prophet Secretary for Combinatorial Auctions and Matroids

2024-11-12
Soheil Ehsani , MohammadTaghi Hajiaghayi , Thomas Keßelheim +1 more
The secretary and the prophet inequality problems are central to the field of Stopping Theory. Recently, there has been a lot of work in generalizing these models to multiple items … The secretary and the prophet inequality problems are central to the field of Stopping Theory. Recently, there has been a lot of work in generalizing these models to multiple items because of their applications in mechanism design. The most important of these generalizations are to matroids and to combinatorial auctions (extends bipartite matching). Kleinberg-Weinberg \cite{KW-STOC12} and Feldman et al. \cite{feldman2015combinatorial} show that for adversarial arrival order of random variables the optimal prophet inequalities give a $1/2$-approximation. For many settings, however, it's conceivable that the arrival order is chosen uniformly at random, akin to the secretary problem. For such a random arrival model, we improve upon the $1/2$-approximation and obtain $(1-1/e)$-approximation prophet inequalities for both matroids and combinatorial auctions. This also gives improvements to the results of Yan \cite{yan2011mechanism} and Esfandiari et al. \cite{esfandiari2015prophet} who worked in the special cases where we can fully control the arrival order or when there is only a single item. Our techniques are threshold based. We convert our discrete problem into a continuous setting and then give a generic template on how to dynamically adjust these thresholds to lower bound the expected total welfare.
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Single-Sample Prophet Inequalities via Greedy-Ordered Selection

2021-01-01
Constantine Caramanis , Paul Dütting , Matthew Faw +6 more
We study single-sample prophet inequalities (SSPIs), i.e., prophet inequalities where only a single sample from each prior distribution is available. Besides a direct, and optimal, SSPI for the basic single … We study single-sample prophet inequalities (SSPIs), i.e., prophet inequalities where only a single sample from each prior distribution is available. Besides a direct, and optimal, SSPI for the basic single choice problem [Rubinstein et al., 2020], most existing SSPI results were obtained via an elegant, but inherently lossy, reduction to order-oblivious secretary (OOS) policies [Azar et al., 2014]. Motivated by this discrepancy, we develop an intuitive and versatile greedy-based technique that yields SSPIs directly rather than through the reduction to OOSs. Our results can be seen as generalizing and unifying a number of existing results in the area of prophet and secretary problems. Our algorithms significantly improve on the competitive guarantees for a number of interesting scenarios (including general matching with edge arrivals, bipartite matching with vertex arrivals, and certain matroids), and capture new settings (such as budget additive combinatorial auctions). Complementing our algorithmic results, we also consider mechanism design variants. Finally, we analyze the power and limitations of different SSPI approaches by providing a partial converse to the reduction from SSPI to OOS given by Azar et al.

Prophet Inequalities via the Expected Competitive Ratio

2022-01-01
Tomer Ezra , Stefano Leonardi , Rebecca Reiffenhäuser +2 more
We consider prophet inequalities under downward-closed constraints. In this problem, a decision-maker makes immediate and irrevocable choices on arriving elements, subject to constraints. Traditionally, performance is compared to the expected … We consider prophet inequalities under downward-closed constraints. In this problem, a decision-maker makes immediate and irrevocable choices on arriving elements, subject to constraints. Traditionally, performance is compared to the expected offline optimum, called the \textit{Ratio of Expectations} (RoE). However, RoE has limitations as it only guarantees the average performance compared to the optimum, and might perform poorly against the realized ex-post optimal value. We study an alternative performance measure, the \textit{Expected Ratio} (EoR), namely the expectation of the ratio between algorithm's and prophet's value. EoR offers robust guarantees, e.g., a constant EoR implies achieving a constant fraction of the offline optimum with constant probability. For the special case of single-choice problems the EoR coincides with the well-studied notion of probability of selecting the maximum. However, the EoR naturally generalizes the probability of selecting the maximum for combinatorial constraints, which are the main focus of this paper. Specifically, we establish two reductions: for every constraint, RoE and the EoR are at most a constant factor apart. Additionally, we show that the EoR is a stronger benchmark than the RoE in that, for every instance (constraint and distribution), the RoE is at least a constant fraction of the EoR, but not vice versa. Both these reductions imply a wealth of EoR results in multiple settings where RoE results are known.
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Revenue-Optimal Deterministic Auctions for Multiple Buyers with Ordinal Preferences over Fixed-Price Items

2022-06-30
Will Ma
In this article, we introduce a Bayesian revenue-maximizing mechanism design model where the items have fixed, exogenously given prices. Buyers are unit-demand and have an ordinal ranking over purchasing either … In this article, we introduce a Bayesian revenue-maximizing mechanism design model where the items have fixed, exogenously given prices. Buyers are unit-demand and have an ordinal ranking over purchasing either one of these items at its given price or purchasing nothing. This model arises naturally from the assortment optimization problem, in that the single-buyer optimization problem over deterministic mechanisms reduces to deciding on an assortment of items to “show.” We study its multi-buyer generalization in the simplest setting of single-winner auctions or, more broadly, any service-constrained environment. Our main result is that if the buyer rankings are drawn independently from Markov chain choice models, then the optimal mechanism is computationally tractable, and structurally a virtual welfare maximizer. We also show that for ranking distributions not induced by Markov chains, the optimal mechanism may not be a virtual welfare maximizer. Finally, we apply our virtual valuation notion for Markov chains, in conjunction with existing prophet inequalities, to improve algorithmic guarantees for online assortment problems.
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Prophet Inequalities with Limited Information

