Estimating an Auction Platform Game with Two-Sided Entry

Estimating an auction platform game with two-sided entry

Marleen Marra∗ January 1, 2021

Abstract I study endogenous entry of bidders and sellers in an auction platform, and how they respond to platform fee changes. As the platform is more valuable to bidders when more sellers enter, and vice versa, welfare impacts of fee changes are theoretically ambiguous. I exploit equilibrium outcomes of the auction platform game to quantify such network effects and estimate the model with data from a wine auction platform. A particular novel element of the model is that it captures entry of heterogeneous sellers and its relation to bidder entry: higher- value sellers exit when raising fees, increasing the platform’s attractiveness to buyers anticipating lower (reservation) prices. I show that relevant model primitives are identified from variation in reserve prices, transaction prices, and the number of bidders. Results highlight the role of selection in redesigning platforms with listing heterogeneity and reveal pricing strategies to increase both platform profitability and user welfare. JEL codes: D44, C57, L10

Keywords: Ascending auctions, Auctions with entry, Seller selection

∗Sciences Po, Department of Economics, 28 Rue des Saints-P`eres, 75007 Paris, France. E-mail: [email protected]. Tel: +33 (0)695 526 146.

1 How should a peer-to-peer auction platform allocate fees between buyers and sellers? What antitrust damages should be awarded when the platform raises fees anticompetitively? Auction platforms facilitating trade between users necessarily generate indirect network effects as they are more valuable to potential bidders when more sellers enter, and vice versa. The theoretical two-sided market literature highlights that both 1) the platform revenue-maximizing fee structure, and 2) welfare impacts of those fees are theoretically ambiguous and depend on the magnitude of network effects.1 This ambiguity has also proved a bottleneck for antitrust policy. Increasing the seller’s listing fee, for instance, could for some magnitudes be beneficial for all parties when paired with a reduction in bidder cost, if bidders exert a larger indirect network effect than sellers. In this paper, I exploit an original data set of online wine auctions and develop a structural model that allows me to quantify network effects arising from endogenous bidder and seller entry. Specifically, I leverage the transparency of payoffs and actions in the auction game to characterize the value to sellers of an additional bidder, and vice versa.2 After recovering the value distributions of potential bidders and sellers and their entry cost, network effects can be estimated directly for any counterfactual fee structure. This approach maps out the platform’s two-sidedness and allows me to provide a tight quantitative analysis of how fee changes affect both platform profitability and user welfare.3 Accounting for seller selection in the auction game furthermore allows me to capture an important interaction effect of this platform. Bidders expect lower (reservation) prices when lower-value sellers are attracted to the platform, so bidder entry depends both on the expected number and type of sellers that enter.4 As the first empirical auction paper to address selective entry of sellers, a separate contribution is therefore my empirically tractable auction platform game with two-sided entry. Conditional on observed auction-level heterogeneity, values are assumed to be of idiosyncratic, private values nature and independent both across bidders and between

1See e.g. Evans(2003), Rochet and Tirole(2006), and Rysman(2009). 2Empirical two-sided market papers have used quasi-experimental designs (e.g. Cullen and Far- ronato(2020) and Li and Netessine(2020)) or demand models where functional form restrictions play a larger role (e.g. Ackerberg and Gowrisankaran(2006) and Lee(2013)) to estimate network effects. 3Fees: buyer / seller commission, buyer entry fee, seller listing fee, and reserve price fee. 4The importance of this dynamic for auction platform profitability was first postulated in Ellison, Fudenberg and Mobius(2004), but never implemented empirically.

