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1 Subadditive Valuations 1.1 The Setup
, 2014
"... In this lecture we study a scenario that generalizes almost all of the ones that we’ve studied in the course. Scenario #9: • A set U of m nonidentical items. • Each bidder i has a private valuation vi: 2U → R+ that is subadditive, meaning that for every pair of sets S, T ⊆ U, vi(S ∪ T) ≤ vi(S) + v ..."
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the articulations of this idea that we’ve seen, subadditivity is the most general; see Figure 1. Proposition 1.1 The set of subadditive valuations strictly contains the set of XOS valuations.
Learning Valuation Functions
 25TH ANNUAL CONFERENCE ON LEARNING THEORY
, 2012
"... A core element of microeconomics and game theory is that consumers have valuation functions over bundles of goods and that these valuations functions drive their purchases. A common assumption is that these functions are subadditive meaning that the value given to a bundle is at most the sum of valu ..."
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Cited by 17 (2 self)
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of values on the individual items. In this paper, we provide nearly tight guarantees on the efficient learnability of subadditive valuations. We also provide nearly tight bounds for the subclass of XOS (fractionally subadditive) valuations, also widely used in the literature. We additionally leverage
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
DOI 10.1287/moor.xxxx.xxxx c○20xx INFORMS Approximation Algorithms for Combinatorial Auctions with ComplementFree Bidders
"... In a combinatorial auction m heterogenous indivisible items are sold to n bidders. This paper considers settings in which the valuation functions of the bidders are known to be complementfree (a.k.a. subadditive). We provide several approximation algorithms for the socialwelfare maximization probl ..."
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compatible. Finally, we present two approximation algorithms for the more restricted class of XOS valuations: A simple deterministic algorithm that provides an approximation ratio of 2 and an e e−1 optimal approximation achieved via randomized rounding. We also present optimal lower bounds for both
Approximation Algorithms for NonSingleminded ProfitMaximization Problems with Limited Supply
"... Abstract. We consider profitmaximization problems for combinatorial auctions with nonsingle minded valuation functions and limited supply. We obtain fairly general results that relate the approximability of the profitmaximization problem to that of the corresponding socialwelfaremaximization (S ..."
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hence submodular, XOS valuations), we obtain a solution with profit OPT SWM /O(log cmax), where OPT SWM is the optimum social welfare and cmax is the maximum itemsupply; thus, this yields an O(log cmax)approximation for the profitmaximization problem. Furthermore, given any class of valuation
Budget feasible mechanism design: from priorfree to bayesian
 In STOC
, 2012
"... Budget feasible mechanism design studies procurement combinatorial auctions in which the sellers have private costs to produce items, and the buyer (auctioneer) aims to maximize a social valuation function on subsets of items, under the budget constraint on the total payment. One of the most importa ..."
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Cited by 7 (1 self)
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connected to the concept of approximate core in cooperative game theory. We provide a mechanism for subadditive functions whose approximation is O(I), via the worst case integrality gap I of this LP. This implies an O(log n)approximation for subadditive valuations, O(1)approximation for XOS valuations
Item Pricing for Revenue Maximization
"... We consider the problem of pricing n items to maximize revenue when faced with a series of unknown buyers with complex preferences, and show that a simple pricing scheme achieves surprisingly strong guarantees. We show that in the unlimited supply setting, a random single price achieves expected rev ..."
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Cited by 41 (4 self)
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. This is the best approximation known for any item pricing scheme for subadditive (or even submodular) valuations, even using multiple prices. We complement this result with a lower bound showing a sequence of subadditive (in fact, XOS) buyers for which any single price has approximation ratio 2 Ω(log1/4 n), thus
On the Complexity of Computing an Equilibrium in Combinatorial Auctions
, 2015
"... We study combinatorial auctions where each item is sold separately but simultaneously via a second price auction. We ask whether it is possible to efficiently compute in this game a pure Nash equilibrium with social welfare close to the optimal one. We show that when the valuations of the bidders ar ..."
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Cited by 1 (0 self)
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that computing an equilibrium requires exponential communication. Finally, for XOS (a.k.a. fractionally subadditive) valuations, we show that if there exists an efficient algorithm that finds an equilibrium, it must use techniques that are very different from our current ones.
Tight bounds for the price of anarchy of simultaneous first price auctions. arXiv:1312.2371
, 2013
"... We study the Price of Anarchy of simultaneous FirstPrice auctions for buyers with submodular and subadditive valuations. The current best upper bounds for the Bayesian Price of Anarchy of these auctions are e/(e − 1) [34] and 2 [16], respectively. We provide matching lower bounds for both cases ev ..."
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Cited by 2 (0 self)
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We study the Price of Anarchy of simultaneous FirstPrice auctions for buyers with submodular and subadditive valuations. The current best upper bounds for the Bayesian Price of Anarchy of these auctions are e/(e − 1) [34] and 2 [16], respectively. We provide matching lower bounds for both cases