Sequencing and Information Revelation in Auctions for Imperfect Substitutes: Understanding eBay’ Market Design s
Eric Budish1 September 12, 2008

Harvard University; Economics Department; Littauer Center; Cambridge, MA 02138. ebudish@hbs.edu. I thank Susan Athey, Lucas Co¤man, Itay Fainmesser, Drew Fudenberg, Ian Jewitt, Adam Juda, Robin Lee, John Lewis, Paul Milgrom, Jerry Orlo¤, Marco Pagnozzi, Ariel Pakes, David Parkes, Robert Ritz, Laura Serban, and especially Paul Klemperer and Al Roth for helpful discussions.

1 Contact:

Abstract An auction marketplace like eBay consists of (i) an individual auction design; and (ii) a multiauction platform design, i.e., the set of rules that organize and convey information about multiple individual auctions. This paper shows that two aspects of eBay’ platform design s – sequencing of auctions by unique ending time, and provision of information about both current and near-future objects for sale –substantially increase the social surplus generated by single-unit second-price auctions when the goods traded are imperfect substitutes. The remaining ine¢ ciency from not using a multi-object auction is surprisingly small. JEL Classi…cation: D44, D82, D83, L13, L81 Keywords: Market Design, Auctions, Information in Auctions, Entry in Auctions, Sequential Auctions, eBay, e-commerce, Platforms

The internet auctioneer eBay was created in 1995 with collectibles traders in mind: its auction format was a natural adaptation of that used by art auctioneers such as Sotheby’ s and Christie’ and the company claimed that the founder’ wife’ desire to trade collectible s, s s “Pez Dispensers” inspired its creation. The character of the goods traded on eBay has changed dramatically. Now, most of the items auctioned are consumer goods that have close substitutes; used automobiles alone constitute around one-quarter of the approximately $60bn of annual trading volume on eBay.1 Modern auction theory suggests that eBay (or an entrant) should consider adopting a multi-object auction format – e.g., the simultaneous ascending auction (SAA) or VickreyClarke-Groves mechanism (VCG) – for the trade of goods with close but imperfect substitutes. Of course, while these mechanisms are e¢ cient (i.e., social-surplus maximizing), they also may be complex for participants to understand and for auctioneers to implement. In high-stakes contexts such as spectrum allocation it may be reasonable to ignore these particular e¢ ciency-complexity tradeo¤s; e.g., bidders can hire sophisticated auction consultants for a presumably negligible fraction of the value of the goods being auctioned. But in lowstakes contexts such as eBay it is worth understanding whether simpler mechanisms perform satisfactorily. To evaluate this question we need to analyze how single-unit auctions aggregate up into a multi-unit auction marketplace. In addition to its single-unit auction design, eBay adapted two aspects of Sotheby’ and Christie’ multi-auction platform design: individual auctions s s are sequenced (by unique ending time), and bidders are allowed to evaluate all of the objects in the sequence before deciding whether and how much to bid in any particular auction. This paper shows that this combination of single-unit auction design and multi-auction platform design may approximate the e¢ ciency performance of a more sophisticated auction design. (The purpose is not to explain eBay’ success; surely, sensible market design is just s one component of the story). Both sequencing auctions and revealing information seem like natural design decisions, perhaps in part because they have been used by the classic auction houses. Both features are also present in the market design used by the largest wholesale used-auto auctioneer,
The Pez Dispenser story turned out to be apocryphal. See Cohen (2002). eBay (2006a) provides trading volume by category for the year 2005. (eBay appears to have subsequently stopped providing this level of category volume detail). The categories with in excess of $2bn are: Motors ($13.6bn), Consumer Electronics ($3.5), Clothing & Accessories ($3.4), Computers ($3.1), Books/Music/Movies ($2.6), Home & Garden ($2.5), Collectibles ($2.2), and Sports ($2.1). Trading volume by category is not available for the early days of eBay. However, the following excerpt from the introductory section of its 1998 IPO prospectus is illustrative of its focus on collectible goods: “eBay . . . [is] . . . a Web-based community in which buyers and sellers are brought together in an e¢ cient and entertaining auction format to buy and sell personal items such as antiques, coins, collectibles, computers, memorabilia, stamps, and toys.” (eBay, 1998).
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Manheim (see Lewis, 2008). But there are alternative auction marketplaces that have made quite di¤erent decisions. For instance, in 2006 the online event-ticket marketplace StubHub ran the …rst ever auction for all of the tickets to a single event. Its single-unit auction design was similar to eBay’ (a unit is a pair of tickets). But it aggregated these auctions into a multi-auction s platform in an importantly di¤erent way: the auctions had identical hard-close ending times (see Harrington, 2006; Smith, 2006). Since most of the bidding in online auctions with hard closes occurs in the …nal moments (see Roth and Ockenfels, 2002), buyers could e¤ectively participate in just a single of the auctions. Rather than Sequencing the auctions, StubHub e¤ectively Separated them. Auctions at charity bene…ts are often organized similarly, which may explain why the …nal moments of so-called “silent auctions”are often anything but. Engelbrecht-Wiggans (1994) describes a sequential auction for restaurant equipment in which future items are hidden behind a curtain until their turn for sale. Some gimmicky online auction sites with very short auction durations (e.g. Bidz.com2 ) also suppress information about future objects for sale. Rather than Revealing information about future objects, these platform designs Hide such information. These alternative designs motivate the study of a simple two-by-two taxonomy of multiauction platforms. The individual auctions are either Sequential or Separated; information about future objects is either Revealed or Hidden. The Separated condition can be interpreted more broadly as a tractable device for modeling an auction platform in which there are obstacles to participating in multiple individual auctions.3 The Hidden condition can be interpreted more broadly as a modeling convention that captures that auction buyers know they have an outside option but are not yet sure of its value. The main theoretical result of this paper (Theorem 1) is that the Sequential Auction with Information Revealed is the most e¢ cient in the taxonomy. This result obtains for any number of unit-demand bidders and any (continuously di¤erentiable) distribution of bidders’ values. A limitation of the theoretical analysis is that we restrict attention to platforms that aggregate exactly two individual auctions, for tractability. We then use numerical analysis to give a sense of magnitudes. The Sequential Auction with Information Revealed generates at least 99% of the e¢ cient social surplus over all simOn Bidz.com "auctions start at $1 every 5 seconds!” $132 million of jewelry was sold on Bidz.com in 2006. This kind of information suppression occurs in the context of non-auction platforms as well; see Hagiu and Jullien (2008). 3 For instance, both Craigslist and Google Base are frequently used for auctions, but listings are sequenced by starting time and search closeness-of-…t, respectively. So, a buyer might participate in one auction, lose, and then learn that some other item he is interested in has meanwhile been sold. The soft closing times of the now-defunct Amazon and Yahoo! auction marketplaces might have the same e¤ect of making it more di¢ cult to participate in the complete sequence of auctions.
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ulated value distributions. By contrast, the other formats can have ine¢ ciency that is an order of magnitude worse. The insight that explains the consistently strong e¢ ciency performance of the Sequential Auction with Information Revealed is that the performance of the alternative mechanisms is in a sense negatively correlated (Theorem 1 then tells us that the eBay design performs better than the best alternative in the taxonomy). Revealing information is especially important for welfare when the distribution of bidder values is heavily left skewed (high values are rare), whereas Sequencing auctions is especially important when the distribution is heavily right skewed (high values are common). Numerical analysis also allows us to provide a simple robustness check to ensure that the results do not depend on the assumption that there are exactly two objects. The results appear to be robust to this extension, both qualitatively and quantitatively. This paper emphasizes e¢ ciency because an auction marketplace that is su¢ ciently inef…cient (e.g., taking switching costs into account) should be vulnerable to entry: a competitor could o¤er both sides of the market strictly more surplus (see Ellison, Fudenberg, and Mobius, 2004 for a study of competition between auction marketplaces). This emphasis is standard for auction markets with multiple buyers and multiple sellers (e.g., assignment, double auctions). However, it may be important to consider revenues as well.4 For most simulated value distributions Sequencing and Revealing Information increase revenues, and it can be shown theoretically that revealing information in a sequential auction increases all types of bidders’…rst-round bids and increases the strength of the second-round bidder pool (competition e¤ects). A metric that is related to revenues, and that eBay emphasizes in its …nancial reporting and marketing materials, is the “seller success rate” In the model of this paper, in which . sellers’costs are strictly lower than the base of the bidder value support and strategic reserve prices are not allowed, the Sequential Auction with Information Revealed achieves a perfect success rate of one, like an e¢ cient multi-object auction and unlike all other members of the taxonomy. Together these results suggest that it is not surprising per se that a non-combinatorial auction marketplace like eBay’ has been successful at facilitating the trade of goods with s imperfect substitutes. A simple single-object auction format, combined with a smart platform design that organizes multiple of these single-object auctions in an economically useful way, performs nearly as a more complicated alternative. These results also suggest that it is not surprising that StubHub’ ticket auction was s
Bulow and Klemperer (2007) emphasize that in the standard single-seller auction context – e.g., the one-time sale of a valuable asset, like a company –revenue is probably the more important metric, because revenue is what will incentivize sellers to create valuable assets in the …rst place.
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unsuccessful, and that StubHub appears to have abandoned the format.5 Nor might it be surprising that Bidz.com has allegedly had to use shill bidding to increase its success rate. (Miniter, 2008) In addition to highlighting a partial explanation for eBay’ success this paper also provides s a framework for understanding an important vulnerability. Some recent empirical studies of eBay auctions suggest that many bidders for substitute goods bid in just a single auction. For instance, Juda (2005) …nds that over half of the bidders in his study (of auctions for a particular model of computer monitor) participate in exactly one auction, are unsuccessful, and then exit the market. Lee and Malmendier (2006), too, document a failure to participate in multiple auctions, and Bajari and Hortacsu (2003) …nd that bidder participation costs dampen entry. In the language of this paper’ taxonomy these empirical …ndings suggest that eBay s auctions may be more Separate than Sequential. There may indeed be ways to meaningfully enhance the e¢ ciency of eBay’ auctions for substitutes, but this paper suggests we look s for them not in multi-object auctions per se, but in technologies that enable bidders more easily to express substitutes preferences over multiple objects. This appears to be eBay’ s understanding as well, as evidenced by its recent enhancements to its bidding proxy to enable bidders to participate in the full sequence of auctions with just a single visit to their computer.6 Related Literature This paper represents a …rst foray into the study of multi-auction platform design: taking the individual auction design as given, we study the rules that organize and convey information about multiple of these individual auctions. Methodologically, it is written in the spirit of the emerging literature on market design that looks at the institutional details of markets, complementing theory with simulation, experimental or case-study evidence to give a sense of magnitudes. (Klemperer, 2002; Roth, 2002) This paper is most closely related to the literatures on combinatorial auctions (e.g., Milgrom, 2004; Cramton et al, 2006) and eBay (e.g., Bajari and Hortacsu 2003, 2004; Lewis, 2006). The literature on eBay as a market institution has focused mainly on the analysis of individual auctions, whereas the combinatorial auctions literature has focused on the design of auction mechanisms to handle allocation problems considerably more complex and
The StubHub auction’ average selling price was $50 per ticket, versus a $148 average aftermarket s value for tickets for that particular tour. Some of the StubHub auctions closed at as low as $3 per ticket. Nevertheless StubHub’ CEO described the auction as a “successful experiment in true dynamic pricing.” s See Cohen and Grossweiner (2006). eBay acquired StubHub in early 2007 and currently operates it as a standalone business. 6 This new feature is called “Bid Assistant” which eBay indicates will save “busy buyers”“valuable time” , . See http://pages.ebay.com/bidassistant.
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high-stakes than the unit-demand environment of interest here. This paper articulates a middle ground between these literatures, by examining how individual single-unit auctions can aggregate up into a multi-object auction marketplace. For the case of interest, such a marketplace may approximate the bene…ts of a more-sophisticated combinatorial auction, and may be easier to understand by low-stakes buyers and sellers. This paper also is related to the auction-theoretic literatures on entry and information disclosure. In this paper, sequencing is of …rst-order importance for a …xed set of entrants; previous results have indicated that entry is of …rst-order importance when the set of participating bidders is endogenous (Athey et al, 2004; Klemperer, 2002; Bulow and Klemperer, 1996). In this paper, providing buyers with additional private information bene…ts the sellers (usually) and social surplus (always). This is distinct from the Linkage Principle (Milgrom and Weber, 1982) which indicates that providing additional public information bene…ts sellers without a¤ecting social surplus, though it has similar implications for auction policy. Finally, this paper is related to the burgeoning literature that studies the design of twosided platform markets (for a recent survey see Rochet and Tirole, 2006). The two-sided platform literature has generally focused on how platforms should set prices (or subsidies) on each side of the market, taking into account that participants on each side generate a network externality for those on the other. This literature typically abstracts from the details on how the interaction between the two sides creates value (notable recent exceptions are Athey and Ellison, 2008, and Hagiu and Jullien, 2008). By contrast, this paper takes the set of participants as exogenous and focuses on how the design of the platform a¤ects how much value the participants are able to create. Work that combines both approaches is likely to be a fruitful avenue for future research. Organization of the Paper The remainder of this paper is organized as follows. Section 1 presents the model. Section 2 considers equilibrium bidding behavior for each of the auction formats in the two-by-two taxonomy, drawing on Zeithammer (2006) for the analysis of the sequential auction with information revealed.7 Section 3 presents the main e¢ ciency results, making novel use of a classic result from the theory of auctions with entry, due to McAfee and McMillan (1987) and Engelbrecht-Wiggans (1993). Section 4 presents revenue results. Section 5 presents numerical simulations for various parameterizations of bidders’values, to give a sense of magnitudes. Section 6 concludes. Proofs and a simple two-value example are
I became aware of Zeithammer (2006) after developing the main results of this paper for a two-value model, for which bid functions can be obtained in closed form. Zeithammer provides an equilibrium existence theorem for Sequential Auctions with Information Revealed (stated herein as Lemma 1) that enabled a generalization of this paper’ analysis of e¢ ciency and revenue di¤erences across market designs. s There is no overlap in our papers’motivations or results. Essentially, the papers are complements, not substitutes.
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contained in appendices.