2 bidders and sellers. I show how the model’s equilibrium distributions of reserve prices, transaction prices and number of bidders per listing are endogenous to the fee structure through optimal entry, bidding and reserve pricing strategies. Observed variation in these outcomes allow for the recovery of model primitives needed to estimate fee impacts. It is highly compelling that sellers enter selectively given that they own the so- called “fine, rare, and vintage wine” before listing. Reduced form evidence supports this idea. Specifically, estimates from a Heckman(1976) selection model suggest that lower value (marginal cost) sellers enter first and also optimally sort into setting a zero reserve price. Initial regression analysis also suggests that listings are independent of each other and that bidders learn their values after entering, consistent with the presence of listing inspection cost. This is a meaningful departure from search models previously applied to homogeneous-good auction platform data.5 Estimates from the structural model indeed reveal significant listing inspection cost between 5-9 percent of the second-highest bid.6 Besides fitting my empirical setting, seller selection and listing inspection cost contribute to the empirical tractability of the auction platform game. I show that the model generates a unique entry equilibrium despite its two-sidedness, characterized by a fixed point in seller value space with nested an equilibrium bidder entry threshold. The auction platform game generates the following network effects, where an (indirect) network effect describes the change in expected surplus for a user when an additional user on the same (other) side enters the platform.7 As such, positive indirect network effects on both sides result from the expected sale price and probability being endogenous to the number of bidders per listing. Due to the constant listing inspection cost the game’s equilibrium entry conditions dictate that this type of platform exhibits no scale effects in the number of listings. Specifically, holding

5See e.g. Backus and Lewis(2016), Hendricks and Sorensen(2018), Bodoh-Creed, Boehnke and Hickman(2020), and Coey, Larsen and Platt(2019). 6Listing inspection cost are associated with understanding the wine’s many idiosyncracies such as its provenance, ullage, delivery cost, storage-, insurance-, and return conditions. 7This follows the definition in e.g. Katz and Shapiro(1985) and Evans and Schmalensee(2013) and crucially it captures changes in expected surplus before any entry equilibrium responses arise. The ex-post change in surplus will be different, especially because the entry equilibrium is dictated by zero-profit conditions on both sides. Network effects in this paper are non-linear depending on the baseline platform composition, are driven by latent value distributions and entry costs, and are different for each of the heterogeneous (potential) sellers. Rochet and Tirole(2006) refer to indirect network effects as cross-group externalities.

3 seller types constant, doubling the number of listings doubles the number of bidders on the platform. Seller selection diminishes the indirect network effect on potential bidders. This is because attracting additional sellers, for instance by lowering the listing fee, results in a platform that is populated with higher-reserve setting sellers. These model predictions are consistent with empirical patterns in the data.8 Counterfactual simulations reveal the magnitude of network effects driven by the estimated latent value distributions and entry costs, by exogenously increasing the number of bidders or sellers. Adding one additional bidder is about twice as profitable for the marginal seller as the reverse. Furthermore, 62 (38) percent of potential bidders’ surplus of adding 100 (10) additional listings evaporates because those sellers set higher reserve prices. On the whole, my empirical results underscore the importance of accounting for seller selection when evaluating mechanism design changes in auction platforms. Perhaps the most telling result is that the negative own-side externality of the selection of higher-value sellers makes that, for sellers who remain on the platform, the reduction in surplus of a unit fee increase is less than one as it excludes “lemons” from the platform.9 In the wine auction platform this effect arises because bidders enter based on the expected distribution of secret reserve prices. For example, I estimate that a one pound increase in the listing fee only lowers expected surplus for sellers who remain on the platform by 77-89 pence. This lemons effect is stronger for lower-value sellers and when sellers are more heterogeneous. In fact, it can be exploited to make all users better off by pairing the one pound higher listing fee with a budget-neutral bidder entry subsidy. These results are of interest beyond the studied setting given that many important two-sided markets feature heterogeneous sellers offering idiosyncratic goods.10 The model also facilitates estimation of currently hard to measure antitrust dam-

8Recent quasi-experimental evidence by Cullen and Farronato(2020) and Li and Netessine(2020) support constant and decreasing returns to scale in other peer-to-peer platforms. The listing inspection cost in my auction platform provide a micro-foundation for the absence of positive scale effects in equilibrium despite the presence of positive indirect network effects. 9I use the term “lemons” after Akerlof(1970) to denote high-value and therefore high-reserve price setting sellers, which make the platform unattractive for potential bidders. 10I use the term “idiosyncratic” as in Einav, Levin, Farronato and Sundaresan(2018) to indicate relatively heterogeneous items; used or vintage goods or individualized services. Other (auction) platforms fit this description, such as for vintage goods (Vinted, ClassicCarAuctions), freelance jobs (Upwork, Uship), or loans to borrowers of varying repayment capacity (Prosper, Bondora), although model extensions might be needed to fit each empirical setting.