2013-01-01
Pablo Azar , Robert Kleinberg , S. Matthew Weinberg
In the classical prophet inequality, a gambler observes a sequence of stochastic rewards $V_1,...,V_n$ and must decide, for each reward $V_i$, whether to keep it and stop the game or … In the classical prophet inequality, a gambler observes a sequence of stochastic rewards $V_1,...,V_n$ and must decide, for each reward $V_i$, whether to keep it and stop the game or to forfeit the reward forever and reveal the next value $V_i$. The gambler's goal is to obtain a constant fraction of the expected reward that the optimal offline algorithm would get. Recently, prophet inequalities have been generalized to settings where the gambler can choose $k$ items, and, more generally, where he can choose any independent set in a matroid. However, all the existing algorithms require the gambler to know the distribution from which the rewards $V_1,...,V_n$ are drawn. The assumption that the gambler knows the distribution from which $V_1,...,V_n$ are drawn is very strong. Instead, we work with the much simpler assumption that the gambler only knows a few samples from this distribution. We construct the first single-sample prophet inequalities for many settings of interest, whose guarantees all match the best possible asymptotically, \emph{even with full knowledge of the distribution}. Specifically, we provide a novel single-sample algorithm when the gambler can choose any $k$ elements whose analysis is based on random walks with limited correlation. In addition, we provide a black-box method for converting specific types of solutions to the related \emph{secretary problem} to single-sample prophet inequalities, and apply it to several existing algorithms. Finally, we provide a constant-sample prophet inequality for constant-degree bipartite matchings. We apply these results to design the first posted-price and multi-dimensional auction mechanisms with limited information in settings with asymmetric bidders.
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Tight Guarantees for Multi-unit Prophet Inequalities and Online Stochastic Knapsack

2022-01-01
Jiashuo Jiang , Will Ma , Jiawei Zhang
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Tight Guarantees for Multi-unit Prophet Inequalities and Online Stochastic KnapsackJiashuo Jiang, Will Ma, and … Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Tight Guarantees for Multi-unit Prophet Inequalities and Online Stochastic KnapsackJiashuo Jiang, Will Ma, and Jiawei ZhangJiashuo Jiang, Will Ma, and Jiawei Zhangpp.1221 - 1246Chapter DOI:https://doi.org/10.1137/1.9781611977073.51PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract Prophet inequalities are a useful tool for designing online allocation procedures and comparing their performance to the optimal offline allocation. In the basic setting of k-unit prophet inequalities, the procedure of Alaei [2] with its celebrated performance guarantee of has found widespread adoption in mechanism design and general online allocation problems in online advertising, healthcare scheduling, and revenue management. Despite being commonly used for implementing a fractional allocation in an online fashion, the tightness of Alaei's procedure for a given k has remained unknown. In this paper we resolve this question, characterizing the tight bound by identifying the structure of the optimal online implementation, and consequently improving the best-known guarantee for k-unit prophet inequalities for all k > 1. We also consider the more general online stochastic knapsack problem where each individual allocation can consume an arbitrary fraction of the initial capacity. Here we introduce a new "best-fit" procedure for implementing a fractionally-feasible knapsack solution online, with a performance guarantee of ≈ 0.319, which we also show is tight with respect to the standard LP relaxation. This improves the previously best-known guarantee of 0.2 for online knapsack. Our analysis differs from existing ones by eschewing the need to split items into "large" or "small" based on capacity consumption, using instead an invariant for the overall utilization on different sample paths. All in all, our results imply tight (non-greedy) Online Contention Resolution Schemes for k-uniform matroids and the knapsack polytope, respectively, which has further implications. Previous chapter Next chapter RelatedDetails Published:2022eISBN:978-1-61197-707-3 https://doi.org/10.1137/1.9781611977073Book Series Name:ProceedingsBook Code:PRDA22Book Pages:xvii + 3771
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MNL-Prophet: Sequential Assortment Selection under Uncertainty

2023-01-01
Vineet Goyal , Salal Humair , Orestis Papadigenopoulos +1 more
Due to numerous applications in retail and (online) advertising the problem of assortment selection has been widely studied under many combinations of discrete choice models and feasibility constraints. In many … Due to numerous applications in retail and (online) advertising the problem of assortment selection has been widely studied under many combinations of discrete choice models and feasibility constraints. In many situations, however, an assortment of products has to be constructed gradually and without accurate knowledge of all possible alternatives; in such cases, existing offline approaches become inapplicable. We consider a stochastic variant of the assortment selection problem, where the parameters that determine the revenue and (relative) demand of each item are jointly drawn from some known item-specific distribution. The items are observed sequentially in an arbitrary and unknown order; upon observing the realized parameters of each item, the decision-maker decides irrevocably whether to include it in the constructed assortment, or forfeit it forever. The objective is to maximize the expected total revenue of the constructed assortment, relative to that of an offline algorithm which foresees all the parameter realizations and computes the optimal assortment. We provide simple threshold-based online policies for the unconstrained and cardinality-constrained versions of the problem under a natural class of substitutable choice models; as we show, our policies are (worst-case) optimal under the celebrated Multinomial Logit choice model. We extend our results to the case of knapsack constraints and discuss interesting connections to the Prophet Inequality problem, which is already subsumed by our setting.
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Prophet Inequality Matching Meets Probing with Commitment.

2021-02-08
Allan Borodin , Calum MacRury , Akash Rakheja
Within the context of stochastic probing with commitment, we consider the online stochastic matching problem for bipartite graphs where edges adjacent to an online node must be probed to determine … Within the context of stochastic probing with commitment, we consider the online stochastic matching problem for bipartite graphs where edges adjacent to an online node must be probed to determine if they exist, based on known edge probabilities. If a probed edge exists, it must be used in the matching (if possible). In addition to improving upon existing stochastic bipartite matching results, our results can also be seen as extensions to multi-item prophet inequalities. We study this matching problem for given constraints on the allowable sequences of probes adjacent to an online node. Our setting generalizes the patience (or time-out) constraint which limits the number of probes that can be made to edges. The generality of our setting leads to some modelling and computational efficiency issues that are not encountered in previous works. We establish new competitive bounds all of which generalize the standard non-stochastic setting when edges do not need to be probed (i.e., exist with certainty). Specifically, we establish the following competitive ratio results for a general formulation of edge constraints, arbitrary edge weights, and arbitrary edge probabilities: (1) A tight $\frac{1}{2}$ ratio when the stochastic graph is generated from a known stochastic type graph where the $\pi(i)^{th}$ online node is drawn independently from a known distribution ${\cal D}_{\pi(i)}$ and $\pi$ is chosen adversarially. We refer to this setting as the known i.d. stochastic matching problem with adversarial arrivals. (2) A $1-1/e$ ratio when the stochastic graph is generated from a known stochastic type graph where the $\pi(i)^{th}$ online node is drawn independently from a known distribution ${\cal D}_{\pi(i)}$ and $\pi$ is a random permutation. This is referred to as the known i.d. stochastic matching problem with random order arrivals.

