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Extremevalue theorems for optimal multidimensional pricing
, 2011
"... We provide a Polynomial Time Approximation Scheme for the multidimensional unitdemand pricing problem, when the buyer’s values are independent (but not necessarily identically distributed.) For all ɛ> 0, we obtain a (1 + ɛ)factor approximation to the optimal revenue in time polynomial, when th ..."
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We provide a Polynomial Time Approximation Scheme for the multidimensional unitdemand pricing problem, when the buyer’s values are independent (but not necessarily identically distributed.) For all ɛ> 0, we obtain a (1 + ɛ)factor approximation to the optimal revenue in time polynomial, when the values are sampled from Monotone Hazard Rate (MHR) distributions, quasipolynomial, when sampled from regular distributions, and polynomial in n poly(log r) , when sampled from general distributions supported on a set [umin, rumin]. We also provide an additive PTAS for all bounded distributions. Our algorithms are based on novel extreme value theorems for MHR and regular distributions, and apply probabilistic techniques to understand the statistical properties of revenue distributions, as well as to reduce the size of the search space of the algorithm. As a byproduct of our techniques, we establish structural properties of optimal solutions. We show that, for all ɛ> 0, g(1/ɛ) distinct prices suffice to obtain a (1+ɛ)factor approximation to the optimal revenue for MHR distributions, where g(1/ɛ) is a quasilinear function of 1/ɛ that does not depend on the number of items. Similarly, for all ɛ> 0 and n> 0, g(1/ɛ · log n) distinct prices suffice for regular distributions,
A Simple and Approximately Optimal Mechanism for an Additive Buyer
"... In this letter we briefly survey our main result from [Babaioff el al. 2014]: a simple and approximately revenueoptimal mechanism for a monopolist who wants to sell a variety of items to a single buyer with an additive valuation. ..."
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Cited by 5 (1 self)
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In this letter we briefly survey our main result from [Babaioff el al. 2014]: a simple and approximately revenueoptimal mechanism for a monopolist who wants to sell a variety of items to a single buyer with an additive valuation.
Bayesian Truthful Mechanisms for Job Scheduling from Bicriterion Approximation Algorithms
 In the 26th Annual ACMSIAM Symposium on Discrete Algorithms (SODA
, 2015
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Simultaneous Bayesian Auctions and Computational Complexity
"... as an alternative to the wellknown complexity issues plaguing combinatorial auctions with incomplete information, and some strong positive results have been shown about their performance. We point out some very serious complexity obstacles to this approach: Computing a Bayesian equilibrium in such ..."
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Cited by 2 (1 self)
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as an alternative to the wellknown complexity issues plaguing combinatorial auctions with incomplete information, and some strong positive results have been shown about their performance. We point out some very serious complexity obstacles to this approach: Computing a Bayesian equilibrium in such auctions is hard for PP — a complexity class between the polynomial hierarchy and PSPACE — and even finding an approximate such equilibrium is as hard as NP, for some small approximation ratio (additive or multiplicative); therefore, the assumption that such equilibria will be arrived at by rational agents is quite problematic. In fact, even recognizing a Bayesian Nash equilibrium is intractable. Furthermore, these results hold even if bidder valuations are quite benign: Only one bidder valuation in our construction is unit demand or monotone submodular, while all others are additive. We also explore the possibility of favorable price of anarchy results for noregret dynamics of the Bayesian simultaneous auctions game, and identify complexity obstacles there as well. 1.
Algorithms for Strategic Agents
, 2014
"... In traditional algorithm design, no incentives come into play: the input is given, and your algorithm must produce a correct output. How much harder is it to solve the same problem when the input is not given directly, but instead reported by strategic agents with interests of their own? The unique ..."