4 ages from (anti-competitive) fee changes. Results show that estimated damages are larger than in simpler models without (seller) entry. About 60 percent of the loss in user surplus after increasing commissions falls on sellers, regardless of which side is targeted. Unlike previously assumed, also winning bidders are affected: their surplus reduces by 7.5 percent of the counterfactual hammer price when doubling the seller commission. This value is estimated to be only 1 percent when shutting down entry on both sides, and 3.9 percent when only holding the set of sellers constant. The described network effects complicate the platform’s fee setting problem. Low- ering fees increases the number of listings and might boost sales, while populating the platform with higher reserve price listings and hence fewer bidders per listing. On top of that, with sunk entry cost, the platform needs to determine how to best allocate fees between the two sides (the canonical two-sided market question in Rochet and Tirole(2006)) and trade off increasing the volume of sales with the share they take from it in the form of commissions. Additional simulations shed light on these otherwise hard to quantify trade-offs and show that alternative fee structures can increase platform revenues by more than 40 percent. It is particularly striking that winning bidders should be given a discount on the transaction price, paired with a higher seller commission or listing fee. A negative buyer commission would certainly be innovative for auction platforms but resemble pricing in other two-sided markets, such as cash-back policies on credit cards or free drinks for early club-goers. Below marginal cost pricing is in fact consistent with subsidizing users that generate larger indirect network effects (Rysman(2009)). To study impacts on platform composition that go beyond heterogeneity in seller tastes, I estimate separate model primitives for a sub-sample of the data with more high-end wines. With pricing for sellers essentially being a two-part tariff, counterfactual results illustrate differences between increasing the unit (listing) or percentage (seller commission) fee. Holding expected platform revenue constant, pairing a buyer discount with an increase in unit fee results in a platform with a larger share of high- end listings but a smaller profit share from that type of listing than when pairing it with an increased unit fee.

Relation to the literature. I build on the large and inﬂuential literature on nonparametric identiﬁcation and estimation of auction models. Especially related are structural models accounting for endogenous bidder entry, including Roberts and

5 Sweeting(2013), Moreno and Wooders(2011), Krasnokutskaya and Seim(2011), Li and Zheng(2009), Fang and Tang(2014), and Gentry and Li(2014). In line with the literature standard, these take the perspective of one seller or assume seller homogene- ity, while Elyakime, Laffont, Loisel and Vuong(1994), Larsen and Zhang(2018), and Larsen(2020) account for seller heterogeneity but no entry. Distinctively, I address endogenous entry of heterogeneous sellers and show how equilibrium entry decisions of bidders and sellers are interconnected in an auction platform setting. In a model without unobserved heterogeneity, I use the distribution of reserve prices to identify latent seller values. In a similar vein, Roberts(2013) uses it to control for unobserved heterogeneity common to buyers and sellers in a model with homogeneous sellers (and no entry). Also related are papers estimating demand in large auction markets. Backus and Lewis(2016) propose a dynamic model that also accounts for bidder substitution across heterogeneous goods and apply it to estimate demand for compact camera’s on eBay. Hendricks and Sorensen(2018) study bidding behavior for iPads with a model of sequential, overlapping auctions. To estimate the demand for Kindle e- readers, Bodoh-Creed et al.(2020) employ a dynamic search model with bidder entry. Coey et al.(2019) model time-sensitive consumer search in the presence of both full (posted) price and discounted (auction) listings of new-in-box items.11 While our emphasis on the auction’s wider platform environment is similar, these papers focus on the dynamics associated with large markets of relatively commoditized goods and rely on steady-state requirements for tractability. Instead, I exploit the constant listing inspection cost inherent to the idiosyncratic nature of the wine, allowing me to estimate a (static) two-sided auction platform model with seller heterogeneity. Other related work uses eBay data to research economic phenomena (e.g. Anwar, McMillan and Zheng(2006), Nekipelov(2007), and Dinerstein, Einav, Levin and Sundaresan (2018)), but none structurally estimate impacts of auction platform fees. Another distinct but relevant literature studies network effects and pricing in two- sided markets (e.g. Ackerberg and Gowrisankaran(2006), Rysman(2007), Lee(2013), Fradkin(2017), Cullen and Farronato(2020), Li and Netessine(2020)), building on an influential theoretical literature. A fundamental difference with these papers is that I use the auction structure to quantify the platform’s attractiveness to bidders when there are more sellers, and vice versa. As such, payoffs from the auction plat-