Matroid Prophet Inequalities

2012-01-01
Robert Kleinberg , S. Matthew Weinberg
Consider a gambler who observes a sequence of independent, non-negative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent … Consider a gambler who observes a sequence of independent, non-negative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent observation. The famous prophet inequality of Krengel, Sucheston, and Garling asserts that a gambler who knows the distribution of each random variable can achieve at least half as much reward, in expectation, as a "prophet" who knows the sampled values of each random variable and can choose the largest one. We generalize this result to the setting in which the gambler and the prophet are allowed to make more than one selection, subject to a matroid constraint. We show that the gambler can still achieve at least half as much reward as the prophet; this result is the best possible, since it is known that the ratio cannot be improved even in the original prophet inequality, which corresponds to the special case of rank-one matroids. Generalizing the result still further, we show that under an intersection of p matroid constraints, the prophet's reward exceeds the gambler's by a factor of at most O(p), and this factor is also tight. Beyond their interest as theorems about pure online algorithms or optimal stopping rules, these results also have applications to mechanism design. Our results imply improved bounds on the ability of sequential posted-price mechanisms to approximate Bayesian optimal mechanisms in both single-parameter and multi-parameter settings. In particular, our results imply the first efficiently computable constant-factor approximations to the Bayesian optimal revenue in certain multi-parameter settings.
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Prophet secretary through blind strategies

2020-08-03
José R. Correa , Raimundo Saona , Bruno Ziliotto
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Tight Guarantees for Multi-unit Prophet Inequalities and Online Stochastic Knapsack

2022-01-01
Jiashuo Jiang , Will Ma , Jiawei Zhang
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Tight Guarantees for Multi-unit Prophet Inequalities and Online Stochastic KnapsackJiashuo Jiang, Will Ma, and … Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Tight Guarantees for Multi-unit Prophet Inequalities and Online Stochastic KnapsackJiashuo Jiang, Will Ma, and Jiawei ZhangJiashuo Jiang, Will Ma, and Jiawei Zhangpp.1221 - 1246Chapter DOI:https://doi.org/10.1137/1.9781611977073.51PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract Prophet inequalities are a useful tool for designing online allocation procedures and comparing their performance to the optimal offline allocation. In the basic setting of k-unit prophet inequalities, the procedure of Alaei [2] with its celebrated performance guarantee of has found widespread adoption in mechanism design and general online allocation problems in online advertising, healthcare scheduling, and revenue management. Despite being commonly used for implementing a fractional allocation in an online fashion, the tightness of Alaei's procedure for a given k has remained unknown. In this paper we resolve this question, characterizing the tight bound by identifying the structure of the optimal online implementation, and consequently improving the best-known guarantee for k-unit prophet inequalities for all k > 1. We also consider the more general online stochastic knapsack problem where each individual allocation can consume an arbitrary fraction of the initial capacity. Here we introduce a new "best-fit" procedure for implementing a fractionally-feasible knapsack solution online, with a performance guarantee of ≈ 0.319, which we also show is tight with respect to the standard LP relaxation. This improves the previously best-known guarantee of 0.2 for online knapsack. Our analysis differs from existing ones by eschewing the need to split items into "large" or "small" based on capacity consumption, using instead an invariant for the overall utilization on different sample paths. All in all, our results imply tight (non-greedy) Online Contention Resolution Schemes for k-uniform matroids and the knapsack polytope, respectively, which has further implications. Previous chapter Next chapter RelatedDetails Published:2022eISBN:978-1-61197-707-3 https://doi.org/10.1137/1.9781611977073Book Series Name:ProceedingsBook Code:PRDA22Book Pages:xvii + 3771
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An O(log log m) Prophet Inequality for Subadditive Combinatorial Auctions

2020-11-01
Paul Dütting , Thomas Keßelheim , Brendan Lucier
Prophet inequalities compare the expected performance of an online algorithm for a stochastic optimization problem to the expected optimal solution in hindsight. They are a major alternative to classic worst-case … Prophet inequalities compare the expected performance of an online algorithm for a stochastic optimization problem to the expected optimal solution in hindsight. They are a major alternative to classic worst-case competitive analysis, of particular importance in the design and analysis of simple (posted-price) incentive compatible mechanisms with provable approximation guarantees. A central open problem in this area concerns subadditive combinatorial auctions. Here n agents with subadditive valuation functions compete for the assignment of m items. The goal is to find an allocation of the items that maximizes the total value of the assignment. The question is whether there exists a prophet inequality for this problem that significantly beats the best known approximation factor of O(log m). We make major progress on this question by providing an O(log log m) prophet inequality. Our proof goes through a novel primal-dual approach. It is also constructive, resulting in an online policy that takes the form of static and anonymous item prices that can be computed in polynomial time given appropriate query access to the valuations. As an application of our approach, we construct a simple and incentive compatible mechanism based on posted prices that achieves an O(log log m) approximation to the optimal revenue for subadditive valuations under an item-independence assumption.
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On submodular prophet inequalities and correlation gap

2024-08-30
Chandra Chekuri , Vasilis Livanos
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Analyzing and Forecasting of COVID-19 Situation Using FbProphet Model Algorithms

2022-05-16
S. Geetha , M. Farida Begam , Ayush Dubey +2 more
SARS-CoV-2 (n-coronavirus) is a global pandemic that has killed millions of people all over the world. In severe situations, it can induce pneumonia and severe acute respiratory syndrome (SARS), which … SARS-CoV-2 (n-coronavirus) is a global pandemic that has killed millions of people all over the world. In severe situations, it can induce pneumonia and severe acute respiratory syndrome (SARS), which can lead to death. It's an asymptomatic sickness that makes life and work more difficult for us. This research focused on the current state of the coronavirus pandemic and forecasted the global situation, as well as its impacts and future status. The authors used the FbProphet model to forecast new covid cases and deaths for the month of August utilizing various information representation and machine learning algorithms. They hope the findings will aid scientists, researchers, and laypeople in predicting and analyzing the effects of the epidemic. Finally, they conclude that the virus's second wave was around four times stronger than the first. They also looked at the trajectory of COVID-19 instances (monthly and weekly) and discovered that the number of cases rises more during the weekdays, which could be due to the weekend lockout.