1

The Model

Basic Setup and Design Taxonomy. Two items, j = 1; 2; are sold by second-price sealed-bid auction to a group of n 3 bidders. The sealed-bid assumption simpli…es the analysis and is motivated by the fact that most bidding on eBay occurs towards the very end of the auction (Roth and Ockenfels, 2002). The restriction to two items is purely for tractability, though it is relaxed in numerical analysis (Section 5.2.5). The auctions are either sequential, which has the usual meaning, or separate, which means that each bidder must choose whether to participate in the auction for 1 or the auction for 2: The entry decisions in the separate auctions are simultaneous, meaning that bidders cannot observe others’entry choices before making their own. We are interested in two informational environments, called revealed and hidden. If information is revealed, each bidder i learns his private value for each object before he makes any bidding or entry decisions. When information is kept hidden, each bidder i learns his private value for object j only at the time of its auction. Speci…cally: in the sequential auction, bidder i learns only his …rst-round value before he must make his …rst-round bid; in the separate auctions, bidder i learns his value for the one auction he enters only after he has made his entry decision. Buyers. Bidders are risk-neutral and have unit demand; in particular, the winner of the …rst of two sequential auctions is assumed to exit the game (this is like assuming highenough resale costs). Bidders who lose both auctions have an outside option normalized to zero. Bidder i’ private value for object j, vij ; is independently and identically distributed s according to the probability distribution f j (v), with support on [v; v], v < v: We assume that the objects are stochastically equivalent, i.e., that f 1 (v) = f 2 (v) f (v) and that f ( ) is a continuously di¤erentiable density with full support on [v; v]. It is possible to add a constant term j (e.g. representing object j’ quality) to all bidders’values for a particular s object j; the only e¤ect this will have in equilibrium is to increase all bids for object j by exactly j without any a¤ect on the allocation or the bidding for the other object. The key assumption about buyers is that bidder i’ value for object j is independent of all s other bidders’idiosyncratic preferences (including his own for the other object). Independence across bidders is required to make use of Zeithammer’ (2006) equilibrium existence s result for the sequential auction with information revealed. Independence within bidders across objects allows the comparison across the two information environments to be the most meaningful. For instance, if the bidders’values are perfectly correlated across objects then 6

the hidden and revealed environments will be outcome equivalent. Sellers. Sellers play a passive role. All they do is set a minimum bid equal to their salvage value of c < v. In particular, they do not use strategic reserve prices or buy prices. This assumption is reasonable in cases where the seller has imprecise information about the distribution of buyers’values. Theory suggests that sellers who have precise information about buyers’values will use buy prices in most natural low-stakes auction markets.8 Indeed, they are used in about one-third of eBay auctions.9 Model Applicability. The model applies best to markets in which goods have some close substitutes, but that are not so thick that the pricing problem becomes trivial. Markets for which the model appears a particularly good approximation are used, discontinued, or supply-constrained goods (e.g., automobiles, business equipment, premium tickets to soldout events). The model is not intended to apply to markets for new goods without supply constraints (e.g., computer parts). In such cases, sellers will typically use …xed prices and may have access to additional supply, and it seems unrealistic to assume that buyers’outside options are symmetric.

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2.1

Equilibrium Bidding
Sequential Auctions

Suppose that bidders submit their …rst-round bids for the sequential auction with information revealed symmetrically according to a function br (v1 ; v2 ), and suppose that the highest bid 1 out of a particular set of n bidders is c1 . Bidder i’ continuation surplus, if he is one of the s n 1 losers, is the expected pro…t of a bidder whose value is vi2 at the dominant strategy equilibrium of a single second-price auction, when his n 2 opponents’ types are random draws from the set f(v1 ; v2 ) : br (v1 ; v2 ) c1 g based on the underlying joint density f (v1 ; v2 ).10 1
A "buy price" is a price which, if bid, ends the auction immediately. Milgrom (2004) shows that a buy price increases expected revenues when there are moderate entry costs. Budish and Takeyama (2001) and Hidvegi et al (2006) show that a buy price increases expected revenues when buyers are risk averse. (In all three of these papers it is assumed that the seller knows the distribution of buyers’values.) Intuition suggests that one of these conditions will typically be satis…ed on eBay: when the stakes are low, entry costs are likely to matter; when the stakes are high, risk aversion will matter. eBay’ Buy It Now di¤ers slightly from a traditional buy price, in that the option to transact at the BIN s price is withdrawn upon the …rst bid greater than the seller’ reserve. s 9 eBay (2006b) indicates that Buy It Now auctions account for 34% of trade by volume. Meeker (2006) indicates that BIN is used in 35% of auction listings, and tends to appear in a greater percentage of listings in conventional goods categories than in traditional collectibles categories. f (v1 )f (v2 ) 10 The conditional density f (v1 ; v2 jc1 ; br ) is simply R r for br (v1 ; v2 ) c1 and zero 1 1 f (x1 )f (x2 )dx1 dx2
b1 (x1 ;x2 ) c1

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otherwise.

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We can write this surplus as S(vi2 ; c1 ; br ): If br is the unique symmetric equilibrium br 1 1 1 11 (discussed in Section 2.1.2), we simply write S(vi2 ; c1 ): As we will see, a single continuation surplus function S(vi2 ; c1 ) will enable us to characterize bidding in the sequential auction for both information environments. 2.1.1 Equilibrium Bidding with Information Hidden

Suppose that bidders submit their …rst-round bids for the sequential auction with information hidden according to a function bh (v1 ; v2 ) = bh (v1 ; ) (with " " indicating a value that is not 1 1 yet known), and so the continuation surplus function is S(vi2 ; c1 ; bh ). Since bidders’values 1 are independent across rounds and bidders do not know their second-round values when they submit their …rst-round bids, it follows that c1 is not informative about the continuing bidders’second-round values (no matter what bh is played). That is, the conditional marginal 1 distribution f2 (v1 ; v2 jc1 ; bh ) is equal to the underlying bidder value distribution f (v2 ): 1 r Observe that b1 must be bounded above by v; in equilibrium no bidder will submit a …rstround bid in excess of his value. So the conditional marginal distribution f2 (v1 ; v2 jv; br ) 1 must also be equal to the underlying value distribution f (v2 ): receiving information that all of one’ second-round opponents bid weakly less than the maximum possible value in s round 1 is equivalent to receiving no information at all about these same opponents. This S(vi2 ; v) for any bh , and so the (unique Perfect implies that S(vi2 ; c1 ; bh ) = S(vi2 ; v; br ) 1 1 1 Bayes-Nash) equilibrium bid function if information is hidden is: (1) bh (v1 ; ) = v1 1 Ev2 S(v2 ; v)