11They also evaluate the impact of changing the listing fee in their model.

6 form game provide a micro-foundation of its network eﬀects that can be simulated under counterfactual (fee) policies. Related are also Athey and Ellison(2011) and Gomes(2014), who model the two-sidedness of position auctions.

The paper proceeds as follows. Section1 presents the data and institutional details. The auction platform model with two-sided entry and its equilibrium strategies are given in Section2. Section3 discusses nonparametric identiﬁcation of model primitives and provides a computationally-feasible estimation strategy. Estimation results, model ﬁt and validation are presented in Section4 and Section5 presents results from counterfactual simulations. Section6 concludes.

1 Institutional details

What is commonly termed “fine, rare, and vintage wine” is sold at auction in secondary markets, run by online wine platforms as well as brick-and-mortar auction houses. Especially the online market is growing rapidly to a 75 million dollar market; global revenues increased by 42 percent between 2011 and 2017 and again 30 percent in 2018.12 Auction data for the empirical analysis in this paper comes from online auction platform www.BidforWine.co.uk (BW). It offers a peer-to-peer marketplace for buyers and sellers to trade their wine and caters (now) to over 20.000 users. BW is one of 8 UK wine auctioneers recognized by The Wine Society.13 Importantly, none of the other 7 intermediaries provide the peer-to-peer format format but instead work with consignment to trade on behalf of sellers. This comes with additional shipping cost and value assessment by the intermediary, worthwhile only for higher-end wine. This naturally positions BW at the lower end of the market.14 BW is therefore taken to be a monopolist in the UK secondary market for lower-end fine wine as its sellers cannot readily substitute to Bonhams or Sotheby’s when BW raises fees. To the extent that there are local P2P marketplaces for those products, their presence is captured by the opportunity cost of trading on BW. Items are sold through an English (ascending) auction mechanism with proxy

12Sources: Financial Times and Wine Spectator. 13The others: Bacchus, Bonhams, Chiswick, Christies, Sotheby’s, Sworders, Tennants. 14Seller-managed P2P listings are the focus of this paper. BW does also oﬀer consignment services when selling a large collection exceeding ﬁve cases, or for exclusive wines.

7 bidding.15 A soft closing rule extends the end time of the auction by two minutes whenever a bid is placed in the ﬁnal two minutes of the auction. Therefore, there is no opportunity for a bid sniping strategy (bidding in the last few seconds, potentially aided by sniping software) on the BW platform. The combination of proxy bidding with a soft closing rule suggests that the data is well approximated by the second- price sealed bid model.

Fees. For successful sales, sellers receive payment from the winning bidder, ship the wine, and are invoiced for the amount of seller commission, listing fee and reserve price fee due. For these seller-managed lots and during the relevant time period, BW

charges no buyer premium (cB) and maintains a seller commission (cS) of 10.2 percent for sale amounts below 200 pounds, and 9 percent for the marginal amount between 200 and 1500 pounds. Regardless of a successful sale, sellers are charged a listing

fee (eS) of 2.1 pounds, a minimum bid fee (0.6 pounds but only when increasing the minimum bid) and reserve price fee (0.3 pounds but only when setting a secret reserve). These fees include 20% VAT. Buyer premium and seller commission are charged as a percent of the transaction price. For example, a seller is charged 7.2 pounds after selling a bottle for 50 pounds without reserve. For completeness, my analysis also includes a buyer entry fee eB that is currently set at 0.

