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17
Sketching Valuation Functions
, 2011
"... Motivated by the problem of querying and communicating bidders ’ valuations in combinatorial auctions, we study how well different classes of set functions can be sketched. More formally, let f be a function mapping subsets of some ground set [n] to the nonnegative real numbers. We say that f ′ is ..."
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Cited by 23 (2 self)
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Motivated by the problem of querying and communicating bidders ’ valuations in combinatorial auctions, we study how well different classes of set functions can be sketched. More formally, let f be a function mapping subsets of some ground set [n] to the nonnegative real numbers. We say that f ′ is an αsketch of f if for every set S, the value f ′ (S) lies between f(S)/α and f(S), and f ′ can be specified by poly(n) bits. We show that for every subadditive function f there exists an αsketch where α = n 1/2 · O(polylog(n)). Furthermore, we provide an algorithm that finds these sketches with a polynomial number of demand queries. This is essentially the best we can hope for since: 1. We show that there exist subadditive functions (in fact, XOS functions) that do not admit an o(n 1/2) sketch. (Balcan and Harvey [3] previously showed that there exist functions belonging to the class of substitutes valuations that do not admit an O(n 1/3) sketch.) 2. We prove that every deterministic algorithm that accesses the function via value queries only cannot guarantee a sketching ratio better than n 1−ɛ. We also show that coverage functions, an interesting subclass of submodular functions, admit arbitrarily good sketches.
Informational limitations of ascending combinatorial auctions
 J. Econom. Theory
"... We study the inherent limitations of natural widelyused classes of ascending combinatorial auctions. Specifically, we show that ascending combinatorial auctions that do not use both nonlinear prices and personalized prices can not achieve social efficiency with general bidder valuations. This cast ..."
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Cited by 8 (3 self)
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We study the inherent limitations of natural widelyused classes of ascending combinatorial auctions. Specifically, we show that ascending combinatorial auctions that do not use both nonlinear prices and personalized prices can not achieve social efficiency with general bidder valuations. This casts doubt on the performance that can be achieved using the simpler auctions suggested, e.g., by Kwasnica et al. (2005),Porter et al. (2003) and Wurman and Wellman (2000) and justifies the added complexity in the auctions suggested by, e.g., Parkes and
Truthfulness via Proxies
, 2011
"... This short note exhibits a truthfulinexpectation log m O( log log m)approximation mechanism for combinatorial auctions with subadditive bidders that uses polynomial communication. 1 ..."
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Cited by 7 (0 self)
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This short note exhibits a truthfulinexpectation log m O( log log m)approximation mechanism for combinatorial auctions with subadditive bidders that uses polynomial communication. 1
Combinatorial Walrasian Equilibrium
"... We study a combinatorial market design problem, where a collection of indivisible objects is to be priced and sold to potential buyers subject to equilibrium constraints. The classic solution concept for such problems is Walrasian Equilibrium (WE), which provides a simple and transparent pricing str ..."
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Cited by 4 (4 self)
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We study a combinatorial market design problem, where a collection of indivisible objects is to be priced and sold to potential buyers subject to equilibrium constraints. The classic solution concept for such problems is Walrasian Equilibrium (WE), which provides a simple and transparent pricing structure that achieves optimal social welfare. The main weakness of the WE notion is that it exists only in very restrictive cases. To overcome this limitation, we introduce the notion of a Combinatorial Walrasian equilibium (CWE), a natural relaxation of WE. The difference between a CWE and a (noncombinatorial) WE is that the seller can package the items into indivisible bundles prior to sale, and the market does not necessarily clear. We show that every valuation profile admits a CWE that obtains at least half of the optimal (unconstrained) social welfare. Moreover, we devise a polytime algorithm that, given an arbitrary allocation X, computes a CWE that achieves at least half of the welfare of X. Thus, the economic problem of finding a CWE with high social welfare reduces to the algorithmic problem of socialwelfare approximation. In addition, we show that every valuation profile admits a CWE that extracts a logarithmic fraction of the optimal welfare as revenue. Finally, these results are complemented by strong lower bounds when the seller is restricted to using item prices only, which motivates the use of bundles. The strength of our results derives partly from their generality — our results hold for arbitrary valuations that may exhibit complex combinations of substitutes and complements.
Algorithmic Mechanism Design Through the lens of Multiunit auctions
, 2013
"... Mechanism Design is a subfield of game theory that aims to design games whose equilibria have desired properties such as achieving high efficiency or high revenue. Algorithmic Mechanism Design is a subfield that lies on the border of Mechanism Design and Computer Science and deals with Mechanism Des ..."