Learning in auctions: Regret is hard, envy is easy

2022-03-09
Constantinos Daskalakis , Vasilis Syrgkanis
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On Submodular Prophet Inequalities and Correlation Gap

2021-01-01
Chandra Chekuri , Vasilis Livanos
Prophet inequalities and secretary problems have been extensively studied in recent years due to their elegance, connections to online algorithms, stochastic optimization, and mechanism design problems in game theoretic settings. … Prophet inequalities and secretary problems have been extensively studied in recent years due to their elegance, connections to online algorithms, stochastic optimization, and mechanism design problems in game theoretic settings. Rubinstein and Singla developed a notion of combinatorial prophet inequalities in order to generalize the standard prophet inequality setting to combinatorial valuation functions such as submodular and subadditive functions. For non-negative submodular functions they demonstrated a constant factor prophet inequality for matroid constraints. Along the way they showed a variant of the correlation gap for non-negative submodular functions. In this paper we revisit their notion of correlation gap as well as the standard notion of correlation gap and prove much tighter and cleaner bounds. Via these bounds and other insights we obtain substantially improved constant factor combinatorial prophet inequalities for both monotone and non-monotone submodular functions over any constraint that admits an Online Contention Resolution Scheme. In addition to improved bounds we describe efficient polynomial-time algorithms that achieve these bounds.

Online Combinatorial Assignment in Independence Systems

2024-01-01
Javier Marinković , José A. Soto , Víctor Verdugo
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Single-Sample Prophet Inequalities Revisited.

2021-03-24
Constantine Caramanis , Matthew Faw , Orestis Papadigenopoulos +1 more
The study of the prophet inequality problem in the limited information regime was initiated by Azar et al. [SODA'14] in the pursuit of prior-independent posted-price mechanisms. As they show, $O(1)$-competitive … The study of the prophet inequality problem in the limited information regime was initiated by Azar et al. [SODA'14] in the pursuit of prior-independent posted-price mechanisms. As they show, $O(1)$-competitive policies are achievable using only a single sample from the distribution of each agent. A notable portion of their results relies on reducing the design of single-sample prophet inequalities (SSPIs) to that of order-oblivious secretary (OOS) policies. The above reduction comes at the cost of not fully utilizing the available samples. However, to date, this is essentially the only method for proving SSPIs for many combinatorial sets. Very recently, Rubinstein et al. [ITCS'20] give a surprisingly simple algorithm which achieves the optimal competitive ratio for the single-choice SSPI problem $-$ a result which is unobtainable going through the reduction to secretary problems. Motivated by this discrepancy, we study the competitiveness of simple SSPI policies directly, without appealing to results from OOS literature. In this direction, we first develop a framework for analyzing policies against a greedy-like prophet solution. Using this framework, we obtain the first SSPI for general (non-bipartite) matching environments, as well as improved competitive ratios for transversal and truncated partition matroids. Second, motivated by the observation that many OOS policies for matroids decompose the problem into independent rank-$1$ instances, we provide a meta-theorem which applies to any matroid satisfying this partition property. Leveraging the recent results by Rubinstein et al., we obtain improved competitive guarantees (most by a factor of $2$) for a number of matroids captured by the reduction of Azar et al. Finally, we discuss applications of our SSPIs to the design of mechanisms for multi-dimensional limited information settings with improved revenue and welfare guarantees.

Tight Guarantees for Static Threshold Policies in the Prophet Secretary Problem

2022-12-29
Nick Arnosti , Will Ma
In the prophet secretary problem, n values are drawn independently from known distributions and presented in a uniformly random order. A decision maker must accept or reject each value when … In the prophet secretary problem, n values are drawn independently from known distributions and presented in a uniformly 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 analyze 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 an appropriate threshold guarantees [Formula: see text] times the value of the offline optimal solution. Note that [Formula: see text], and by Stirling’s approximation, [Formula: see text]. This represents the best-known guarantee for the prophet secretary problem for all k > 1 and is tight for all k for the class of static threshold policies. We provide two simple methods for setting the threshold. Our first method sets a threshold such that [Formula: see text] values are accepted in expectation, and offers an optimal guarantee for all k. Our second sets a threshold such that the expected number of values exceeding the threshold is equal to k. This approach gives an optimal guarantee if k > 4 but gives suboptimal guarantees for [Formula: see text]. Our proofs use a new result for optimizing sums of independent Bernoulli random variables, which extends a result of Hoeffding from 1956 and could be of independent interest. Supplemental Material: The online appendices are available at https://doi.org/10.1287/opre.2022.2419 .
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Tight Guarantees for Multiunit Prophet Inequalities and Online Stochastic Knapsack

2024-06-03
Jiashuo Jiang , Will Ma , Jiawei Zhang
In “Tight Guarantees for Multiunit Prophet Inequalities and Online Stochastic Knapsack,” Jiang, Ma, and Zhang analyze the multiunit prophet inequalities and online knapsack problems. They formulate the problems as the … In “Tight Guarantees for Multiunit Prophet Inequalities and Online Stochastic Knapsack,” Jiang, Ma, and Zhang analyze the multiunit prophet inequalities and online knapsack problems. They formulate the problems as the online contention resolution scheme problems. Then, they develop linear programming formulations to represent the online contention resolution scheme problems. They characterize the optimal solution of the linear programming and derive the corresponding optimal policies. For the multiunit prophet inequalities, they are able to obtain the tight ratios, which improved over the previous ones. For the online stochastic knapsack problem, their ratio is also optimal and the best up to date. They finally generalize their results to the multiresource setting and expand the applicability of their approach.
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Max-Weight Online Stochastic Matching: Improved Approximations Against the Online Benchmark

2022-07-12
Mark Braverman , Mahsa Derakhshan , Antonio Molina Lovett
In this paper, we study max-weight stochastic matchings on online bipartite graphs under both vertex and edge arrivals. We focus on designing polynomial time approximation algorithms with respect to the … In this paper, we study max-weight stochastic matchings on online bipartite graphs under both vertex and edge arrivals. We focus on designing polynomial time approximation algorithms with respect to the online benchmark, which was first considered by Papadimitriou, Pollner, Saberi, and Wajc [EC'21].
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Online Stochastic Max-Weight Bipartite Matching: Beyond Prophet Inequalities

