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11
Optimal multidimensional mechanism design: Reducing revenue to welfare maximization
, 2012
"... Bayesian auctions with arbitrary (possibly combinatorial) feasibility constraints and independent bidders with arbitrary (possibly combinatorial) demand constraints, appropriately extending Myerson’s singledimensional result [24] to this setting. We also show that every feasible Bayesian auction ca ..."
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Cited by 38 (13 self)
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Bayesian auctions with arbitrary (possibly combinatorial) feasibility constraints and independent bidders with arbitrary (possibly combinatorial) demand constraints, appropriately extending Myerson’s singledimensional result [24] to this setting. We also show that every feasible Bayesian auction can be implemented as a distribution over virtual VCG allocation rules. A virtual VCG allocation rule has the following simple form: Every bidder’s type ti is transformed into a virtual type fi(ti), via a bidderspecific function. Then, the allocation maximizing virtual welfare is chosen. Using this characterization, we show how to find and run the revenueoptimal auction given only black box access to an implementation of the VCG allocation rule. We generalize this result to arbitrarily correlated bidders, introducing the notion of a secondorder VCG allocation rule. We obtain our reduction from revenue to welfare optimization via two algorithmic results on reduced form auctions in settings with arbitrary feasibility and demand constraints. First, we provide a separation oracle for determining feasibility of a reduced form auction. Second, we provide a geometric algorithm to decompose any feasible reduced form into a distribution over virtual VCG allocation rules. In addition, we show how to execute both algorithms given only
Reducing revenue to welfare maximization: Approximation algorithms and other generalizations
 IN SODA
, 2013
"... It was recently shown in [12] that revenue optimization can be computationally efficiently reduced to welfare optimization in all multidimensional Bayesian auction problems with arbitrary (possibly combinatorial) feasibility constraints and independent additive bidders with arbitrary (possibly comb ..."
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Cited by 13 (6 self)
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It was recently shown in [12] that revenue optimization can be computationally efficiently reduced to welfare optimization in all multidimensional Bayesian auction problems with arbitrary (possibly combinatorial) feasibility constraints and independent additive bidders with arbitrary (possibly combinatorial) demand constraints. This reduction provides a polytime solution to the optimal mechanism design problem in all auction settings where welfare optimization can be solved efficiently, but it is fragile to approximation and cannot provide solutions to settings where welfare maximization can only be tractably approximated. In this paper, we extend the reduction to accommodate approximation algorithms, providing an approximation preserving reduction from (truthful) revenue maximization to (not necessarily truthful) welfare maximization. The mechanisms output by our reduction choose allocations via blackbox calls to welfare approximation on randomly selected inputs, thereby generalizing also our earlier structural results on optimal multidimensional mechanisms to approximately optimal mechanisms. Unlike [12], our results here are obtained through novel uses of the Ellipsoid algorithm and other optimization techniques over nonconvex regions.
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,
Understanding incentives: Mechanism design becomes algorithm design
, 2013
"... We provide a computationally efficient blackbox reduction from mechanism design to algorithm design in very general settings. Specifically, we give an approximationpreserving reduction from truthfully maximizing any objective under arbitrary feasibility constraints with arbitrary bidder types to ( ..."
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Cited by 10 (6 self)
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We provide a computationally efficient blackbox reduction from mechanism design to algorithm design in very general settings. Specifically, we give an approximationpreserving reduction from truthfully maximizing any objective under arbitrary feasibility constraints with arbitrary bidder types to (not necessarily truthfully) maximizing the same objective plus virtual welfare (under the same feasibility constraints). Our reduction is based on a fundamentally new approach: we describe a mechanism’s behavior indirectly only in terms of the expected value it awards bidders for certain behavior, and never directly access the allocation rule at all. Applying our new approach to revenue, we exhibit settings where our reduction holds both ways. That is, we also provide an approximationsensitive reduction from (nontruthfully) maximizing virtual welfare to (truthfully) maximizing revenue, and therefore the two problems are computationally equivalent. With this equivalence in hand, we show that both problems are NPhard to approximate within any polynomial factor, even for a single monotone submodular bidder. We further demonstrate the applicability of our reduction by providing a truthful mechanism maximizing fractional maxmin fairness. This is the first instance of a truthful mechanism that optimizes a nonlinear objective.
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.
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.
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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Cited by 1 (0 self)
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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