1.1 Data description

I construct a dataset of wine auctions by web-scraping all open auctions on BW at 30-minute intervals between January 2017 and May 2018. At these intervals, I observe most of what bidders observe as well. Observed wine characteristics include the type of wine (red, white, rosé,sparkling, or fortified), grapes, vintage, region of origin, delivery and payment information, storage conditions, returns and insurance, seller ratings and feedback, fill level of the bottle, and the seller’s textual description. Summary statistics are reported in table1. Only a third of listings is created by a seller with feedback from previous transactions, pointing to the consumer-to- consumer nature of the platform. Seller identities are observable but bidder identities

15Bidders submit a maximum bid and the algorithm places bids to keep the current price one increment above the second-highest bid. When the highest bid is less than one increment above the second highest bid, the transaction price remains the second highest bid. This is diﬀerent from the rule at eBay; see Hickman, Hubbard and Paarsch(2017).

8 are unobserved except for those bidders who have left feedback after winning the auction. I also scraped the profile pages of all users ever registered. When defining a potential seller as a member who has listed a wine for sale at least once, only 279 out of 2,591 potential sellers created a listing during the sample period. Labelling all 13,176 remaining users as potential bidders, this also points to entry on the bidder side. Even under the extreme assumption that all auctions are populated by different bidders, a total of 10,856 actual bidders would be counted. In my analysis I treat bidders and sellers as distinct groups of users, but this is an abstraction: the data shows that 44 out of the 247 feedback-leaving winning bidders have also listed a wine for sale. The sample includes 3, 500 auctions after excluding auctions that are consigned, sell spirits, or sell multiple lots at once. While there is a significant range in sale prices, 81 percent of auctions fall in the lowest seller commission bracket (≤ 200 pounds). I focus on these auctions (“main sample”) and estimate the model separately for “high- end” auctions with prices between 200 and 1500 pounds, to assess the importance of fee changes for the platform’s composition. The repetitive recording of bids for ongoing auctions was necessary to approximate the reserve price distribution. When the seller sets a reserve price without making it public in the form of a minimum bid amount, the notifications “reserve not met” or “reserve almost met” accompany any standing price that does not exceed the reserve. I approximate the reserve price as the average between the highest standing price for which the reserve price is not met and the lowest for which it is met.16 While only 26 percent of listings has an increased minimum bid amount, 45 percent has a (secret) reserve price, and 4 percent has both. The use of secret reserve prices in auction platforms remains a puzzle in the empirical auction literature and solving that puzzle is beyond the scope of this paper (see Jehiel and Lamy(2015)). In the rest of this paper I group them together and refer to the “reserve price” as the maximum of: the minimum bid amount and the approximated secret reserve price. Of larger consequence is the choice made by a third of sellers to refrain from setting any form of reserve. This is observable to bidders by a “no reserve price” button -

16If all bids would be recorded in real time, this approximation would be accurate up to half a bidding increment due to the proxy bidding system. Also the 30-minute scraping interval results in a good approximation of the reserve price distribution (see appendix).

9 Table 1: Descriptive statistics

N Mean St. Dev. Min Pctl(25) Median Pctl(75) Max Hammer price 3,500 148.10 302.45 0 40 82.1 170 6,400 Number of bidders 3,500 3.10 2.51 0 1 3 5 13 Number bottles 3,500 3.71 4.23 1 1 2 6 72 Is sold 3,500 0.64 0.48 0 0 1 1 1 Price per bottle if sold 2,235 76.40 129.52 0.33 17.12 35.00 83.33 2,200.00 Has reserve price 3,500 0.67 0.47 0 0 1 1 1 Seller has feedback 3,500 0.29 0.46 0 0 0 1 1 Seller has ratings 3,500 0.86 0.35 0 1 1 1 1 even before they enter the listing. Correspondingly, the model results in equilibrium number of bidder distributions that diﬀer between these two listing types.