2024-08-01
Christos H. Papadimitriou , Tristan Pollner , Amin Saberi +1 more
The rich literature on online Bayesian selection problems has long focused on so-called prophet inequalities, which compare the gain of an online algorithm to that of a “prophet” who knows … The rich literature on online Bayesian selection problems has long focused on so-called prophet inequalities, which compare the gain of an online algorithm to that of a “prophet” who knows the future. An equally natural, though significantly less well-studied, benchmark is the optimum online algorithm, which may be omnipotent (i.e., computationally unbounded), but not omniscient. What is the computational complexity of the optimum online? How well can a polynomial-time algorithm approximate it? Motivated by applications in ride hailing, we study the above questions for the online stochastic maximum-weight matching problem under vertex arrivals. For this problem, a number of [Formula: see text]-competitive algorithms are known. This is the best possible ratio for this problem, as it generalizes the original single-item prophet inequality problem. We present a polynomial-time algorithm, which approximates the optimal online algorithm within a factor of 0.51—strictly more than the best-possible constant against a prophet. In contrast, we show that it is PSPACE-hard to approximate this problem within some universal constant [Formula: see text]. Funding: Financial support from the National Science Foundation [Grants CCF1763970, CCF1812919, and CCF191070], the Office of Naval Research [Grant N000141912550], and Cisco Research is gratefully acknowledged.
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Optimal Stopping with Interdependent Values

2024-07-08
Simon Mauras , Divyarthi Mohan , Rebecca Reiffenhäuser
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Truthful Online Scheduling of Cloud Workloads under Uncertainty

2022-04-25
Moshe Babaioff , Ronny Lempel , Brendan Lucier +3 more
Cloud computing customers often submit repeating jobs and computation pipelines on approximately regular schedules, with arrival and running times that exhibit variance. This pattern, typical of training tasks in machine … Cloud computing customers often submit repeating jobs and computation pipelines on approximately regular schedules, with arrival and running times that exhibit variance. This pattern, typical of training tasks in machine learning, allows customers to partially predict future job requirements. We develop a model of cloud computing platforms that receive statements of work (SoWs) in an online fashion. The SoWs describe future jobs whose arrival times and durations are probabilistic, and whose utility to the submitting agents declines with completion time. The arrival and duration distributions, as well as the utility functions, are considered private customer information and are reported by strategic agents to a scheduler that is optimizing for social welfare.
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The Stationary Prophet Inequality Problem

2022-07-12
Kristen Kessel , Ali Shameli , Amin Saberi +1 more
We study a continuous and infinite time horizon counterpart to the classic prophet inequality, which we term the stationary prophet inequality problem. Here, copies of a good arrive and perish … We study a continuous and infinite time horizon counterpart to the classic prophet inequality, which we term the stationary prophet inequality problem. Here, copies of a good arrive and perish according to Poisson point processes. Buyers arrive similarly and make take-it-or-leave-it offers for unsold items. The objective is to maximize the (infinite) time average revenue of the seller. Our main results are pricing-based policies which (i) achieve a 1/2-approximation of the optimal offline policy, which is best possible, and (ii) achieve a better than (1-1/e)-approximation of the optimal online policy. Result (i) improves upon bounds implied by recent work of Collina et al. (WINE'20), and is the first optimal prophet inequality for a stationary problem. Result (ii) improves upon a 1-1/e bound implied by recent work of Aouad and Sarita (EC'20), and shows that this prevalent bound in online algorithms is not optimal for this problem.
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Static Pricing for Multi-unit Prophet Inequalities

2024-07-01
Shuchi Chawla , Nikhil R. Devanur , Thodoris Lykouris
Characterizing the Efficiency of Static Pricing Schemes as a Function of the Supply The problem of selling a supply of k units to a stream of customers constitutes one of … Characterizing the Efficiency of Static Pricing Schemes as a Function of the Supply The problem of selling a supply of k units to a stream of customers constitutes one of the cornerstones in revenue management. Static pricing schemes (that output the same price to all customers) are commonly used because of their simplicity and their many desirable properties; they are anonymous, nonadaptive, and order oblivious. Although the efficiency of those schemes should improve as the supply k increases, prior work has only focused either on algorithms that aim for a constant approximation that is independent of k or on the setting where k becomes really large. In contrast, this paper characterizes the efficiency of static pricing schemes as a function of the supply. Our approach stems from identifying a “sweet spot” between selling enough items and obtaining enough utility from customers with high valuations. Subsequent work shows that our pricing scheme is the optimal static pricing for every value of k.
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An O(log log m) prophet inequality for subadditive combinatorial auctions

2020-12-02
Paul Dütting , Thomas Keßelheim , Brendan Lucier
We survey the main results from [Dütting, Kesselheim, and Lucier 2020]: 1 a simple posted-price mechanism for subadditive combinatorial auctions with m items that achieves an O (log log m … We survey the main results from [Dütting, Kesselheim, and Lucier 2020]: 1 a simple posted-price mechanism for subadditive combinatorial auctions with m items that achieves an O (log log m ) approximation to the optimal welfare, plus a variant with entry fees that approximates revenue. These are based on a novel subadditive prophet inequality.
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Unknown I.I.D. prophets: better bounds, streaming algorithms, and a new impossibility

2020-07-12
José R. Correa , Paul Dütting , Felix Fischer +2 more
A prophet inequality states, for some α ∈ [0, 1], that the expected value achievable by a gambler who sequentially observes random variables X1, . . . , Xn and … A prophet inequality states, for some α ∈ [0, 1], that the expected value achievable by a gambler who sequentially observes random variables X1, . . . , Xn and selects one of them is at least an α fraction of the maximum value in the sequence. We obtain three distinct improvements for a setting that was first studied by Correa et al. (EC, 2019) and is particularly relevant to modern applications in algorithmic pricing. In this setting, the random variables are i.i.d. from an unknown distribution and the gambler has access to an additional βn samples for some β ≥ 0. We first give improved lower bounds on α for a wide range of values of β; specifically, α ≥ (1 + β)/e when β ≤ 1/(e − 1), which is tight, and α ≥ 0.648 when β = 1, which improves on a bound of around 0.635 due to Correa et al. (SODA, 2020). Adding to their practical appeal, specifically in the context of algorithmic pricing, we then show that the new bounds can be obtained even in a streaming model of computation and thus in situations where the use of relevant data is complicated by the sheer amount of data available. We finally establish that the upper bound of 1/e for the case without samples is robust to additional information about the distribution, and applies also to sequences of i.i.d. random variables whose distribution is itself drawn, according to a known distribution, from a finite set of known candidate distributions. This implies a tight prophet inequality for exchangeable sequences of random variables, answering a question of Hill and Kertz (Contemporary Mathematics, 1992), but leaves open the possibility of better guarantees when the number of candidate distributions is small, a setting we believe is of strong interest to applications.