At this bid amount, rational bidders are indi¤erent between winning and losing at the margin.12 S(v2 ; v) can be calculated explicitly as n 1 1 E(F(1:n 1) (v) F(2:n 1) (v)) - the expected di¤erence between the highest and second-highest of n 1 bidder values, multiplied by the probability of being the bidder whose value is highest. If v Ev2 S(v2 ; v) is strictly less than the seller’ salvage value c, then some bidders will abstain from the …rst auction. s
Zeithammer (2006) contains explicit expressions for S(v2 ; c1 ) in terms of the underlying joint density f (v1 ; v2 ). 12 Perfectness eliminates any asymmetric equilibria here. Participation in the second-round auction has the same expected value to all bidders, independent of the play of the …rst-round auction. So, the …rst-round auction is like a one-shot second-price sealed-bid auction in which losers receive a set payment.
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2.1.2

Equilibrium Bidding with Information Revealed

When information is revealed di¤erent types will shade their bids by di¤erent amounts; all bidders bid so as to be indi¤erent between winning and losing at the margin. Equilibrium bidding, in the …rst of two second-price auctions with information revealed, is characterized by the following equation: (2) br (v1 ; v ) = |1 {z 2}
Bid

1st pd value

The key to equation (2) is that the bidder’ own equilibrium bid is the argument in the s S( ) function de…ning which types of …rst-round bidders continue into the second-round. A necessary condition for a bid to be a component of an equilibrium strategy is that it satisfy (2). The logic is a simple generalization of Vickrey’ (1961). Given a highest outside s r bid x, i’ bid bi1 a¤ects neither the price he pays if he wins (x) nor his continuation value s if he loses (he faces a set of opponents who each bid less than x in the …rst round). If br > vi1 S(vi2 ; br ) then i prefers to marginally lose rather than marginally win. Such a i1 i1 bi1 cannot be an equilibrium bid since i can improve his surplus by bidding a small amount less, i.e., by breaking ties in the way he prefers. A similar argument indicates that br < i1 r vi1 S(vi2 ; bi1 ) cannot be an equilibrium. Zeithammer (2006) provides su¢ cient conditions for the existence of a unique symmetric equilibrium bidding strategy br . 1 Lemma 1 (Zeithammer 2006; Proposition 3). If f ( ) is a continuously di¤erentiable density with full support on [v; v]; then there exists a unique symmetric pure-strategy Perfect BayesNash equilibrium in which types with v1 > v bid according to a unique br (v1 ; v2 ); and the 1 zero-measure set of types with v1 = v abstain from the …rst auction. The continuationsurplus function S(v2 ; c1 ) is unique, and continuous in c1 with dS(v21;c1 ) > 1: dc

v1 |{z}

Surplus as a marginal loser

S(v2 ; br (v1 ; v2 )) 1 | {z }

2.2

Separate Auctions

Strategies in the Separate auctions consist of an entry decision followed by a bidding decision. Since bidders enter a single second-price sealed-bid auction the bidding decision is trivial in any Perfect equilibrium: bidders simply bid their values. The entry decision is a bit more subtle, because bidders may want to coordinate. If we restrict attention to symmetric Perfect Bayes-Nash equilibria, then if information is revealed the unique13 equilibrium entry decision of each bidder i is to enter auction j i¤ vij > vi(3 j) ,
13 Bidders can break ties, which occur with probability zero, however they like. Otherwise, the equilibrium is unique.

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and if information is hidden the unique equilibrium entry decision of each bidder i is to randomize uniformly between the two auctions. There also exist asymmetric PBNEs which address the coordination problem, for both informational environments. For instance, if n = 4 and information is hidden, then it is an equilibrium for bidders 1 and 2 always to enter 1 and bidders 3 and 4 always to enter 2.

3

E¢ ciency

In the present context, in which payments are pure transfers and all agents are risk neutral, social surplus is de…ned as the sum of the winners’values for the objects they receive (if an object is unallocated, the seller, whose value is c, is the "winner"). A fully e¢ cient auction marketplace is one that achieves the maximum possible expected social surplus, i.e., realizes the social-surplus maximizing allocation across all realizations of bidder values. In order to make e¢ ciency comparisons across auction formats we need to translate individuals’self-interested bidding decisions into aggregate social surplus contributions. The basic insight that enables this is a classic result from the theory of auctions with entry, due to McAfee and McMillan (1987) and Engelbrecht-Wiggans (1993). The latter statement is more general, and is summarized for the case relevant to the present analysis: Lemma 2 (Engelbrecht-Wiggans, 1993: Proposition 1) Let bidders’private values for a single object be drawn from a symmetric joint probability distribution. At the dominant strategy equilibrium of the second-price sealed bid auction, each bidder has an expected pro…t equal to his marginal contribution to social surplus. This result is useful here because it implies that the marginal loser surplus term S(v2 ; c1 ) is equivalent to the expected contribution to second-round surplus of a bidder with value v2 ; when the set of n 2 second-round opponents is drawn from those who bid weakly less than c1 in the …rst auction in equilibrium.

3.1

Comparison Results

Our …rst comparison is across information environments for sequential auctions: Proposition 1 The unique symmetric PBNE of the Sequential Auction with Information Revealed generates greater expected social surplus than the unique PBNE of the Sequential Auction with Information Hidden. The idea of the proof is as follows. If information is hidden the allocation is "greedy" – the …rst object is allocated to whomever of the n bidders has the highest value for it, and 10

then the second object is allocated to whomever of the n 1 remaining bidders has the highest value for it. If information is revealed the allocation is non-greedy whenever the bidder (i) with the highest …rst-round value shades his bid su¢ ciently that he loses the …rst auction to some other bidder (j) who has a lower …rst-round value but shades less. Sometimes i’ losing the …rst auction turns out ex-post to be social-surplus maximizing, s and sometimes not (e.g., if some third bidder k has a very high second-round value). What Lemma 2 tells us is that i’ and j’ shading decisions in round 1 are directly related s s to their expected social-surplus contribution in round 2. The proof combines this insight with the bid functions (2) to create a surplus-crediting scheme that yields the desired result; essentially, given the equilibrium bidding functions, the fact that i loses round 1 despite having the highest value is more than compensated by the fact that he participates in round 2 instead of j.14 Our next comparison is across auction formats, for either information environment: Proposition 2 For either Hidden or Revealed information, the unique symmetric PBNE of the Sequential Auction generates greater expected social surplus than any PBNE of the Separate Auctions. There exist value realizations for either information environment in which Separate auctions generate greater social surplus than Sequential auctions. Nevertheless, the proof for the Hidden information case is trivial: expected surplus is the largest of n and n 1 draws in the Sequential auction, versus n k and k draws (for some random k) in the Separate auctions. The proof for the Revealed information case is more subtle, and has a similar structure to that of Proposition 1. Instead of focusing on the outcomes of the bidder with the highest value for the …rst object, it focuses on the outcomes of the bidder with the highest bid for the …rst object in the sequential auction, again using (2), Lemma 2, and a careful surpluscrediting scheme. Notice that Proposition 2 allows for any PBNE of the Separate Auctions, not just the unique symmetric PBNE. This is important because asymmetric equilibria seem particularly plausible for Separate Auctions, due to the presence of coordination problems. Note too that no analogue of Proposition 1 is stated for the Separate Auctions. It is of course true
The equilibrium bidding strategy implicit in (2) requires that bidders understand the informational content of their opponents’ bids. Eyster and Rabin (2005) argue that in various economic environments this is unrealistic. Reassuringly, if bidders are "naive" and bid according to bnaive = v1 S(v2 ; v) – that 1 is, they ignore the informational content of marginally losing –then an appropriately restated Proposition 1 still obtains. The proof is analogous, and is omitted.
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that if we limit attention to symmetric PBNEs, then revealing information increases the expected social surplus of the Separate Auctions. Together Propositions 1 and 2 imply that the eBay multi-auction platform design is the most e¢ cient of the taxonomy (though of course not fully e¢ cient): Theorem 1 The unique symmetric PBNE of the Sequential Auction with Information Revealed generates greater expected social surplus than any PBNE of any of the alternate formats in the taxonomy. Remark 1 The Sequential Auction with Information Revealed is not fully e¢ cient. There exist multi-object auction formats (e.g., the Simultaneous Ascending Auction or VickreyClarke-Groves mechanism) that are fully e¢ cient in this environment. If n = 3 and values are distributed uniformly on [0; 1] then the worst-case e¢ ciency loss for the Sequential Auctions with Information Revealed occurs when the values are f(1; :99); (0; 1); (:58; 0)g: The bidder whose values are (1; :99) just loses the …rst auction to the (:58; 0) bidder, for 0:42 of squandered social surplus. The worst-case e¢ ciency loss of each of the other formats in the taxonomy is 1. (This point is elaborated substantially in Section 5.2.1). This ine¢ ciency is in contrast to results in Weber (1983) and Peters and Severinov (2006) for homogeneous objects, and Beggs and Graddy (1997) for vertically di¤erentiated objects. In these environments: 1) one-dimensional bids are su¢ cient statistics for multi-item demand pro…les; and 2) social surplus is always maximized by the greedy allocation.

4

Revenues

There is no uniquely best market design in the taxonomy from the sellers’ perspective, in contrast to the …nding in Theorem 1 that the eBay design is the unique best from society’ perspective. What follows are several results that are suggestive of the Sequential s Auction with Information Revealed having the strongest revenue performance, as well as an examination of cases in which other designs outperform.

4.1

Revenue Consequences of Revealing Information

We begin with two results that indicate that second-round competition in the sequential auction with information revealed is "stronger than random". Proposition 3 takes the perspective of the second-round seller, whereas Lemma 3 takes the perspective of buyers anticipating their second-round surplus. 12

Proposition 3 (Second-Round Competition E¤ect) If information is revealed in a Sequential Auction then the distribution of participating buyers’second-round values …rst-order stochastically dominates f (v2 ), the unconditional distribution. Consequently, expected secondround revenue is higher in the Sequential Auction with Information Revealed than in the Sequential Auction with Information Hidden. Lemma 3 For any c1 , v2 : S(v2 ; c1 ) S(v2 ; v):

db The stochastic dominance in Proposition 3 follows immediately from dv1 < 0 and the 2 independence of values across objects. Revenue is higher because …rst-order stochastic dominance of the participating bidder-value distribution implies …rst-order stochastic dominance of the second-highest of n 1 draws from the bidder-value distribution. Our next result, which derives from Lemma 3, shows that …rst-round competition in the sequential auction with information revealed is strong as well.15

Proposition 4 (First-Round Competition E¤ect) Revealing Information in a Sequential Auction increases each bidder’ …rst-round bid in expectation. That is, for any vi1 ; Evi2 [br (v1 ; v2 )] s 1 h b1 (v1 ; ): Note that Proposition 4 is not a statement about …rst-round revenues, which depend on the second-highest bid. While adding positive-mean noise to a distribution always increases the expectation of the …rst (highest) order statistic, it need not increase the expectation of the second order statistic. Again, we do not make a revenue comparison across information environments for the Separate Auctions due to the multiplicity of equilibria. If we limit attention to the unique symmetric PBNEs, then revealing information increases expected revenue in the Separate Auctions.

