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Welfare Guarantees for Combinatorial Auctions with Item Bidding
, 2010
"... We analyze the price of anarchy (POA) in a simple and practical nontruthful combinatorial auction when players have subadditive valuations for goods. We study the mechanism that sells every good in parallel with separate secondprice auctions. We first prove that under a standard “no overbidding ” ..."
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Cited by 19 (2 self)
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We analyze the price of anarchy (POA) in a simple and practical nontruthful combinatorial auction when players have subadditive valuations for goods. We study the mechanism that sells every good in parallel with separate secondprice auctions. We first prove that under a standard “no overbidding ” assumption, for every subadditive valuation profile, every pure Nash equilibrium has welfare at least 50 % of optimal — i.e., the POA is at most 2. For the incomplete information setting, we prove that the POA with respect to BayesNash equilibria is strictly larger than 2 — an unusual separation from the fullinformation model — and is at most 2 ln m, where m is the number of goods.
The Price of Anarchy . . .
, 2012
"... We define smooth games of incomplete information. We prove an “extension theorem ” for such games: price of anarchy bounds for pure Nash equilibria for all induced fullinformation games extend automatically, without quantitative degradation, to all mixedstrategy BayesNash equilibria with respect ..."
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Cited by 5 (0 self)
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We define smooth games of incomplete information. We prove an “extension theorem ” for such games: price of anarchy bounds for pure Nash equilibria for all induced fullinformation games extend automatically, without quantitative degradation, to all mixedstrategy BayesNash equilibria with respect to a product prior distribution over players ’ preferences. We also note that, for BayesNash equilibria in games with correlated player preferences, there is no general extension theorem for smooth games. We give several applications of our definition and extension theorem. First, we show that many games of incomplete information for which the price of anarchy has been studied are smooth in our sense. Thus our extension theorem unifies much of the known work on the price of anarchy in games of incomplete information. Second, we use our extension theorem to prove new bounds on the price of anarchy of BayesNash equilibria in congestion games with incomplete information.
Simultaneous Auctions are (almost) Efficient
, 2012
"... Simultaneous item auctions are simple procedures for allocating items to bidders with potentially complex preferences over different item sets. In a simultaneous auction, every bidder submits bids on all items simultaneously. The allocation and prices are then resolved for each item separately, base ..."
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Cited by 2 (1 self)
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Simultaneous item auctions are simple procedures for allocating items to bidders with potentially complex preferences over different item sets. In a simultaneous auction, every bidder submits bids on all items simultaneously. The allocation and prices are then resolved for each item separately, based solely on the bids submitted on that item. Such procedures occur in practice (e.g. eBay) but are not truthful. We study the efficiency of Bayesian Nash equilibrium (BNE) outcomes of simultaneous first and secondprice auctions when bidders have complementfree (a.k.a. subadditive) valuations. We show that the expected social welfare of any BNE is at least 1 2 of the optimal social welfare in the case of firstprice auctions, and at least 1 4 in the case of secondprice auctions. These results improve upon the previouslyknown logarithmic bounds, which wereestablished by Hassidim et al. (2011) for firstpriceauctions and by Bhawalkar and Roughgarden (2011) for secondprice auctions. 1
The Hebrew University
"... We consider two alternatives to optimal auctions: postedprice mechanisms and dynamic auctions. In postedprice mechanisms, the seller posts a single price and sells the item at this price to a bidder that accepts it. In a dynamic auction, bidders arrive sequentially and each bidder leaves the marke ..."
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We consider two alternatives to optimal auctions: postedprice mechanisms and dynamic auctions. In postedprice mechanisms, the seller posts a single price and sells the item at this price to a bidder that accepts it. In a dynamic auction, bidders arrive sequentially and each bidder leaves the market before the next bidder arrives. We establish an exact asymptotic characterization of the optimal revenue in each of these mechanisms for general distributions, under a mild condition taken from extremevalue theory. We also devise postedprice and dynamic mechanisms that achieve this optimal revenue. Intuitively, one would expect auctions to perform better compared to posted prices as the values of the bidders are more dispersed; We show that this intuition holds up to a point where more value dispersion causes an opposite phenomenon. Our results also imply that such mechanisms may lose a nontrivial share of the optimalauction revenue, even in large markets.
Do Externalities Degrade GSP’s Efficiency?
, 2012
"... We consider variants of the cascade model of externalities in sponsored search auctions introduced independently by Aggrawal et al. and Kempe and Mahdian in 2008, where the clickthrough rate of a slot depends also on the ads assigned to earlier slots. Aggrawal et al. and Kempe and Mahdian give a dy ..."
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We consider variants of the cascade model of externalities in sponsored search auctions introduced independently by Aggrawal et al. and Kempe and Mahdian in 2008, where the clickthrough rate of a slot depends also on the ads assigned to earlier slots. Aggrawal et al. and Kempe and Mahdian give a dynamic programming algorithm for finding the efficient allocation in this model. We give worstcase efficiency bounds for a variant of the classical Generalized Second Price (GSP) auction in this model. Our technical approach is to first consider an idealized version of the model where an unlimited number of ads can be displayed on the same page; here, Aggrawal et al. and Kempe and Mahdian show that a greedy algorithm finds the optimal allocation. The game theoretic analog of this greedy algorithm can be thought of as a variant of the classical GSP auction. We give the first nontrivial worstcase efficiency bounds for GSP in this model. In the more general model with limited slots, greedy algorithms like GSP can compute extremely bad allocations. Nonetheless, we show that an appropriate extension of the greedy algorithm is approximately optimal, and that the worstcase equilibrium inefficiency in the corresponding analog of GSP also remains bounded. In the context of these models, the GSP mechanisms suffer from two forms of suboptimality: that from using a simple allocation rule (the greedy algorithm) rather than an optimal one (based on dynamic programming), and that from the strategic behavior of the bidders (caused by using the GSP’s critical bid pricing rule rather than one leading to a dominantstrategy implementation). Our results show that for this class of problems, the two causes of efficiency loss can be analyzed separately.