Non-Adaptive Matroid Prophet Inequalities

2020-01-01
Shuchi Chawla , Kira Goldner , Anna R. Karlin +1 more
We investigate non-adaptive algorithms for matroid prophet inequalities. Matroid prophet inequalities have been considered resolved since 2012 when [KW12] introduced thresholds that guarantee a tight 2-approximation to the prophet; however, … We investigate non-adaptive algorithms for matroid prophet inequalities. Matroid prophet inequalities have been considered resolved since 2012 when [KW12] introduced thresholds that guarantee a tight 2-approximation to the prophet; however, this algorithm is adaptive. Other approaches of [CHMS10] and [FSZ16] have used non-adaptive thresholds with a feasibility restriction; however, this translates to adaptively changing an item's threshold to infinity when it cannot be taken with respect to the additional feasibility constraint, hence the algorithm is not truly non-adaptive. A major application of prophet inequalities is in auction design, where non-adaptive prices possess a significant advantage: they convert to order-oblivious posted pricings, and are essential for translating a prophet inequality into a truthful mechanism for multi-dimensional buyers. The existing matroid prophet inequalities do not suffice for this application. We present the first non-adaptive constant-factor prophet inequality for graphic matroids.

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

2022-01-01
Saeed Alaei , Ali Makhdoumi , Azarakhsh Malekian +1 more
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 … 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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Matroid Secretary is Equivalent to Contention Resolution.

2021-03-06
Shaddin Dughmi
We show that the matroid secretary problem is equivalent to correlated contention resolution in the online random-order model. Specifically, the matroid secretary conjecture is true if and only if every … We show that the matroid secretary problem is equivalent to correlated contention resolution in the online random-order model. Specifically, the matroid secretary conjecture is true if and only if every matroid admits an online random-order contention resolution scheme which, given an arbitrary (possibly correlated) prior distribution over subsets of the ground set, matches the balance ratio of the best offline scheme for that distribution up to a constant. We refer to such a scheme as universal. Our result indicates that the core challenge of the matroid secretary problem lies in resolving contention for positively correlated inputs, in particular when the positive correlation is benign in as much as offline contention resolution is concerned. Our result builds on our previous work which establishes one direction of this equivalence, namely that the secretary conjecture implies universal random-order contention resolution, as well as a weak converse, which derives a matroid secretary algorithm from a random-order contention resolution scheme with only partial knowledge of the distribution. It is this weak converse that we strengthen in this paper: We show that universal random-order contention resolution for matroids, in the usual setting of a fully known prior distribution, suffices to resolve the matroid secretary conjecture in the affirmative. Our proof is the composition of three reductions. First, we use duality arguments to reduce the matroid secretary problem to the matroid prophet secretary problem with arbitrarily correlated distributions. Second, we introduce a generalization of contention resolution we term labeled contention resolution, to which we reduce the correlated matroid prophet secretary problem. Finally, we combine duplication of elements with limiting arguments to reduce labeled contention resolution to classical contention resolution.

Single-Sample Prophet Inequalities via Greedy-Ordered Selection

2022-01-01
Constantine Caramanis , Paul Dütting , Matthew Faw +6 more
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Single-Sample Prophet Inequalities via Greedy-Ordered SelectionConstantine Caramanis, Paul Dütting, Matthew Faw, Federico Fusco, Philip … Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Single-Sample Prophet Inequalities via Greedy-Ordered SelectionConstantine Caramanis, Paul Dütting, Matthew Faw, Federico Fusco, Philip Lazos, Stefano Leonardi, Orestis Papadigenopoulos, Emmanouil Pountourakis, and Rebecca ReiffenhäuserConstantine Caramanis, Paul Dütting, Matthew Faw, Federico Fusco, Philip Lazos, Stefano Leonardi, Orestis Papadigenopoulos, Emmanouil Pountourakis, and Rebecca Reiffenhäuserpp.1298 - 1325Chapter DOI:https://doi.org/10.1137/1.9781611977073.54PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract We study single-sample prophet inequalities (SSPIs), i.e., prophet inequalities where only a single sample from each prior distribution is available. Besides a direct, and optimal, SSPI for the basic single choice problem [Rubinstein et al., 2020], most existing SSPI results were obtained via an elegant, but inherently lossy reduction to order-oblivious secretary (OOS) policies [Azar et al., 2014]. Motivated by this discrepancy, we develop an intuitive and versatile greedy-based technique that yields SSPIs directly rather than through the reduction to OOSs. Our results can be seen as generalizing and unifying a number of existing results in the area of prophet and secretary problems. Our algorithms significantly improve on the competitive guarantees for a number of interesting scenarios (including general matching with edge arrivals, bipartite matching with vertex arrivals, and certain matroids), and capture new settings (such as budget additive combinatorial auctions). Complementing our algorithmic results, we also consider mechanism design variants. Finally, we analyze the power and limitations of different SSPI approaches by providing a partial converse to the reduction from SSPI to OOS given by Azar et al. Previous chapter Next chapter RelatedDetails Published:2022eISBN:978-1-61197-707-3 https://doi.org/10.1137/1.9781611977073Book Series Name:ProceedingsBook Code:PRDA22Book Pages:xvii + 3771
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Worst-Case Welfare of Item Pricing in the Tollbooth Problem