4.2

Revenue Consequences of Sequencing

1 Sequencing increases participation: bidders participate in 2 n auctions on average, versus one for the Separate auctions. Proposition 2 shows that this increase in participation unambiguously bene…ts social surplus. It is not always the case, however, that Sequencing increases revenues. There is a tradeo¤: Sequencing increases participation, but bidders in the …rst auction of the sequence shade their bids. Whether participation or shading is more important depends on the order statistics of the bidder value distribution. Let F (v) denote the cumulative distribution function of the
15 Note that the naive bidders discussed in footnote 20 avoid the …rst-round competition e¤ect of Proposition 4: Eyster and Rabin’ "cursed" bidders are "blessed" in this environment. s

13

bidder value distribution f (v), and let V F (v) and V 2 [F (v)]2 . V 2 is the cumulative distribution function for the highest of two independent draws from V . We write E(Vk:n ) 2 and E(Vk:n )for the expectation of the k th highest of n draws from V and V 2 , respectively. If information is revealed, the revenue bene…t from increased participation is closely 2 related to E(V2:n ) E(V2:(n=2)) ) > 0: This expression indicates the expected increase in the second-highest bidder value from adding back n=2 lower-of-2 draws to the n=2 higherof-2 draws that constitute the bidding pool for separate with information revealed. If information is hidden, the relevant expression is instead E(V2:n ) E(V2:(n=2)) ) > 0, because the n=2 participants in each separate auction are random.16 The revenue costs from shading are more closely related to the expected di¤erence between the …rst- and second-highest of n bidder values, E(V1:n ) E(V2:n ) < 0. The amount by which bidders shade is their conditional expected pro…t in the second auction, which is their expectation of the di¤erence between their value and the second-highest value, conditional on their value being the highest of the n 1 remaining bidders. An example of a distribution for which sequencing harms total revenues for either information environment is the two-value distribution (de…ned formally in Appendix B) with n = 4; and p, the probability that a bidders’ value is Low, equal to 0:9. The uniform distribution is an example for which sequencing increases revenues.

4.3

Seller Success Rates

A metric that eBay emphasizes as of central importance is the proportion of auctions that result in a successful sale, which eBay calls the "conversion rate".17 Though it has not been possible to compare revenues across the taxonomy, it is possible to conclude that eBay’ s auction format has the highest conversion rate in the taxonomy. Proposition 5 For any n; c and any distribution of bidder values, the unique PBNE of Sequential Auctions with Information Revealed has a seller success rate of one. For each of the other auction formats in the taxonomy there exist parameters for which the success rate in symmetric PBNE is strictly less than one. The success rate of one for eBay auctions follows immediately from Proposition 1 of Zeithammer (2006). The success rate of the Sequential Auction with Information Hidden
16 These expressions ignore the variance in the number of participants in each separate auction, and so may understate the bene…t from participation. This variance plays a central role in Section 5.2.1. 17 In its 2006 10-K …ling, in the Introductory section in which eBay formally describes its business strategy, it writes: "We have aggregated a signi…cant number of buyers, sellers, and items listed for sale, which, in turn, has resulted in a vibrant online commerce environment. Our sellers generally enjoy high conversion rates and our buyers enjoy an extensive selection of broadly-priced goods and services." (eBay 2006c, emphasis added).

14

is strictly less than one whenever there is a positive probability of bidders’abstaining from s the …rst auction, i.e., whenever v Ev2 S(v2 ; v) < c (roughly, the seller’ salvage value is high relative to the minimum bidder value). The success rate of the Separate Auctions is always strictly less than one in symmetric PBNE, though there exist asymmetric PBNEs with perfect success rates. The perfect success rate of course should not be taken literally. The assumption of the model that is unrealistic in practice is the restriction against sellers setting reserve prices in excess of v:18

5

Simulation Results: Magnitude of E¤ects

On the one hand, eBay made two auction-platform-design decisions that each increased the e¢ ciency of its single-unit auction design (Theorem 1). On the other hand, the eBay market design is not e¢ cient (Remark 1) – only a multi-object auction design can e¢ ciently solve the multi-object allocation problem. So should we admire eBay’ clever design or rue its s failure to implement simultaneous ascending auctions? The primary purpose of this section is to calculate the magnitudes, for many di¤erent distributions of bidder values, of (i) the e¢ ciency gains from Sequencing and Information, and (ii) the remaining ine¢ ciency from not using an e¢ cient multi-object auction. A secondary purpose of this section is to compare revenues across the various auction formats.

5.1

Simulation Methodology

We assume that bidders’ values are distributed according to a Beta distribution. The advantages of the Beta distribution are (i) it is parsimonious (it is fully speci…ed by two shape parameters and ); (ii) it is continuous and di¤erentiable with full support on [0; 1]; and (iii) it allows for many di¤erent kinds of plausible value distributions, including uniform, [ shaped, and \ shaped. A second assumption is that the seller’ salvage value, and hence s minimum allowable bid, is c = v = 0: Given a particular ; ; and n the methodology is as follows. First, I use Zeithammer’ s 19 (2006) procedure to compute S(v2 ; c1 ) on a …ne grid. I then can use (1) to directly calculate …rst-round bids for the sequential auction with information hidden,20 and (2) plus the strict
Meeker (2006) provides a recent estimate of eBay’ average success rate as 38%, and states that eBay’ s s success rate had been around 50% in 1998 (its IPO). 19 Speci…cally, I use a step size of 0.005 and an error tolerance of 0.001. 20 Types for whom v1 Ev2 S(v2 ; v) < c = 0 will wish to randomize between abstaining and bidding zero. Rather than simulate the randomization, I calculated outcomes under two polar cases: …rst with all such types abstaining, and second with all such types bidding zero, with the winner determined based on the
18

15

monotonicity of v1 S(v2 ; b1 ) b1 in b1 to solve for …rst-round bids for the sequential auction with information revealed.21 In the second round of the sequential auctions, and in the separate auctions, bidders bid their exact values. Once I have obtained the bidding functions for some ( ; ; n) tuple I draw 1,000,000 sets of n bidders’ values from Beta( ; ) and calculate revenues and surplus for each. The results reported in Table 1 are the averages over these 1,000,000 sets of values. Additionally, for each set of values I calculate the revenues and surplus from an e¢ cient Vickrey-Clarke-Groves auction (in which bidders have a dominant strategy of stating their true values).

5.2

Simulation Results
2

The following table summarizes the results of numerical calculations for n = 3 and ; f0:2; 0:5; 0:8; 1; 2; 5g. [Insert Table 1: E¢ ciency and Revenue Performance] 5.2.1 E¢ ciency Performance

The main thing to note is that the Sequential Auction with Information Revealed generates 98.9% of the e¢ cient surplus or greater in all speci…cations.22 Either separating the auctions or hiding information risks ine¢ ciency an order of magnitude larger. How can we explain the consistently-strong e¢ ciency performance of the Sequential Auction with Information Revealed? First, recalling that E(V1:z ) denotes the expectation of the 1st-highest of z draws from the bidder value distribution f ( ); and recalling that the Sequential Auction with Hidden Information implements the greedy allocation, we can write: (3) E(V1:n ) + E(V1:(n
1) )

= Surplus(Seq/Hidden)

Proposition 1

Second, let K be a random variable distributed Binomial(n; 1 ), with Pr(K = k) indicat2
highest …rst-round value. The two cases di¤er only when all n bidders wish to abstain, and so the average di¤erence is quite small. (For the reported simulation parameters, the maximum surplus di¤erence is 0:0009 and the maximum revenue di¤erence is 0:008). Reported results are for the case where all types abstain. 21 Speci…cally, bids are calculated as a weighted average of the two grid points between which v1 b1 S(v2 ; b1 ) crosses zero. 22 Note too that the 98.9% is a measurement of the social surplus created from the idiosyncratic portion of bidders’value functions. As an extreme illustration, if bidders’values are distributed Beta on [100,101], then surplus is greater than 99.98% for all speci…cations.

< |{z}

Surplus (Seq/Revealed)

16

ing the probability that exactly k of n bidders in the Separate Auctions with Information Revealed enter the …rst auction (i.e., their …rst value is higher than their second value). Conditional on k entrants, the highest value in the …rst auction is the highest of 2k random draws from f ( ), and so we can write: (4) Ek [E(V1:2k )+E(V1:2(n
k) )]

= Surplus(Sep/Revealed)

Proposition 2

Finally, because bidders have unit demand we can provide a simple upper bound to the surplus creation of an e¢ cient VCG auction: (5) Surplus (Seq/Revealed) |{z} Surplus (VCG) <
Remark 1

< |{z}

Surplus (Seq/Revealed)

unit demand

Combining (3)-(5) we can provide the following bound on the ine¢ ciency of the eBay auction format: "eBay Ine¢ ciency" = Surplus (VCG) - Surplus (Seq/Revealed) 0
1) ) ; E(V1:n ) Ine¢ ciency Bound: Seq/Hdn

< |{z}

2E(V1:n )

The Sequential Auction with Hidden Information squanders a large amount of social surplus when a bidder with a much-higher second-round value than the other bidders also has a marginally-higher …rst-round value than the other bidders, and so ends up winning the …rst auction instead. This risk is greatest when E(V1:n ) E(V1:(n 1) ); i.e., the marginal expected value of including the nth bidder in the second auction, is large (large …rst di¤erences, e.g., because high values are rare). The Separate Auction with Revealed Information squanders a large amount of social surplus when a bidder with high values for both objects enters the "wrong" auction: the object for which he has a marginally higher value has very strong competition, while the other has very weak competition. This risk is greatest when E(V1:n ) Ek (E(V1:2k )) is large, i.e., when the marginal expected value of including the z th bidder in an auction is highly concave in z (large second di¤erences, e.g., because high values are common). Theorem 1 tells us that the eBay auction format –Sequential Auctions with Information Revealed – performs better than the better of these two alternatives. The insight that explains the eBay format’ robust performance is that the performance of these two alternas tives is in a sense negatively correlated: distributions of bidder values that yield large values of E(V1:n ) E(V1:(n 1) ) yield small values of E(V1:n ) Ek (E(V1:2k )); and vice versa. For any 17