1.2 Network eﬀects of entry into platform

Before presenting a theoretical model of the auction platform with two-sided entry, it is important to get a sense of the type of network effects generated by its user interactions. The ideal type of variation to do so would be exogenous variation in platform fees or in the populations of potential users. Such variation is not available in the data, also motivating why I rely on structural analysis to derive network effects in counterfactual simulations. In my view, the structural approach is a plus as it reveals precisely how network effects are generated while resulting in a richer set (a continuum) than what could have been obtained with a limited number of exogenous shocks. However, lack of such shocks to the equilibrium mean that the empirical patterns presented in this section are based on sub-optimal variation in the data over time. I first exploit the time dimension of the data, considering months as separate realizations of the auction platform game’s equilibrium. The first empirical fact to highlight is a positive correlation between the total number of bidders on the platform and the number of listings. A simple linear regression shows that each additional listing is associated with about 3 additional bidders (columns 1 and 2 of table2). For the purpose of this reduced form analysis a market is thus defined as a month, and a product is defined as the combination of high-level filters used on the platform: type of wine, region of origin, and vintage decade.17 The positive correlation is also

17For example, red Bordeaux from the 1960s and non-vintage Champagne are diﬀerent products.

10 Table 2: Empirical evidence network eﬀects

Dependent variable: Total bidders product/market Number bidders per listing (1) (2) (3) (4) Number listings product/market 2.920 3.095 −0.011 0.001 (0.073) (0.138) (0.008) (0.015) No-reserve only No Yes No Yes Observations 1,230 451 3,500 1,149 Adjusted R2 0.874 0.933 0.306 0.291

Standard errors in parenthesis. Results from OLS regressions including product fixed effects, defined as: (region x wine type x vintage decade) corresponding to high-level filters. present without controlling for product fixed effects and when exclusively looking at auctions without reserve price. While basic, the empirical fact is consistent with the presence of positive indirect network effects on the platform.18 This is not surprising: such effects should arise mechanically in auction platforms from the mere fact that transaction prices are endogenous to the number of bidders per listing. As bidders sort over listings, a platform with more listings increases expected payoffs for potential bidders, and vice versa. However, the average number of bidders per listing turns out to be independent of the number of listings (columns 3 and 4 of table2). This suggests the absence of scale effects in the number of listings when holding seller types constant, because if additional listings create some sort of additional payoff the equilibrium number of bidders per listing would be higher as well. Constant returns to scale might be surprising to readers having different two-sided market models in mind, such as one in which buyers look for a specific item and more listings increase the probability of finding that item.19 But it is consistent with quasi-experimental empirical evidence in other peer-to-peer platforms (Cullen and Farronato(2020) and Li and Netessine(2020)), driven by market frictions. In my setting, with unvetted listings of vintage wines, it is reasonable that bidders need to inspect each listing’s many product idiosyncracies

18Indirect network eﬀects describe that a product is more valuable to users when it is more widely adopted by another group (Katz and Shapiro(1985), Evans and Schmalensee(2013)), also referred to as cross-group externalities (Rochet and Tirole(2006)). 19For settings where match quality is considered an important property of the market, it would introduce additional complexities to the structural analysis. Deltas and Jeitschko(2007) show that the platform’s proﬁt are in that case discontinuous in the listing fee, leading to unstable equilibria.