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Cited by 3 (0 self)
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Mechanism Design is a subfield of game theory that aims to design games whose equilibria have desired properties such as achieving high efficiency or high revenue. Algorithmic Mechanism Design is a subfield that lies on the border of Mechanism Design and Computer Science and deals with Mechanism Design in algorithmicallycomplex scenarios that are often found in computational settings such as the Internet. The central challenge in Algorithmic Mechanism Design is the tension between the computational constraints and the gametheoretic ones. This survey demonstrates both the tension and ways of addressing it by focusing on a single simple problem: multiunit auctions. A variety of issues will be discussed: representation, computational hardness, communication, convexity, approximations, VCG mechanisms and their generalizations, singleparameter settings vs. multiparameter settings, and the power of randomness. 1
Iterative Auction Design for Graphical Valuations
, 2012
"... Multiitem iterative auctions are a class of mechanisms that are commonly employed in practice, for instance, in the context of spectrum and procurement auctions. However, the iterative auction formats that are provably efficient are either limited to restrictive environments that do not allow for ..."
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Cited by 2 (0 self)
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Multiitem iterative auctions are a class of mechanisms that are commonly employed in practice, for instance, in the context of spectrum and procurement auctions. However, the iterative auction formats that are provably efficient are either limited to restrictive environments that do not allow for value complementarity between different items, or they require complex pricing structures. Consequently, the iterative auctions that are used in practice often lack efficiency guarantees. In this work, we develop practical and efficient iterative auctions for multiitem settings that exhibit both value complementarity and substitutability. We obtain such auctions by focusing on a class of valuation functions that admit a compact representation, which we refer to as graphical valuations. Value functions that belong to this class are associated with a value graph, nodes of which correspond to the items that are sold by the auctioneer. The edges of this graph capture the value complementarity and substitutability exhibited by the items. The value a bidder has for a given set of items can be expressed using the edge and node weights
Welfare Maximization and the Supermodular Degree
, 2013
"... Given a set of items and a collection of players, each with a nonnegative monotone valuation set function over the items, the welfare maximization problem requires that every item be allocated to exactly one player, and one wishes to maximize the sum of values obtained by the players, as computed by ..."
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Given a set of items and a collection of players, each with a nonnegative monotone valuation set function over the items, the welfare maximization problem requires that every item be allocated to exactly one player, and one wishes to maximize the sum of values obtained by the players, as computed by applying the respective valuation function to the bundle of items allocated to the player. This problem in its full generality is NPhard, and moreover, at least as hard to approximate as set packing. Better approximation guarantees are known for restricted classes of valuation functions. In this work we introduce a new parameter, the supermodular degree of a valuation function, which is a measure for the extent to which the function exhibits supermodular behavior. We design an approximation algorithm for the welfare maximization problem whose approximation guarantee is linear in the supermodular degree of the underlying valuation functions.
VCGequivalent in Expectation Mechanism: General Framework for Constructing Iterative Combinatorial Auction Mechanisms
"... ABSTRACT In this paper, we develop a new class of iterative mechanisms called a VCGequivalent in expectation mechanism. Iterative auctions are preferred over their sealedbid counterparts in practical settings, since they can avoid full revelation of type information. However, to guarantee that si ..."
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ABSTRACT In this paper, we develop a new class of iterative mechanisms called a VCGequivalent in expectation mechanism. Iterative auctions are preferred over their sealedbid counterparts in practical settings, since they can avoid full revelation of type information. However, to guarantee that sincere strategies are an ex post equilibrium, the mechanism needs to achieve exactly the same outcome as the VickreyClarkeGroves (VCG) mechanism. To guarantee that a mechanism is VCGequivalent, it inevitably asks an irrelevant query, in which a participant has no incentive to answer the query sincerely. Such an irrelevant query causes unnecessary leakage of private information and a different incentive issue. In a VCGequivalent in expectation mechanism, the mechanism achieves the same allocation as VCG, but the transfers are the same as VCG only in expectation. We show that in a VCGequivalent in expectation mechanism, sincere strategies constitute a sequential equilibrium. Also, we develop a general procedure for constructing a VCGequivalent in expectation mechanism that does not ask any irrelevant query. To demonstrate the applicability of this idea in a practical application, we develop a VCGequivalent in expectation mechanism that can be applied to the Japanese 4G spectrum auction.
Equilibria of Generalized Cut and Choose Protocols
"... Classic cake cutting protocols — which fairly allocate a divisible good among agents with heterogeneous preferences — are susceptible to manipulation. Do their strategic outcomes still guarantee fairness? We model the interaction among agents as a game and study its Nash equilibria. We show that eac ..."
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Classic cake cutting protocols — which fairly allocate a divisible good among agents with heterogeneous preferences — are susceptible to manipulation. Do their strategic outcomes still guarantee fairness? We model the interaction among agents as a game and study its Nash equilibria. We show that each protocol in the novel class of generalized cut and choose protocols — which includes the most important discrete cake cutting protocols — is guaranteed to have an εequilibrium for all ε> 0. Moreover, we observe that the (approximate) equilibria of proportional protocols — which guarantee each of the n agents a 1/nfraction of the cake — must be (approximately) proportional. Finally, we design a generalized cut and choose protocol where all equilibrium outcomes satisfy the stronger fairness notion of envyfreeness.