B < min @ E(V1:n ) E(V1:(n | {z

}|

Ine¢ ciency Bound: Sep/Rev

C Ek (E(V1:2k ))A {z }

1

bidder value distribution f ( ), the function E(V1:z ) of z has to satisfy: (i) decreasing …rstdi¤erences (the …rst bidder is more valuable than the second, etc.); and (ii) lim E(V1:z ) = v z!1 (values are bounded above by v). So if E(V1:z ) is very concave in z (has large second di¤erences) it must quickly become quite ‡ in z (has small …rst-di¤erences). at To see this in Table 1, consider the set of rows with n = 3 and = :2. As we increase the distribution becomes more left skewed, i.e., higher values become rarer and E(V1:z ) becomes more ‡ in z. The result is that the e¢ ciency performance of the Separate at Auctions with Information Revealed improves from 89:7% to 98:0%, while the performance of the Sequential Auctions with Information Hidden weakens from 96:0% to 90:1%. By contrast, the performance of the Sequential Auctions with Information Revealed –which is better than the better of these two alternatives – stays pretty constant, at between 98:9% and 99:1%. 5.2.2 Revenue Performance

Referring now to the revenues columns of Table 1, we see that the sequential auction with information revealed generates at least 93.7% of the VCG revenues over the parameters simulated. Revealing information increase total expected revenues over all parameters simulated. Though there exist parameters for which, with information revealed, separating the two auctions increases expected revenues, the overall picture is that the eBay format has the best revenue performance of the taxonomy. 5.2.3 Discussion of Magnitudes

To put the e¤ects of Information and Sequencing in perspective, we note that in a classic paper Riley and Samuelson (1981) calculate that the revenue bene…t of utilizing an optimal reserve price in the sale of a single item to two bidders with values uniform on [0; 1] is 1 5 = 0:0833 = 25%: Here, the e¢ ciency bene…t of Sequencing and Information in the 12 3 sale of two items to three unit-demand bidders with values drawn from the uniform on [0; 1] is 1:44 1:06 = 0:38 = 36%, and the revenue bene…t is 0:753 0:375 = 0:38 = 100%: By contrast, the welfare loss from not using an e¢ cient multi-object auction is 1:45 1:44 = 0:01 < 1%. 5.2.4 Changing the Number of Bidders

Increasing the number of bidders from n = 3 improves the e¢ ciency and revenue performance of all auction formats. All formats bene…t from the increase in the expectation of the highest and second-highest bidder values in a given auction. The Separate auctions bene…t additionally because the risk of an item going unallocated drops quickly in n (the probability 18

of an item going unallocated is 2n1 1 ), and so the di¤erence in performance between the Sequential and Separate auctions gets smaller. For instance, if n = 8, all formats but the Sequential Auction with Information Hidden achieve e¢ ciency performance of 98%. See Table 2. [Insert Table 2: Changing the Number of Bidders] 5.2.5 Changing the Number of Rounds

For robustness, it is desirable to check that the main e¢ ciency results do not depend on the assumption that there are exactly two items auctioned. Increasing the number of rounds beyond two makes calculating equilibrium behavior for the Sequential Auction with Information Revealed di¢ cult. What makes the case of two rounds tractable is that a bidder’ continuation surplus depends on a single auction in s which it is a dominant strategy to bid his value. So we can compute S(v2 ; c1 ) directly: the continuous state variable c1 conveys information about the set of continuing bidders (see fn. 16), and given this set and v2 we can calculate expected surplus. If there are R 3 rounds, then in all rounds but the last continuation surplus needs to be computed recursively, which is complex because there are continuous value inputs and state variables. An additional problem is that bidders’information sets may be asymmetric after the …rst round: if, as on eBay, the second-highest bid is disclosed, then the bidder whose bid this was has a di¤erent information set than do his opponents. Thus, to provide this robustness check we will have to make simplifying assumptions. First, suppose that each value is drawn from a discrete value distribution with V possible values. This introduces atoms and gaps, and so departs from the main model, but it reduces the set of possible types to V R . Second, assume that only the winning bid is disclosed, and not the price this bidder actually paid (i.e., not the second-highest bid). This reduces the number of possible histories (states) to (V R )R 1 . Finally, loosely inspired by Pakes and McGuire (1994), restrict attention to equilibria in which a bidders’bid in round r is a symmetric function of (i) the state; (ii) his value in round r; and (iii) the distribution of his values in rounds r 1:::R; but not the order in which these future values occur. If we set V = 2, then this reduces the number of possible equilibrium bids in round r to 2(R r + 1), and so the set of possible histories is reduced to 2R 1 R!. Given these assumptions, for small enough R the set of possible histories and types is small enough that we can solve recursively for equilibrium behavior. (For R = 2 we can solve the game analytically; see Appendix B). Results for R up to 7 ( 3x105 histories), and for various values of p, the likelihood that a particular bidder value is the lower of the

19

two possible values, are reported in Table 3. [Insert Table 3: Robustness Check: Additional Rounds] Reassuringly, the same qualitative and quantitative features emerge from this analysis. The Sequential Auction with Information Revealed has the strongest e¢ ciency performance of the taxonomy across all parameter values, and always has e¢ ciency of at least 98%.23 Hiding information is most harmful when high-values are rare (p = 0:8), whereas Separating auctions is most harmful when high-values are plentiful (p = 0:2). One new pattern that emerges is that the e¢ ciency performance of the separate auctions weakens with market size (holding the buyer-seller ratio constant). This is because the coordination problem amongst bidders with multiple high values is exacerbated.

6

Conclusion

The original hypothesis of my investigation into internet auctions for imperfect substitutes was: (i) eBay auctions are ine¢ cient because they elicit single-dimensional bids from bidders with multi-dimensional value information (Remark 1); (ii) multi-object auctions are e¢ cient; (iii) therefore eBay should use some kind of multi-object auction format; (iv) but market forces have not corrected this failure because of eBay’ network externalities; (v) which were s won because eBay’ auction design is e¢ cient for collectibles, its original focus. s This paper does not falsify my original hypothesis per se, but does suggest there’ more to s the story than Remark 1. The extent to which the lack of simultaneous ascending auctions on the internet should count as a "market failure" depends on the magnitudes of the potential welfare gains. What emerges from this paper is the understanding that the allocative ine¢ ciency of eBay’ auction marketplace may be quite small. Theorem 1 indicates that s the eBay format is the least ine¢ cient out of a real-world-motivated taxonomy of market designs. The simulation results suggest that the eBay format robustly achieves 99% of the maximum expected surplus, whereas each of the other formats can perform an order of magnitude worse. As Klemperer (2002) and Roth (2002) each emphasize, sometimes it is the tiny details – here, Sequencing and Information – that make the largest di¤erence in market design. A promising direction for future research is to explore the importance of participation costs in auction-marketplace design. In a sense, this paper handles these costs in a highly
23 For R = 2 the Sequential Auction with Information Revealed is fully e¢ cient when there are two values. There can be ine¢ cient allocations for any R 3. For instance, if there are 4 bidders whose types are fHHL; HLH; HLH; LLLg and the HHL type wins round one (i.e., he wins the coin toss against the HLH bidders who bid the same as he) then the allocation is ine¢ cient.

20

reduced form: Separate auctions correspond to a high-cost regime; Sequential auctions to a low-cost regime; and Combinatorial auctions correspond to a regime in which even complex participation is costless. With respect to eBay in particular, there is some evidence that participation costs are quite high (Juda, 2005), and that eBay itself is aware of this: a recent feature, called Bid Assistant, is claimed to save “busy buyers”“valuable time” precisely by , making it easier to participate in a sequence of auctions (with information revealed).

A

Proofs

Proof of Proposition 1. The argument proceeds in two steps. First we show the result assuming that no bidders abstain from the sequential auction with information hidden (that is, v Ev2 S(v2 ; v) > c): Second, we show that the argument is easily adapted to the case where some types may abstain. Case 1: No types abstain from the …rst of two sequential auctions with information hidden. With future-object information hidden, the equilibrium allocation is greedy: whichever bidder has the highest value for the …rst object wins the …rst auction, and whichever remaining bidder has the highest value for the second object wins the second auction. This follows from the independence of values across objects. To prove that revealing information increases e¢ ciency we only need to look at cases where revealing information alters the allocation, i.e. causes a non-greedy allocation. Consider the bidder, i, with the highest …rst-round value: vi1 vk1 ; 8k. Of i’ n 1 s opponents, let bidder j be the one with the highest …rst-round bid in equilibrium with information revealed: br br ; 8k 6= i.24 From (2), we know that for both i and j : j1 k1 (6) br = vi1 i1 S(vi2 ; br ) i1

(7)

br = vj1 j1

S(vj2 ; br ) j1

If br > br , then the allocation is greedy and e¢ ciency has not been a¤ected by the release i1 j1 of future-object information. If br br ; then revealing information has altered the allocation (in the case of ties, it i1 j1 alters the allocation when j wins the coin toss). We need to show that this alteration is e¢ ciency enhancing in expectation.
In the event of a tie for the role of either i or j (which occur with probability zero), we randomize, taking care to select j after the determination of i.
24

21

De…ne the random variable vK2 (x) as the largest second-round value out of n 2 bidders randomly drawn from the set of types who bid weakly less than x in the …rst round. So, given the way we chose j, the largest second-round value of bidders other than i and j is the random variable vK2 (br ). j1 We use the following crediting scheme for social surplus creation for the case where revealing information changes the allocation: Contribution to Social Surplus Seq / Hidden i j k 6= i; j (combined) max(vj2 vi1 vK2 (br )); 0) j1 vK2 (br ) j1 Seq / Revealed max(vi2 vK2 (br ); 0) j1 vj1 vK2 (br ) j1

Notice that the crediting scheme treats the two rounds di¤erently. In the …rst round, the winner (i or j) is credited with his full value. In the second round, the set of bidders k 6= i; j, each of whom bids weakly less than br by construction, is credited with its maximum value. j1 If whichever of i or j loses the …rst round then has a higher value still, he is credited just with his marginal contribution over and above that of the set k 6= i; j: So we need to show that: (8) E max(vi2 vK2 (br ); 0) + vj1 + vK2 (br ) E vi1 + max(vj2 vK2 (br ); 0) + vK2 (br ) j1 j1 j1 j1 By Lemma 2, S(vj2 ; br ) = E[max(vj2 vK2 (br ); 0)] and S(vi2 ; br ) = E[max(vi2 j1 j1 j1 r vK2 (bj1 ); 0)] : a bidder’ expected pro…ts in a single unit auction against a set of n 2 s bidders each of whom bids weakly less than br (the de…nition of the S( ) function) is equal j1 to his expected marginal contribution to social surplus against that same group. So (8) reduces to: (9) S(vi2 ; br ) + vj1 j1 S(vj2 ; br ) + vi1 j1