11 before knowing how much to value the wine.20 The structural model presented below shows that such listing inspection cost indeed generate the constant returns to scale property of the platform when holding seller types constant. Constant listing inspection cost also imply that listings are independent of each other, even when being similar in product space or ending in close proximity of each other. A regression analysis supports this empirically. The presence of more competing listings does not systematically affect the average: i) number of bidders per listing (as documented above), ii) number of bids per bidder, iii) transaction prices, and iv) reserve prices.21 Results are consistent across the 18 different definitions of what constitutes a competing listing, employed to avoid missing dependencies by focusing on erroneous cuts of the data.22 A logical consequence is non-selective bidder entry where bidders learn their values after inspecting the auction characteristics. Selective bidder entry would show up in the data as markets with more listings of a certain product, attracting more total bidders, having (stochastically) lower bids as high-value bidders enter first. OLS regression results are consistent with non-selective bidder entry: while an extra bidder in an auction is associated with a transaction price that is roughly 10 pounds higher, more total bidders / a larger market does not significantly affect the price.23 It is worth emphasizing that constant listing inspection cost associated to the idiosyncratic nature of the products also dissipate any continuation value of bidding in an auction. Besides being the best representation of the data and being consistent with the absence of scale economies on other peer-to-peer platforms, this assumption delivers empirical tractability to my auction platform model that is complicated with entry of both bidders and sellers. This feature clearly sets the environment apart from auction models with dynamic or static search elements, and without seller selection (or entry), that have fittingly been estimated for more commodity-like products.24 One

20To wit, certain wines are better when mature and, conditional on ageing well, increase in value as fewer of them remain uncorked over time. But being perishable, storage conditions and provenance are key to deliver this potential quality. Listing pages report the wine’s storage conditions, provenance, ullage, and other quality indicators that are not provided in the brief landing page excerpts. In wine auctions, ullage refers to visible oxidation: “Base of Neck” is better than “Top Shoulder” in Bordeaux-style bottles. Burgundy-style bottles have a metric classification (see appendix). 21See the appendix for the complete estimation results. 22Specifically, I define a competing listing as one whose auction ends within a rolling window of i) 30 days, ii) 7 days, or iii) 2 days of the listing in question and that offers the same product, with product definitions ranging from any wine to one of five combinations of high-level filters. 23See the appendix for the complete estimation results. 24Structural auction (platform) models have been applied to study compact cameras (Backus and

12 Table 3: Thin markets

Variables: — Percentiles: 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Times product listed, 4 weeks: 1 1 1 1 1 2 2 3 6 37 Times product listed, 15 mths: 1 1 1 1 2 3 4 8 16 230 Times title occurs, 15 mths: 1 1 1 1 1 1 1 1 2 17 Conservative product definition: (region x wine type x vintage decade), corresponding to high-level filters. distinguishing feature of an auction platform with idiosyncratic goods is therefore that, at each point in time, it contains a low number of highly similar listings. That is a valid description of my data. Even with coarse product definitions being the high-level filters discussed above, for 50 percent of listings on BW there is only one such product offered in the same month and for another 20 percent there are only three of these products available (table3). Sellers on this peer-to-peer platform are individual collectors with private values (marginal cost) for the wine, and a subset of them sets a (secret) reserve price for the wine they list for sale. Attracting lower cost sellers to the platform should therefore increase the expected payoff to bidders and increase their entry, so that both the number and type of sellers on the platform generate indirect network effects on bidders.25 Estimation results from a Heckman two-stage selection model suggest that lower cost potential sellers are indeed more likely to enter the platform, and that those who enter are more likely to have cost draws low enough to set a zero reserve price. The exclusion restriction central to the test is that the number of potential sellers might influence sellers’ entry decisions in a given market (month), but conditional on entry should not affect their cost and hence the reserve prices they set.26 Assuming that

Lewis(2016)), Kindle e-readers (Bodoh-Creed et al.(2020)), iPads (Hendricks and Sorensen(2018)), pop CD’s (Nekipelov(2007)), CPU’s (Anwar et al.(2006)), and iPods (Adachi(2016)). 25Ellison et al.(2004) were the first to mention such an effect, hypothesizing that seller selection contributed to why Amazon and Yahoo! struggled in some countries: having no listing fees attracted non-serious (high reserve price) sellers, in turn shunning bidders. Providing an empirically tractable model of a platform with seller selection is a particularly novel aspect of my paper. 26This is the seller equivalent of the exclusion restriction in Roberts and Sweeting(2013), who test for bidder selection into USFS timber auctions. Other variables included in the first-stage entry decision that are plausibly excluded from sellers’ cost conditional on entry are: the share of markets the potential seller entered, the number of listings it has in other markets, the date it created an account on the platform, and the number of other users that created an account in the same month. Estimation results from the first stage entry model are reported in the appendix.