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In traditional algorithm design, no incentives come into play: the input is given, and your algorithm must produce a correct output. How much harder is it to solve the same problem when the input is not given directly, but instead reported by strategic agents with interests of their own? The unique challenge stems from the fact that the agents may choose to lie about the input in order to manipulate the behavior of the algorithm for their own interests, and tools from Game Theory are therefore required in order to predict how these agents will behave. We develop a new algorithmic framework with which to study such problems. Specifically, we provide a computationally efficient blackbox reduction from solving any optimization problem on "strategic input, " often called algorithmic mechanism design to solving a perturbed version of that same optimization problem when the input is directly given, traditionally called algorithm design. We further demonstrate the power of our framework by making significant progress on several longstanding open problems. First, we extend Myerson's celebrated characterization of single item auctions [Mye8l] to multiple items, providing also a computationally efficient implementation of optimal auctions. Next, we design a computationally efficient 2approximate mechanism
MultiItem Auctions Defying Intuition?
, 2013
"... The best way to sell n items to a buyer who values each of them independently and uniformly randomly in [c, c+ 1] is to bundle them together, as long as c is large enough. Still, for any c, the grand bundling mechanism is never optimal for large enough n, despite the sharp concentration of the buyer ..."
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The best way to sell n items to a buyer who values each of them independently and uniformly randomly in [c, c+ 1] is to bundle them together, as long as c is large enough. Still, for any c, the grand bundling mechanism is never optimal for large enough n, despite the sharp concentration of the buyer’s total value for the items as n grows. Optimal multiitem mechanisms are rife with unintuitive properties, making multiitem generalizations of Myerson’s celebrated mechanism a daunting task. We survey recent work on the structure and computational complexity of revenueoptimal multiitem mechanisms, providing structural as well as algorithmic generalizations of Myerson’s result to multiitem settings.
Optimal Multiparameter Auction Design
, 2014
"... This thesis studies the design of Bayesian revenueoptimal auctions for a class of problems in which buyers have general (nonlinear and multiparameter) preferences. This class includes the classical linear singleparameter problem considered by Myerson (1981), for which he provided a simple chara ..."
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This thesis studies the design of Bayesian revenueoptimal auctions for a class of problems in which buyers have general (nonlinear and multiparameter) preferences. This class includes the classical linear singleparameter problem considered by Myerson (1981), for which he provided a simple characterization of the optimal mechanism, leading to numerous applications in theory and practice. However, for fully general preferences no generic and practical solution is known (various negative computational or structural results exist for special cases), even for the problem of designing a mechanism for a single agent. With general preferences, the optimal mechanism can be complex and impractical. This thesis identifies key conditions implying that the optimal mechanism is practical. Our main results are different in that they identify different conditions implying different notions of practicality, but are all similar in adopting a modular view to the problem that separates the task of designing a solution for the singleagent problem as the main module, from the task of combining these modules to form an optimal multiagent mechanism. First, for multiparameter linear settings, we specify a large class of distributions over values that implies that the optimal singleagent mechanism is posted pricing, and the optimal multiagent mechanism maximizes virtual values for players ’ favorite items. When agents are identical, the mechanism becomes the second price 4auction with reserve for favorite items. Second, and more generally, we specify a condition called revenuelinearity, defined beyond multiparameter linear settings, that implies that optimizing agents ’ marginal revenue maximizes revenue. Finally, adopting efficient computability as the notion of practicality, we show that for any setting in which singleagent solutions are efficiently computable, multiagent solutions are also computable.
NearOptimal and Robust Mechanism Design for Covering Problems with Correlated Players
, 2013
"... We consider the problem of designing incentivecompatible, expost individually rational (IR) mechanisms for covering problems in the Bayesian setting, where players ’ types are drawn from an underlying distribution and may be correlated, and the goal is to minimize the expected total payment made ..."
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We consider the problem of designing incentivecompatible, expost individually rational (IR) mechanisms for covering problems in the Bayesian setting, where players ’ types are drawn from an underlying distribution and may be correlated, and the goal is to minimize the expected total payment made by the mechanism. We formulate a notion of incentive compatibility (IC) that we call robust Bayesian IC (robust BIC) that is substantially more robust than BIC, and develop blackbox reductions from robustBIC mechanism design to algorithm design. For singledimensional settings, this blackbox reduction applies even when we only have an LPrelative approximation algorithm for the algorithmic problem. Thus, we obtain nearoptimal mechanisms for various covering settings including singledimensional covering problems, multiitem procurement auctions, and multidimensional facility location.