Using the equilibrium bids (6) and (7), adding and subtracting S(vi2 ; br ), and rearranging i1 terms, this becomes (10) br j1 br i1 S(vi2 ; br ) i1 S(vi2 ; br ) j1

which obtains since by assumption br br and by Lemma 1, dS(vi2 ;x) > 1: So for any j1 i1 dx set of bidders, whenever future-object information induces a non-greedy allocation, surplus is higher in expectation. The e¢ ciency improvement is strict whenever br > br . j1 i1 Case 2: Some types abstain from the …rst of two sequential auctions with information hidden. 22

If there is at least a single bidder who does not abstain from the …rst auction, we can use the argument for Case 1 to show that revealing information increases social surplus. The allocation with information hidden is greedy, and we showed above that the allocation with information revealed improves upon the greedy allocation. We need to worry however about the case where all bidders abstain from the …rst auction, since there will be n rather than n 1 participants in the second auction. Let q be the bidder who submits the highest …rst-round bid in equilibrium with information revealed: br br ; 8j (that such a q exists almost always follows from Zeithammer’ s q1 j1 (2006; Proposition 1) no abstentions result). Adapting the notation from case 1, we will n write vK21 (x) for the random variable equal to the largest second-round value out of n 1 bidders randomly drawn from the set of types who bid weakly less than x: We credit the n 1 bidders other than q; each of whom would bid weakly less than n br with information revealed, with vK21 (br ) of surplus for either information environment. q1 q1 With information hidden, then by construction q also abstains, and we credit him with n max(vq2 vK21 (br ); 0) of surplus. With information revealed, q wins the …rst auction, and q1 we credit him with vq1 of surplus creation. Using (2) and Lemma 2 we have: (11) vq1 = br + S(vq2 ; br ) = br + E(max(vq2 q1 q1 q1 vK2 (br ); 0)) q1

Since the largest of n 1 draws from a distribution is larger than the largest of n 2 n draws from the same distribution E(max(vq2 vK2 (br ); 0) E(max(vq2 vK21 (br ); 0)): q1 q1 r Zeithammer’ (2006) Proposition 1 gives that bq1 is weakly greater than the seller’ salvage s s value. So for any set of n bidders who abstain if information is hidden, revealing information increases expected surplus, as required. Proof of Proposition 2. First, note that if information is hidden, then any equilibrium of the separate auctions must be less e¢ cient than the equilibrium of the sequential auctions. The two sequential auctions have n and n 1 participants, whereas the two separate auctions have just n k and k for some k = 1; :::; n 1, and in any of these auctions the participating bidder with the highest value wins. What remains is to show that the sequential auction with information revealed generates more surplus than the separate auctions with information revealed. Let i be the bidder who wins object one when the auction format is Seq / Revealed. We consider three cases. Case 1: i also wins object one under Sep / Revealed. Clearly the Seq / Revealed generates weakly more welfare than the Sep / Revealed in this case because the set of participants in the auction for the second object under Sep / Revealed is a subset of the set of participants for Seq / Revealed, and in either auction 23

format the bidder with the highest valuation wins the object in equilibrium. Case 2: i enters the auction for object one under Sep / Revealed, but some other bidder j wins object one. This argument is simple once we de…ne an appropriate scheme for crediting social surplus creation to bidders (similar to that used in the proof of Proposition 1). We know that each of the bidders k 6= i; j bids weakly less than br in the Seq / Revealed. i1 Sep Credit this set of bidders with surplus vK2 (br ) in the Seq / Revealed , and with vK2 (br ) in i1 i1 Sep the Sep / Revealed, where vK2 (x) is the random variable equal to the largest second-item value out of n 2 bidders randomly drawn from the set of types who (1) bid weakly less than x in the …rst round, and also (2) value the second object more highly than the …rst. Now credit social surplus according to the following table: Contribution to Social Surplus Seq / Revealed i j k 6= i; j (combined) vi1 max(vj2 vK2 (br ); 0) i1 r vK2 (bi1 ) Sep / Revealed 0 vj1
Sep vK2 (br ) i1 Sep vK2 (br ), what we need to i1

Using Lemma 2 and the observation that clearly vK2 (br ) i1 show is: (12) vi1 + S(vj2 ; br ) i1 vj1

Adding and subtracting both S(vi2 ; br ) and S(vj2 ; br ) and using the equilibrium bids for i1 j1 i and j gives (13) br + S(vi2 ; br ) i1 i1 br j1 S(vj2 ; br ) j1 S(vj2 ; br ) i1

which inequality is seen to obtain using Lemma 1, br br and the observation that i1 j1 r r r S(vi2 ; bi1 ) 0. The e¢ ciency gain is strict whenever bi1 > bj1 . Case 3: i enters the auction for object two under Sep / Revealed, and so some other bidder j wins object one. We use a counting scheme similar to that for the previous case; indeed all of the cells of the surplus crediting scheme are the same but for one. The di¤erence is i’ contribution s to social surplus in Sep / Revealed, which now will be positive since he enters the second Sep auction. This cell is now equal to max(vi2 vK2 (br ); 0). i1 Sep r r Since vK2 (bi1 ) vK2 (bi1 ) for any realized set of bidders k 6= i; j; it follows that E[max(vi2 Sep r Sep vK2 (bi1 ); 0) + vK2 (br )] E[max(vi2 vK2 (br ); 0) + vK2 (br )]: Using Lemma 2, this reduces i1 i1 i1 24

r the problem to showing that S(vi2 ; bi1 ) + vj1 S(vj2 ; br ) + vi1 : Add and subtract S(vj2 ; br ) j1 i1 r r r r and use the equilibrium bid formulas, and we have bi1 bj1 > S(vj2 ; bi1 ) S(vj2 ; bj1 ) which is true by Lemma 1. The e¢ ciency gain is strict whenever br > br . i1 j1 Since these 3 cases are exhaustive, the proof is complete.

Rv It is easy to see that F (v2 jc1 ; v1 ) F (v2 jv; v1 ), and so v F (v2 jc1 ; x)f (x)dx F (v2 jc1 ) F (v2 jv), i.e., the distribution of a second-round opponent’ bid given a …rst-round winning s bid of c1 …rst-order stochastically dominates that given a …rst-round winning bid of v. From this it follows that the distribution of the maximum second-round opponent bid given c1 …rst-order stochastically dominates that given v, and so a bidder with value v2 has a greater expected contribution to social surplus in the second round given c1 than given v. Hence, by Lemma 2, S(v2 ; c1 ) S(v2 ; v): Proof of Proposition 4. First, we consider bidders for whom v1 Ev2 S(v2; v) c; such bidders never abstain. With information hidden, these bidders all shade by a common amount Ev2 S(v2 ; v): We need to show that, for any v1 : (14) bh (v1 ; ) = v1 1 Ev2 S(v2; v) v1 Ev2 S(v2; br (v1 ; v2 )) = Ev2 br (v1 ; v2 ) 1 1

Proof of Lemma 3. Lemma 2 and the symmetry of bidders implies 0 < @S(v22;c1 ) 1 for @v v2 > v because such bidders win the second auction with strictly positive probability. Total dbr (v1 ;v ) di¤erentiation of (2) with respect to v2 , using Lemma 1, then yields 1 dv2 2 < 0 for v2 > v, with weak inequality for v2 = v: Pick an arbitrary v1 . De…ne v2 (v1 ; c1 ) v as the least v2 for which br (v1 ; v2 ) c1 : Now 1 de…ne R v2 f (x)dx min(v2 (v1 ;c1 );v2 ) F (v2 jc1 ; v1 ) = R1 f (x)dx v2

That is, the …rst round bid of a bidder with arbitrary …rst-round value v1 with information hidden is less than the expectation of this same bidder’ …rst-round bid with information s r revealed over all possible realizations of v2 . Since b1 (v1 ; v2 ) v, the result follows directly from Lemma 3. Bidders for whom v1 Ev2 S(v2; v) < c either always abstain or randomize between abstaining and bidding exactly c when information is hidden. Zeithammer (2006, Proposition 1) shows that with information revealed almost all of these bidders bid strictly greater than c. The zero-measure set for whom v1 = v abstains in either information environment.

25

B

A Simple Two-Value Example

To build intuition about the Sequential Auction with Information Revealed we provide a simple two-value example that is solvable in closed form. All is as described in Section 1, except let f ( ) be a probability mass function with f (L) = p and f (H) = 1 p; and v < L < H < v: Note that the two-value model is not a special case of the main model, because it has atoms and gaps. There are four types of bidders, (L; L); (L; H); (H; L); and (H; H): First note that (L; L) and (H; L) types will shade their …rst-round bids by zero, since their second-round pro…ts are zero whatever the outcome of the …rst round. (H; H)0 s bid will only be relevant at the margin against other (H; H) bidders, in the case where there also are not any (H; L) bidders. If there is at least one (L; H) bidder or at least one additional (H; H) bidder, his second round pro…ts if he loses the …rst round are zero. Thus we can explicitly calculate (H; H)’ unique symmetric PBNE …rst-round bid as:25 s (15) br (H; H) = H 1 (n 1)[(1 p)2 ][p2 ]n [1 p(1 p)]n 1 pn
2 1

(H

L) = H

S(H; br (H; H)) 1

Finally, consider type (L; H). His strategy is surprising. We might expect type (L; H) to bid quite cautiously in the …rst round, since his future is attractive and his present bleak. But if he bids any amount less than L, his bid is only relevant at the margin when he faces exclusively (L; H) opponents, i.e., when his future is actually worth zero. Relative to unconditionally losing, marginally losing as an (L; H) conveys especially bad news about second-round pro…t opportunities: S(H; x) = 0 for any x < L: Consequently, for equilibrium to exist in this model we need to assume that there exists an L strictly less than L such that bid amounts in the interval (L ; L) are not permitted. The (L; H) types bid L in equilibrium.26 Example 1 Let L = 10; H = 20; n = 3; and p = 1=2. Then Ev2 S(v2 ; H) = 2:5 and S(H; br (H; H)) = 4: That is, an (H; H) bidder is indi¤erent between marginally winning 1 and marginally losing the …rst auction at his equilibrium bid of 16. E¢ ciency and Revenues It follows directly from the bidding functions that for the twovalue model the eBay auction format is e¢ cient, and it is easy to see that each of the other
This can be found by noting that br (H; H) = H Pr(Y jZ)(H L) = H Pr(Y ) (H L); where: 1 Pr(Z) Y ="I face exactly one (H; H) opponent and n 2 (L; L) opponents: I want to lose Round 1" Z ="I face at least one (H; H) opponent and no (H; L) opponents: My Round 1 bid matters" 26 The analogous result without atoms in the bidder value distribution is that a zero-measure set of bidders – those whose …rst-round values are equal to v – may abstain from the …rst auction (Zeithammer 2006b; Proposition 1). Also note that if L L is large enough then the (L; H) types may randomize.
25

26

auction formats can be ine¢ cient: Proposition A1 For the two-value model, the unique symmetric PBNE of the Sequential Auction with Information Revealed is e¢ cient. Any PBNE of any of the alternate formats in the taxonomy is ine¢ cient. The …rst-round competition e¤ect is surprisingly large in the two-value model. (L; L) and (H; L) types bid their full value instead of shading, and (L; H) types shade by the minimum possible amount. There even exist parameter values for which (H; H) types with Information Revealed bid more than (H; ) types with Information Hidden! For example, if n = 3; p = 0:25, then S(H; br (H; H)) = 11:8% (H L), whereas Ev2 S(v2 ; v) = 18:75% (H L). 1 (This e¤ect obtains for p low enough or n high enough.)27 Proposition A2 (Magnitude of First-Round Competition E¤ect) For the two-value model, for low enough p or high enough n, revealing information in a sequence of two auctions unambiguously increases all types’…rst-round bids, in all states of the world.

