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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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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.
Sampling and Representation Complexity of Revenue Maximization
, 2014
"... We consider (approximate) revenue maximization in mechanisms where the distribution on input valuations is given via “black box” access to samples from the distribution. We analyze the following model: a single agent, m outcomes, and valuations represented as mdimensional vectors indexed by the ou ..."
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Cited by 5 (0 self)
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We consider (approximate) revenue maximization in mechanisms where the distribution on input valuations is given via “black box” access to samples from the distribution. We analyze the following model: a single agent, m outcomes, and valuations represented as mdimensional vectors indexed by the outcomes and drawn from an arbitrary distribution presented as a black box. We observe that the number of samples required – the sample complexity – is tightly related to the representation complexity of an approximately revenuemaximizing auction. Our main results are upper bounds and an exponential lower bound on these complexities. We also observe that the computational task of “learning ” a good mechanism from a sample is nontrivial, requiring careful use of regularization in order to avoid overfitting the mechanism to the sample. We establish preliminary positive and negative results pertaining to the computational complexity of learning a good mechanism for the original distribution by operating on a sample from said distribution.
A dynamic axiomatic approach to firstprice auctions
 In Proceedings of the 14th ACM Conference on Electronic Commerce
, 2013
"... ar ..."
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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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
On the Complexity of Optimal Lottery Pricing and Randomized Mechanisms
"... We study the optimal lottery problem and the optimal mechanism design problem in the setting of a single unitdemand buyer with item values drawn from independent distributions. Optimal solutions to both problems are characterized by a linear program with exponentially many variables. For the menu s ..."
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We study the optimal lottery problem and the optimal mechanism design problem in the setting of a single unitdemand buyer with item values drawn from independent distributions. Optimal solutions to both problems are characterized by a linear program with exponentially many variables. For the menu size complexity of the optimal lottery problem, we present an explicit, simple instance with distributions of support size 2, and show that exponentially many lotteries are required to achieve the optimal revenue. We also show that, when distributions have support size 2 and share the same high value, the simpler scheme of item pricing can achieve the same revenue as the optimal menu of lotteries. The same holds for the case of two items with support size 2 (but not necessarily the same high value). For the computational complexity of the optimal mechanism design problem, we show that unless the polynomialtime hierarchy collapses (more exactly, PNP = P#P), there is no universal efficient randomized algorithm to implement an optimal mechanism even when distributions have support size 3.
The complexity of optimal mechanism design
, 2013
"... Myerson's seminal work provides a computationally efficient revenueoptimal auction for selling one item to multiple bidders. Generalizing this work to selling multiple items at once has been a central question in economics and algorithmic game theory, but its complexity has remained poorly und ..."
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Myerson's seminal work provides a computationally efficient revenueoptimal auction for selling one item to multiple bidders. Generalizing this work to selling multiple items at once has been a central question in economics and algorithmic game theory, but its complexity has remained poorly understood. We answer this question by showing that a revenueoptimal auction in multiitem settings cannot be found and implemented computationally efficiently, unless ZPP = P. This is true even for a single additive bidder whose values for the items are independently distributed on two rational numbers with rational probabilities. Our result is very general: we show that it is hard to compute any encoding of an optimal auction of any format (direct or indirect, truthful or nontruthful) that can be implemented in expected polynomial time. In particular, under wellbelieved complexitytheoretic assumptions, revenueoptimization in very simple multiitem settings can only be tractably approximated. We note that our hardness result applies to randomized mechanisms in a very simple setting, and is not an artifact of introducing combinatorial structure to the problem by
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.