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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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Cited by 11 (3 self)
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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,
Duality and optimality of auctions for uniform distributions
 IN PROCEEDINGS OF THE 15TH ACM CONFERENCE ON ECONOMICS AND COMPUTATION, EC ’14
, 2014
"... We derive exact optimal solutions for the problem of optimizing revenue in singlebidder multiitem auctions for uniform i.i.d. valuations. We give optimal auctions of up to 6 items; previous results were only known for up to three items. To do so, we develop a general duality framework for the gene ..."
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Cited by 6 (1 self)
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We derive exact optimal solutions for the problem of optimizing revenue in singlebidder multiitem auctions for uniform i.i.d. valuations. We give optimal auctions of up to 6 items; previous results were only known for up to three items. To do so, we develop a general duality framework for the general problem of maximizing revenue in manybidders multiitem additive Bayesian auctions with continuous probability valuation distributions. The framework extends linear programming duality and complementarity to constraints with partial derivatives. The dual system reveals the geometric nature of the problem and highlights its connection with the theory of bipartite graph matchings. The duality framework is used not only for proving optimality, but perhaps more importantly, for deriving the optimal auction; as a result, the optimal auction is defined by natural geometric constraints.
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.
The Complexity of Optimal Multidimensional Pricing
"... We resolve the complexity of revenueoptimal deterministic auctions in the unitdemand singlebuyer Bayesian setting, i.e., the optimal item pricing problem, when the buyer’s values for the items are independent. We show that the problem of computing a revenueoptimal pricing can be solved in poly ..."
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Cited by 3 (0 self)
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We resolve the complexity of revenueoptimal deterministic auctions in the unitdemand singlebuyer Bayesian setting, i.e., the optimal item pricing problem, when the buyer’s values for the items are independent. We show that the problem of computing a revenueoptimal pricing can be solved in polynomial time for distributions of support size 2 and its decision version is NPcomplete for distributions of support size 3. We also show that the problem remains NPcomplete for the case of identical distributions. 1
Bounding Optimal Revenue in MultipleItems Auctions”, arXiv 1404.2832
, 2014
"... We use a weakduality technique from the dualitytheory framework for optimal auctions developed in [Giannakopoulos and Koutsoupias, 2014] and we derive closedform upperbound formulas for the optimal revenue of singlebidder multiitem additive Bayesian auctions, in the case that the items ’ valu ..."
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Cited by 2 (0 self)
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We use a weakduality technique from the dualitytheory framework for optimal auctions developed in [Giannakopoulos and Koutsoupias, 2014] and we derive closedform upperbound formulas for the optimal revenue of singlebidder multiitem additive Bayesian auctions, in the case that the items ’ valuations come i.i.d. from a uniform distribution and in the case where they follow independent (but not necessarily identical) exponential distributions. Using this, we are able to get in both settings specific approximation ratio bounds for the simple deterministic auctions studied by Hart and Nisan [2012], namely the one that sells the items separately and the one that sells them all in a full bundle. These bounds are constant, strictly below 2 for uniform priors and strictly below e for the exponential ones, for arbitrary number of items. We also propose and study the performance of a very simple randomized auction for exponential valuations, called Proportional. As a corollary, for the special case where the exponential distributions are also identical, we can derive that selling deterministically in a full bundle is optimal for any number of items. 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.
Public Projects, Boolean Functions, and the Borders of Border’s Theorem
, 2015
"... Border’s theorem gives an intuitive linear characterization of the feasible interim allocation rules of a Bayesian singleitem environment, and it has several applications in economic and algorithmic mechanism design. All known generalizations of Border’s theorem either restrict attention to relat ..."
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Border’s theorem gives an intuitive linear characterization of the feasible interim allocation rules of a Bayesian singleitem environment, and it has several applications in economic and algorithmic mechanism design. All known generalizations of Border’s theorem either restrict attention to relatively simple settings, or resort to approximation. This paper identifies a complexitytheoretic barrier that indicates, assuming standard complexity class separations, that Border’s theorem cannot be extended significantly beyond the stateoftheart. We also identify a surprisingly tight connection between Myerson’s optimal auction theory, when applied to public project settings, and some fundamental results in the analysis of Boolean functions.
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.