References
[1] Athey, Susan and Glenn Ellison, 2008. "Position Auctions with Consumer Search." Working Paper. [2] Athey, Susan, Jonathan Levin and Enrique Seira, 2004. "Comparing Open and Sealed Bid Auctions: Theory and Evidence from Timber Auctions." Working Paper. [3] Bajari, Patrick and Ali Hortacsu, 2003. "The Winner’ Curse, Reserve Prices, and s Endogenous Entry: Empirical Insights from eBay Auctions." RAND Journal of Economics, 34(2), 329-55. [4] Bajari, Patrick and Ali Hortacsu, 2004. "Economic Insights from Internet Auctions: A Survey." Journal of Economic Literature, 42(2), 457-86.
The fact that all types always bid more relies on the atom at (H; H). In the two-value model, when an (H; H) bidder is marginal, he does not know if he is tied at the margin with one additional (H; H) bidder, in which case his second round pro…ts might be H L, or with multiple (H; H) bidders, in which case his second round pro…ts will be zero. Suppose that instead of the atom at (H; H) there is (1 p)2 of density in the region [H ; H + ]2 . Consider type (H ; H + ) : his …rst-round bid is lowest out of all "(H; H)" types, and in particular, if his …rst-round bid is marginal, then the probability that there are other "(H; H)" types who will continue to the second-round is zero. So he will shade his …rst-round bid by approximately S(H; v) = pn 2 (H L), which is more than he would have shaded by if information were hidden. In the limit when = 0, he instead bids according to (15) –the di¤erence is that now when he is marginal he may face other (H; H) types in the second round.
27

27

[5] Beggs, Alan and Kathryn Graddy, 1997. "Declining Values and the Afternoon E¤ect: Evidence from Art Auctions." RAND Journal of Economics, 28(3), 544-65. [6] Budish, Eric and Lisa Takeyama, 2001. "Buy Prices in Online Auctions. Irrationality on the Internet?" Economics Letters, 72, 325-33. [7] Bulow, Jeremy and Paul Klemperer, 1996. "Auctions versus Negotiations." American Economic Review, 86(1), 180-94. [8] Bulow, Jeremy and Paul Klemperer, 2007. "When are Auctions Best?" Working Paper. [9] Cohen, Adam, 2002. The Perfect Store: Inside eBay. Little, Brown and Company. [10] Cohen, Jane and Bob Grossweiner, 2006. "Industry Pro…le: Je¤ Fluhr." Celebrity Access, October 9, accessed on www.stubhub.com. [11] Cramton, Peter, Yoav Shoham and Combinatorial Auctions. The MIT Press. [12] eBay, 1998. eBay Inc. form S-1, July 15. [13] eBay, 2006a. "2006 Analyst Day Presentation." http://investor.ebay.com/downloads/AnalystDay_2006.pdf. Accessed at Richard Steinberg (eds.), 2006.

[14] eBay, 2006b. "eBay Inc. Announces Fourth Quarter and Full-Year 2005 Financial Results." Accessed at http://investor.ebay.com/…nancial.cfm. [15] eBay, 2006c. eBay Inc. form 10-K. Accessed at http://investor.ebay.com/sec.cfm. [16] Ellison, Glenn, Drew Fudenberg and Markus Mobius, 2004. "Competing Auctions." Journal of the European Economic Association, 2(1), 30-66. [17] Engelbrecht-Wiggans, Richard, 1993. "Optimal Auctions Revisited." Games and Economic Behavior, 5, 227-39. [18] Engelbrecht-Wiggans, Richard, 1994. "Sequential Auctions of Stochastically Equivalent Objects." Economics Letters, 44, 87-90. [19] Eyster, Erik and Matthew Rabin, 2005. "Cursed Equilibrium" Econometrica, 73(5), 1623-72. [20] Hagiu, Andrea and Bruno Jullien, 2007. "Designing a Two-Sided Platform: When to Increase Search Costs?" Working Paper.

28

[21] Harrington, Richard, 2006. "Ticket Auction Trend May Cost You." Washington Post, June 2, WE27. [22] Hidvegi, Zoltan, Wenli Wang and Andrew B. Whinston, 2006. "Buy-price English Auction." Journal of Economic Theory, 129(1), 31-56. [23] Juda, Adam I., 2005. "The Sequential Auction Problem: An Analysis and Solution." Harvard University Working Paper. [24] Kaiser, Laura Fisher and Michael Kaiser, 1999. The O¢ cial eBay Guide to Buying, Selling, and Collecting Just About Anything. Simon & Schuster. [25] Klemperer, Paul, 2002. "What Really Matters in Auction Design." Journal of Economic Perspectives, 16, 169-89. [26] Lee, Hanh Ah and Ulrike Malmendier, 2006. "Do Consumers Know their Willingness to Pay? Evidence from eBay Auctions." Stanford University Working Paper. [27] Lewis, Gregory, 2006. "Asymmetric Information, Adverse Selection, and Seller Reputation on eBay Motors." Working Paper. [28] Lewis, Gregory, 2008. "What’ in a Name? s Preparation. Online versus O- ine Repuations." In

[29] McAfee, Preston and John McMillan, 1987. "Auctions with Entry." Economics Letters, 23, 343-7. [30] Meeker, Mary, 2006. "eBay: Listings Data –Reiterate Overweight." Morgan Stanley Research, May 18. [31] Milgrom, Paul, 2004. Putting Auction Theory to Work. Cambridge University Press. [32] Milgrom, Paul and Robert Weber, 1982. "A Theory of Auctions and Competitive Bidding." Econometrica, 50(5), 1089-1122 [33] Miniter, Paulette, 2008. "Bidz.com Battles to Restore Credibility." Accessed at http://www.smartmoney.com/undertheradar/ index.cfm?story=20080320Bidz&a‡ =yahoo&hpadref=1 [34] Pakes, Ariel and Paul McGuire, 1994. "Computing Markov-Perfect Nash Equilibria: Numerical Implications of a Dynamic Di¤erentiated Product Model." RAND Journal of Economics, 25(4), 555-589. 29

[35] Peters, Michael and Sergei Severinov, 2006. "Internet Auctions with Many Traders." Journal of Economic Theory, 130(1), 220-245. [36] Riley, John G. and William F. Samuelson, 1981. "Optimal Auctions." American Economic Review, 71, 381-92. [37] Rochet, Jean-Charles and Jean Tirole, 2006. "Two-Sided Markets: A Progress Report." Rand Journal of Economics, 37(3), 645-67. [38] Roth, Alvin E., 2002. "The Economist as Engineer: Game Theory, Experimentation, and Computation as Tools for Design Economics." Econometrica, 70(4), 1341-78. [39] Roth, Alvin E. and Axel Ockenfels, 2002. "Last-Minute Bidding and the Rules for Ending Second-Price Auctions: Evidence from eBay and Amazon Auctions on the Internet." American Economic Review, 92(4), 1093-1103. [40] Smith, Ethan, 2006. "Ticketmaster Wall Street Journal, Sept 12. Adapts to Rivals’ Online Threat."

[41] Vickrey, William, 1961. “Counterspeculation, Auctions, and Competitive Sealed Tenders.”Journal of Finance, 16(1), 8-37. [42] Vickrey, William, 1962. “Auction and Bidding Games.” Recent Advances in Game Theory. The Princeton University Conference, 15-27. In

[43] Weber, Robert J., 1983. “Multi-Object Auctions.”In Engelbrecht-Wiggans, Richard et al (eds.) Auctions, Bidding and Contracting. New York University Press, 165-91. [44] Zeithammer, Robert, 2006. Goods." Working Paper. "Sequential Auctions with Information about Future

30

Table 1. Efficiency and Revenue Performance of Market Designs for Allocating Imperfect Substitutes
Social Surplus Sequential Revealed Hidden 99.1% 98.9% 98.9% 98.9% 99.0% 99.1% 99.5% 99.1% 99.0% 99.1% 99.1% 99.2% 99.6% 99.3% 99.2% 99.2% 99.2% 99.3% 99.7% 99.4% 99.3% 99.3% 99.3% 99.3% 99.9% 99.6% 99.5% 99.5% 99.4% 99.4% 99.9% 99.8% 99.8% 99.7% 99.7% 99.6% 96.0% 94.3% 93.0% 92.5% 91.1% 90.1% 97.9% 96.9% 96.3% 96.0% 94.8% 93.7% 98.6% 97.8% 97.3% 97.1% 96.3% 95.4% 98.9% 98.1% 97.7% 97.5% 96.8% 96.0% 99.5% 98.9% 98.6% 98.5% 98.0% 97.3% 99.8% 99.5% 99.4% 99.3% 99.0% 98.5% Revenues Sequential Revealed Hidden 94.1% 94.0% 94.3% 94.5% 95.5% 98.1% 94.7% 93.9% 93.7% 93.7% 93.9% 94.1% 95.7% 94.6% 94.3% 94.3% 94.2% 94.3% 96.1% 95.0% 94.8% 94.7% 94.5% 94.5% 97.6% 96.6% 96.3% 96.1% 95.9% 95.7% 98.9% 98.3% 98.0% 97.9% 97.6% 97.3% 82.0% 74.7% 69.8% 67.9% 63.9% 61.4% 85.4% 81.9% 78.6% 77.0% 72.1% 68.4% 88.4% 85.7% 83.3% 82.0% 77.5% 73.5% 89.9% 87.3% 85.5% 84.4% 80.4% 76.2% 93.9% 92.0% 91.0% 90.4% 88.2% 84.8% 97.2% 96.1% 95.5% 95.2% 94.1% 92.4%

n

Simulation Parameters alpha beta 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0.2 0.2 0.2 0.2 0.2 0.2 0.5 0.5 0.5 0.5 0.5 0.5 0.8 0.8 0.8 0.8 0.8 0.8 1 1 1 1 1 1 2 2 2 2 2 2 5 5 5 5 5 5 0.2 0.5 0.8 1 2 5 0.2 0.5 0.8 1 2 5 0.2 0.5 0.8 1 2 5 0.2 0.5 0.8 1 2 5 0.2 0.5 0.8 1 2 5 0.2 0.5 0.8 1 2 5