2023-04-26
Zihan Tan , Yifeng Teng , Mingfei Zhao
We study the worst-case welfare of item pricing in the tollbooth problem. The problem was first introduced by Guruswami et al. [27], and is a special case of the combinatorial … We study the worst-case welfare of item pricing in the tollbooth problem. The problem was first introduced by Guruswami et al. [27], and is a special case of the combinatorial auction in which (i) each of the m items in the auction is an edge of some underlying graph; and (ii) each of the n buyers is single-minded and only interested in buying all edges of a single path. We consider the competitive ratio between the hindsight optimal welfare and the optimal worst-case welfare among all item-pricing mechanisms, when the order of the arriving buyers is adversarial. We assume that buyers own the tie-breaking power, i.e. they can choose whether or not to buy the demand path at 0 utility. We prove a tight competitive ratio of 3/2 when the underlying graph is a single path (also known as the highway problem), whereas item-pricing can achieve the hindsight optimal if the seller is allowed to choose a proper tie-breaking rule to maximize the welfare [6, 11]. Moreover, we prove an O(1) upper bound of competitive ratio when the underlying graph is a tree.
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Worst-Case Welfare of Item Pricing in the Tollbooth Problem

2021-01-01
Zihan Tan , Yifeng Teng , Mingfei Zhao
We study the worst-case welfare of item pricing in the \emph{tollbooth problem}. The problem was first introduced by Guruswami et al, and is a special case of the combinatorial auction … We study the worst-case welfare of item pricing in the \emph{tollbooth problem}. The problem was first introduced by Guruswami et al, and is a special case of the combinatorial auction in which (i) each of the $m$ items in the auction is an edge of some underlying graph; and (ii) each of the $n$ buyers is single-minded and only interested in buying all edges of a single path. We consider the competitive ratio between the hindsight optimal welfare and the optimal worst-case welfare among all item-pricing mechanisms, when the order of the arriving buyers is adversarial. We assume that buyers own the \emph{tie-breaking} power, i.e. they can choose whether or not to buy the demand path at 0 utility. We prove a tight competitive ratio of $3/2$ when the underlying graph is a single path (also known as the \emph{highway} problem), whereas item-pricing can achieve the hindsight optimal if the seller is allowed to choose a proper tie-breaking rule to maximize the welfare. Moreover, we prove an $O(1)$ upper bound of competitive ratio when the underlying graph is a tree. For general graphs, we prove an $\Omega(m^{1/8})$ lower bound of the competitive ratio. We show that an $m^{\Omega(1)}$ competitive ratio is unavoidable even if the graph is a grid, or if the capacity of every edge is augmented by a constant factor $c$. The results hold even if the seller has tie-breaking power.
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Bayesian and Randomized Clock Auctions

2022-07-12
Michal Feldman , Vasilis Gkatzelis , Nick Gravin +1 more
In a single-parameter mechanism design problem, a provider is looking to sell some service to a group of potential buyers. Each buyer i has a private value vi for receiving … In a single-parameter mechanism design problem, a provider is looking to sell some service to a group of potential buyers. Each buyer i has a private value vi for receiving this service, and some feasibility constraint restricts which subsets of buyers can be served simultaneously. Recent work in economics introduced (deferred-acceptance) clock auctions as a superior class of auctions for this problem, due to their transparency, simplicity, and very strong incentive guarantees. Subsequent work in computer science focused on evaluating these auctions with respect to their social welfare approximation guarantees, leading to strong impossibility results: in the absence of prior information regarding the buyers' values, no deterministic clock auction can achieve a bounded approximation, even for simple feasibility constraints with only two maximal feasible sets.
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Prophet Inequalities with Cancellation Costs

2024-06-10
Farbod Ekbatani , Rad Niazadeh , Pranav Nuti +1 more
Most of the literature on online algorithms and sequential decision-making focuses on settings with "irrevocable decisions" where the algorithm's decision upon arrival of the new input is set in stone … Most of the literature on online algorithms and sequential decision-making focuses on settings with "irrevocable decisions" where the algorithm's decision upon arrival of the new input is set in stone and can never change in the future. One canonical example is the classic prophet inequality problem, where realizations of a sequence of independent random variables X1, X2,… with known distributions are drawn one by one and a decision maker decides when to stop and accept the arriving random variable, with the goal of maximizing the expected value of their pick. We consider "prophet inequalities with recourse" in the linear buyback cost setting, where after accepting a variable Xi, we can still discard Xi later and accept another variable Xj, at a buyback cost of f × Xi. The goal is to maximize the expected net reward, which is the value of the final accepted variable minus the total buyback cost. Our first main result is an optimal prophet inequality in the regime of f ≥ 1, where we prove that we can achieve an expected reward 1+f/1+2f times the expected offline optimum. The problem is still open for 0<f<1 and we give some partial results in this regime. In particular, as our second main result, we characterize the asymptotic behavior of the competitive ratio for small f and provide almost matching upper and lower bounds that show a factor of 1−Θ(flog(1/f)). Our results are obtained by two fundamentally different approaches: One is inspired by various proofs of the classical prophet inequality, while the second is based on combinatorial optimization techniques involving LP duality, flows, and cuts.
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An $O(\log \log m)$ Prophet Inequality for Subadditive Combinatorial Auctions

2022-05-19
Paul Dütting , Thomas Keßelheim , Brendan Lucier
Prophet inequalities compare the expected performance of an online algorithm for a stochastic optimization problem to the expected optimal solution in hindsight. They are a major alternative to classic worst-case … Prophet inequalities compare the expected performance of an online algorithm for a stochastic optimization problem to the expected optimal solution in hindsight. They are a major alternative to classic worst-case competitive analysis, of particular importance in the design and analysis of simple (posted-price) incentive compatible mechanisms with provable approximation guarantees. A central open problem in this area concerns subadditive combinatorial auctions. Here $n$ agents with subadditive valuation functions compete for the assignment of $m$ items. The goal is to find an allocation of the items that maximizes the total value of the assignment. The question is whether there exists a prophet inequality for this problem that significantly beats the best known approximation factor of $O(\log m)$. We make major progress on this question by providing an $O(\log \log m)$ prophet inequality. Our proof goes through a novel primal-dual approach. It is also constructive, resulting in an online policy that takes the form of static and anonymous item prices that can be computed in polynomial time given appropriate query access to the valuations. As an application of our approach, we construct a simple and incentive compatible mechanism based on posted prices that achieves an $O(\log \log m)$ approximation to the optimal revenue for subadditive valuations under an item-independence assumption.
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From Contention Resolution to Matroid Secretary and Back