APPROXIMATION IN ALGORITHMIC GAME THEORY: ROBUST APPROXIMATION BOUNDS FOR EQUILIBRIA AND AUCTIONS
, 2011
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Simultaneous SingleItem Auctions
"... Abstract. In a combinatorial auction (CA) with item bidding, several goods are sold simultaneously via singleitem auctions. We study how the equilibrium performance of such an auction depends on the choice of the underlying singleitem auction. We provide a thorough understanding of the price of an ..."
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Abstract. In a combinatorial auction (CA) with item bidding, several goods are sold simultaneously via singleitem auctions. We study how the equilibrium performance of such an auction depends on the choice of the underlying singleitem auction. We provide a thorough understanding of the price of anarchy, as a function of the singleitem auction payment rule. When the payment rule depends on the winner’s bid, as in a firstprice auction, we characterize the worstcase price of anarchy in the corresponding CAs with item bidding in terms of a sensitivity measure of the payment rule. As a corollary, we show that equilibrium existence guarantees broader than that of the firstprice rule can only be achieved by sacrificing its property of having only fully efficient (pure) Nash equilibria. For payment rules that are independent of the winner’s bid, we prove a strong optimality result for the canonical secondprice auction. First, its set of pure Nash equilibria is always a superset of that of every other payment rule. Despite this, its worstcase POA is no worse than that of any other payment rule that is independent of the winner’s bid. 1
The Dining Bidder Problem: à la russe et à la française
"... Item bidding auctions are a line of research which provides a simple and often efficient alternative to traditional combinatorial auction design in particular, they were inspired by real world auction houses, like eBay and Sotheby’s. We survey the literature from a culinary perspective, offering an ..."
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Item bidding auctions are a line of research which provides a simple and often efficient alternative to traditional combinatorial auction design in particular, they were inspired by real world auction houses, like eBay and Sotheby’s. We survey the literature from a culinary perspective, offering an intuitive illustration of the welfare in simultaneous and sequential auctions. Welfare in simultaneous first and second price auctions is high when bidders have complementfree valuations. In contrast, sequential second price auctions can lead to bad outcomes due to signaling problems and even in the case of first price, a good outcome is only guaranteed for unit demand bidders. We give an intuitive interpretation of an example with bad welfare in sequential first price auctions with submodular bidders from Paes Leme, Syrgkanis and Tardos (SODA’12).
Simultaneous Bayesian Auctions and Computational Complexity
"... as an alternative to the wellknown complexity issues plaguing combinatorial auctions with incomplete information, and some strong positive results have been shown about their performance. We point out some very serious complexity obstacles to this approach: Computing a Bayesian equilibrium in such ..."
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as an alternative to the wellknown complexity issues plaguing combinatorial auctions with incomplete information, and some strong positive results have been shown about their performance. We point out some very serious complexity obstacles to this approach: Computing a Bayesian equilibrium in such auctions is hard for PP — a complexity class between the polynomial hierarchy and PSPACE — and even finding an approximate such equilibrium is as hard as NP, for some small approximation ratio (additive or multiplicative); therefore, the assumption that such equilibria will be arrived at by rational agents is quite problematic. In fact, even recognizing a Bayesian Nash equilibrium is intractable. Furthermore, these results hold even if bidder valuations are quite benign: Only one bidder valuation in our construction is unit demand or monotone submodular, while all others are additive. We also explore the possibility of favorable price of anarchy results for noregret dynamics of the Bayesian simultaneous auctions game, and identify complexity obstacles there as well. 1.
Inefficiency of Standard MultiUnit Auctions
"... Abstract. We study two standard multiunit auction formats for allocating multiple units of a single good to multidemand bidders. The first one is the Discriminatory Auction, which charges every winner his winning bids. The second is the Uniform Price Auction, which determines a uniform price to ..."
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Abstract. We study two standard multiunit auction formats for allocating multiple units of a single good to multidemand bidders. The first one is the Discriminatory Auction, which charges every winner his winning bids. The second is the Uniform Price Auction, which determines a uniform price to be paid per unit. Variants of both formats find applications ranging from the allocation of state bonds to investors, to online sales over the internet. For these formats, we consider two bidding interfaces: (i) standard bidding, which is most prevalent in the scientific literature, and (ii) uniform bidding, which is more popular in practice. In this work, we evaluate the economic inefficiency of both multiunit auction formats for both bidding interfaces, by means of upper and lower bounds on the Price of Anarchy for pure Nash equilibria and mixed BayesNash equilibria. Our developments improve significantly upon bounds that have been obtained recently for submodular valuation functions. Also, for the first time, we consider bidders with subadditive valuation functions under these auction formats. Our results signify nearefficiency of these auctions, which provides further justification for their use in practice. 1