Multi-Object VCG or SAA 1.648 1.114 0.829 0.708 0.405 0.177 1.891 1.553 1.282 1.143 0.734 0.350 1.941 1.710 1.486 1.360 0.940 0.481 1.956 1.767 1.569 1.453 1.042 0.554 1.981 1.884 1.763 1.683 1.349 0.824 1.993 1.954 1.900 1.860 1.665 1.233

Separate Revealed Hidden 89.7% 94.5% 96.2% 96.7% 97.6% 98.0% 87.5% 90.4% 92.2% 93.0% 94.6% 95.5% 87.3% 89.0% 90.5% 91.2% 92.9% 94.1% 87.3% 88.6% 89.9% 90.5% 92.1% 93.3% 87.4% 87.9% 88.6% 89.0% 90.2% 91.4% 87.4% 87.6% 87.9% 88.1% 88.7% 89.5% 69.1% 64.0% 62.0% 61.2% 59.5% 58.4% 76.5% 71.0% 68.8% 67.8% 65.6% 63.9% 79.7% 74.7% 72.4% 71.5% 69.0% 67.1% 81.0% 76.4% 74.1% 73.2% 70.6% 68.5% 84.0% 80.8% 79.0% 78.1% 75.6% 73.2% 86.0% 84.4% 83.4% 82.8% 81.1% 78.8%

Multi-Object VCG or SAA 0.724 0.308 0.191 0.152 0.074 0.029 1.326 0.753 0.521 0.433 0.234 0.098 1.568 1.047 0.781 0.668 0.389 0.173 1.655 1.184 0.915 0.795 0.484 0.223 1.832 1.534 1.315 1.201 0.843 0.449 1.935 1.800 1.681 1.611 1.338 0.896

Separate Revealed Hidden 85.2% 100.3% 102.2% 102.6% 102.8% 102.9% 63.9% 78.9% 84.8% 86.7% 90.3% 91.8% 58.0% 68.7% 74.1% 76.3% 80.9% 83.7% 56.2% 64.8% 69.8% 71.8% 76.7% 79.9% 52.8% 57.2% 60.1% 61.5% 65.5% 69.1% 51.0% 52.7% 54.0% 54.6% 56.8% 59.5% 44.8% 43.3% 41.9% 41.1% 39.8% 39.2% 44.8% 46.2% 46.4% 46.2% 45.5% 44.8% 45.4% 46.4% 46.8% 47.0% 46.9% 46.5% 45.8% 46.5% 47.0% 47.2% 47.3% 47.0% 47.4% 47.3% 47.4% 47.6% 47.9% 47.9% 48.8% 48.5% 48.5% 48.5% 48.5% 48.6%

Max Median Min

1.993 1.469 0.177

99.9% 99.3% 98.9%

99.8% 97.4% 90.1%

98.0% 90.3% 87.3%

86.0% 73.2% 58.4%

1.935 0.788 0.029

98.9% 94.7% 93.7%

97.2% 84.6% 61.4%

102.9% 70.8% 51.0%

48.8% 46.9% 39.2%

Table 2. Efficiency and Revenue Performance - Effect of Changing the Number of Bidders
Social Surplus Sequential Revealed Hidden 99.3% 99.4% 99.7% 99.7% 99.8% 99.8% 99.1% 99.4% 99.5% 99.7% 99.7% 99.8% 99.4% 99.6% 99.7% 99.7% 99.8% 99.8% 99.1% 99.3% 99.5% 99.6% 99.6% 99.7% 99.6% 99.8% 99.9% 99.9% 99.9% 99.9% 97.5% 98.6% 99.2% 99.4% 99.5% 99.6% 96.9% 98.4% 98.9% 99.3% 99.5% 99.7% 98.0% 98.8% 99.2% 99.4% 99.5% 99.6% 94.8% 97.0% 98.1% 98.5% 98.9% 99.1% 98.9% 99.5% 99.7% 99.8% 99.9% 99.9% Revenues Sequential Revealed Hidden 94.7% 97.2% 98.1% 98.7% 99.0% 99.3% 93.9% 96.5% 97.5% 98.4% 98.8% 99.1% 95.9% 97.8% 98.7% 99.0% 99.2% 99.4% 93.9% 96.7% 97.9% 98.3% 98.7% 98.8% 96.6% 98.4% 99.0% 99.4% 99.6% 99.6% 84.4% 92.9% 95.9% 97.2% 98.0% 98.6% 81.9% 91.6% 94.9% 96.8% 97.6% 98.3% 88.2% 94.4% 96.7% 97.8% 98.3% 98.7% 72.1% 84.9% 91.1% 93.9% 95.5% 96.5% 92.0% 96.8% 98.3% 99.0% 99.3% 99.5%

n

Simulation Parameters alpha beta 3 4 5 6 7 8 3 4 5 6 7 8 3 4 5 6 7 8 3 4 5 6 7 8 3 4 5 6 7 8 1 1 1 1 1 1 0.5 0.5 0.5 0.5 0.5 0.5 2 2 2 2 2 2 0.5 0.5 0.5 0.5 0.5 0.5 2 2 2 2 2 2 1 1 1 1 1 1 0.5 0.5 0.5 0.5 0.5 0.5 2 2 2 2 2 2 2 2 2 2 2 2 0.5 0.5 0.5 0.5 0.5 0.5

Combinatorial VCG 1.453 1.578 1.646 1.702 1.737 1.770 1.553 1.695 1.777 1.831 1.867 1.894 1.349 1.446 1.506 1.554 1.588 1.615 0.734 0.852 0.943 1.010 1.071 1.119 1.884 1.929 1.952 1.964 1.973 1.979

Separate Revealed Hidden 90.5% 93.9% 95.9% 97.1% 98.0% 98.5% 90.4% 93.3% 95.2% 96.5% 97.5% 98.2% 90.2% 94.2% 96.2% 97.5% 98.3% 98.6% 94.6% 95.7% 96.5% 97.4% 98.0% 98.2% 87.9% 93.1% 96.1% 97.7% 98.7% 99.2% 73.2% 77.8% 81.4% 84.4% 86.2% 87.8% 71.0% 76.1% 80.5% 83.4% 86.1% 88.0% 75.6% 80.2% 83.3% 85.7% 87.4% 88.9% 65.6% 68.1% 71.2% 72.9% 74.7% 76.0% 80.8% 87.0% 91.0% 93.5% 95.3% 96.4%

Combinatorial VCG 0.795 1.094 1.268 1.385 1.467 1.533 0.753 1.131 1.367 1.508 1.614 1.689 0.843 1.069 1.191 1.279 1.342 1.389 0.234 0.375 0.494 0.587 0.664 0.730 1.534 1.748 1.837 1.882 1.913 1.933

Separate Revealed Hidden 71.8% 76.4% 82.8% 87.9% 91.2% 93.6% 78.9% 78.4% 82.5% 86.4% 89.9% 92.1% 65.5% 74.6% 83.2% 88.5% 91.9% 94.2% 90.3% 86.9% 88.8% 90.4% 92.3% 93.5% 57.2% 70.3% 80.4% 87.7% 92.2% 95.2% 47.2% 52.4% 59.3% 64.6% 69.5% 73.1% 46.2% 48.9% 54.3% 60.5% 65.2% 69.1% 47.9% 56.0% 64.0% 69.7% 74.0% 77.6% 45.5% 45.8% 49.6% 52.6% 56.0% 58.9% 47.3% 59.9% 70.4% 78.0% 83.2% 87.3%

Table 3. Robustness Check for Efficiency Performance: More Rounds
Social Surplus Sequential Revealed Hidden 100.0% 100.0% 100.0% 99.3% 99.6% 99.9% 99.9% 99.8% 99.9% 100.0% 99.7% 99.6% 100.0% 99.3% 99.2% 100.0% 99.8% 99.2% 100.0% 99.7% 98.8% 100.0% 99.7% 98.5% 100.0% 99.8% 98.3% 98.6% 95.4% 91.5% 98.5% 93.7% 90.3% 99.7% 96.4% 92.9% 99.8% 96.1% 91.9% 99.8% 96.2% 91.2% 100.0% 98.0% 92.5% 100.0% 98.2% 91.8% 100.0% 98.3% 91.3% 100.0% 99.2% 92.5%

n

Simulation Parameters R p 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 10 10 10 11 11 11 2 2 2 3 3 3 3 3 3 4 4 4 5 5 5 5 5 5 6 6 6 7 7 7 7 7 7 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8

Multi-Object VCG or SAA 19.805 17.033 9.260 29.946 27.471 16.357 29.990 28.867 19.094 39.996 39.244 27.706 50.000 49.544 37.013 50.000 49.786 39.828 60.000 59.874 49.824 70.000 69.928 60.177 70.000 69.965 62.481

Separate Revealed Hidden 86.6% 88.6% 96.8% 80.1% 81.6% 93.3% 86.5% 85.2% 92.5% 82.1% 81.4% 88.9% 79.0% 78.6% 86.0% 83.3% 82.5% 86.1% 80.6% 80.2% 83.5% 78.6% 78.4% 81.2% 81.7% 81.5% 82.1% 79.1% 67.9% 58.5% 71.1% 56.5% 44.2% 78.7% 62.1% 45.9% 73.8% 56.1% 38.4% 70.5% 52.7% 33.6% 75.2% 57.2% 35.0% 72.4% 54.3% 31.7% 70.3% 52.4% 29.3% 73.7% 55.8% 30.7%

Notes: Calculations for two-value distribution with High = 10, Low = 0, and seller's cost c << 0 n = number of bidders; R = number of rounds; p = probability that each value is Low



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