2025-05-09
Shaddin Dughmi

A Bridge between Liquid and Social Welfare in Combinatorial Auctions with Submodular Bidders

2019-07-17
Dimitris Fotakis , Kyriakos Lotidis , Chara Podimata
We study incentive compatible mechanisms for Combinatorial Auctions where the bidders have submodular (or XOS) valuations and are budget-constrained. Our objective is to maximize the liquid welfare, a notion of … We study incentive compatible mechanisms for Combinatorial Auctions where the bidders have submodular (or XOS) valuations and are budget-constrained. Our objective is to maximize the liquid welfare, a notion of efficiency for budgetconstrained bidders introduced by Dobzinski and Paes Leme (2014). We show that some of the known truthful mechanisms that best-approximate the social welfare for Combinatorial Auctions with submodular bidders through demand query oracles can be adapted, so that they retain truthfulness and achieve asymptotically the same approximation guarantees for the liquid welfare. More specifically, for the problem of optimizing the liquid welfare in Combinatorial Auctions with submodular bidders, we obtain a universally truthful randomized O(log m)-approximate mechanism, where m is the number of items, by adapting the mechanism of Krysta and Vöcking (2012).Additionally, motivated by large market assumptions often used in mechanism design, we introduce a notion of competitive markets and show that in such markets, liquid welfare can be approximated within a constant factor by a randomized universally truthful mechanism. Finally, in the Bayesian setting, we obtain a truthful O(1)-approximate mechanism for the case where bidder valuations are generated as independent samples from a known distribution, by adapting the results of Feldman, Gravin and Lucier (2014).
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Prophet Inequalities over Time

2023-07-07
Andreas Abels , Elias Pitschmann , Daniel Schmand
In this paper, we introduce an over-time variant of the well-known prophet inequality with i.i.d. random variables. Instead of stopping with one realized value at some point in the process, … In this paper, we introduce an over-time variant of the well-known prophet inequality with i.i.d. random variables. Instead of stopping with one realized value at some point in the process, we decide for each step how long we select the value. Then we cannot select another value until this period is over. The goal is to maximize the expectation of the sum of selected values. We describe the structure of the optimal stopping rule and give upper and lower bounds on the prophet inequality. In online algorithms terminology, this corresponds to bounds on the competitive ratio of an online algorithm.
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Prophet Inequalities for Matching with a Single Sample.

2021-04-05
Paul Dütting , Federico Fusco , Philip Lazos +2 more
We consider the prophet inequality problem for (not necessarily bipartite) matching problems with independent edge values, under both edge arrivals and vertex arrivals. We show constant-factor prophet inequalities for the … We consider the prophet inequality problem for (not necessarily bipartite) matching problems with independent edge values, under both edge arrivals and vertex arrivals. We show constant-factor prophet inequalities for the case where the online algorithm has only limited access to the value distributions through samples. First, we give a $16$-approximate prophet inequality for matching in general graphs under edge arrivals that uses only a single sample from each value distribution as prior information. Then, for bipartite matching and (one-sided) vertex arrivals, we show an improved bound of $8$ that also uses just a single sample from each distribution. Finally, we show how to turn our $16$-approximate single-sample prophet inequality into a truthful single-sample mechanism for online bipartite matching with vertex arrivals.

Comparison of Threshold Stop Rules and Maximum for Independent Nonnegative Random Variables

1984-11-01
Ester Samuel‐Cahn
Let $X_i \geq 0$ be independent, $i = 1, \cdots, n$, and $X^\ast_n = \max(X_1, \cdots, X_n)$. Let $t(c) (s(c))$ be the threshold stopping rule for $X_1, \cdots, X_n$, defined … Let $X_i \geq 0$ be independent, $i = 1, \cdots, n$, and $X^\ast_n = \max(X_1, \cdots, X_n)$. Let $t(c) (s(c))$ be the threshold stopping rule for $X_1, \cdots, X_n$, defined by $t(c) = \text{smallest} i$ for which $X_i \geq c(s(c) = \text{smallest} i$ for which $X_i > c), = n$ otherwise. Let $m$ be a median of the distribution of $X^\ast_n$. It is shown that for every $n$ and $\underline{X}$ either $EX^\ast_n \leq 2EX_{t(m)}$ or $EX^\ast_n \leq 2EX_{s(m)}$. This improves previously known results, [1], [4]. Some results for i.i.d. $X_i$ are also included.
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Bayesian Combinatorial Auctions: Expanding Single Buyer Mechanisms to Many Buyers

2014-01-01
Saeed Alaei
We present a general framework for approximately reducing the mechanism design problem for multiple agents to single agent subproblems in the context of Bayesian combinatorial auctions. Our framework can be … We present a general framework for approximately reducing the mechanism design problem for multiple agents to single agent subproblems in the context of Bayesian combinatorial auctions. Our framework can be applied to any setting which roughly satisfies the following assumptions: (i) agents' types are distributed independently (not necessarily identically), (ii) objective function is additively separable over the agents, and (iii) there are no interagent constraints except for the supply constraints (i.e., that the total allocation of each item should not exceed the supply). Our framework is general in the sense that it makes no direct assumption about agents' valuations, type distributions, or single agent constraints (e.g., budget, incentive compatibility, etc.). We present two generic multiagent mechanisms which use single agent mechanisms as black boxes. If an $\alpha$-approximate single agent mechanism is available for each agent, and assuming no agent ever demands more than $\frac{1}{k}$ of all units of each item, our generic multiagent mechanisms are $\gamma_{k}\alpha$-approximations of the optimal multiagent mechanism, where $\gamma_{k}$ is a constant which is at least $1-\frac{1}{\sqrt{k+3}}$. As a byproduct of our construction, we present a generalization of prophet inequalities where both gambler and prophet are allowed to pick $k$ numbers each to receive a reward equal to their sum. Finally, we use our framework to obtain multiagent mechanisms with improved approximation factor for several settings from the literature.
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