13 Table 4: Heckman selection model: second-stage results

Dependent variable: Estimated residual reserve Setting a zero reserve (1) (2) (3) (4) Inverse Mills Ratio −64.646 −57.205 0.083 0.104 (14.027) (14.046) (0.019) (0.016) First stage entry model basic elaborate basic elaborate Observations 3,500 3,500 3,500 3,500 F Statistic 21.241 16.587 17.926 18.196 Standard errors in parenthesis. Estimation results from first stage reported in the appendix. the reserve price increases in a seller’s value (lemma2), the second stage regresses the reserve price on the estimated Inverse Mills Ratio from a first stage Probit regression of the decision to enter. A confounding factor is that conditional on entry, seller values driving reserve prices are left-censored if sellers optimally sort themselves into setting no reserve price (as in e.g. Jehiel and Lamy(2015) and my theoretical model). I propose two solutions. In columns 1 and 2 of table4 the dependent variable is the residual reserve price after conditioning on auction observables, imputed to the lowest estimated value for all sellers who set a zero reserve.27 As this might lead a somewhat mechanical selection result, in columns 3 and 4 the dependent variable is a dummy variable indicating that the seller sets a zero reserve price. Estimates in columns 2 and 4 are based on a more elaborate first stage entry model, as detailed in the appendix. All four regression models suggest that sellers enter BW selectively.28

2 An auction platform with two-sided entry

This section develops an empirically tractable structural auction platform model, based on the presented empirical patterns in the BW data, and solves for the game’s equilibrium strategies. Speciﬁcally, the model captures: selective entry of sellers, positive indirect network eﬀects generated by entry, which result in constant returns to scale in equilibrium when holding the seller type distribution constant. These features are based on a parsimonious set of assumptions that are expected to describe

27The residual is obtained from a linear regression of the reserve price on auction observables. 28Due to the fact that sellers own the wine (sometimes for decades), less pertinent than when thinking about bidder selection into auctions is the question how selective seller entry is; e.g. how much of their value is already known and how much needs to be learned from inspecting the good.

14 uniform sorting bidders over listings Entry stage Auction stage

pot. sellers: pot. sellers: pot. bidders: sellers: bidders: learn cost entry decision entry decision set reserve learn value, bid

Figure 1: Timing of the game

ﬁrst-order aspects of other (auction) platforms for idiosyncratic goods as well.

2.1 Model

Consider a monopoly platform with second-price sealed bid auctions to allocate indi- visible goods among bidders with unit demands. Risk-neutral users face homogeneous opportunity cost of time spent on the platform, on top of any monetary fees charged. For bidders, these entry cost are referred to as “listing inspection cost” associated with each listing they enter. Sellers set non-negative secret reserve prices. The presence of a positive reserve price is the only thing that bidders observe before entering, motivated by highly visible “no reserve price” button attached to such auctions on the landing page that lists the various ongoing auctions. The incomplete information structure and strategic interaction makes this setting suitable to study with a game-theoretic framework. It is worth noting that the two-sided entry feature with seller selection are particularly novel elements of the model and the focus of this paper.

Timing. I model this setting as a two-stage game with in the ﬁrst stage sequential entry where listings are created before bidders enter. Listings are ex-ante identical up to the reserve price button, so conditional on this event bidders are sorted with some constant probability over listings.29 The auction stage is standard: sellers set a secret reserve price and bidders bid after learning their values. The timing of the game is represented schematically in ﬁgure1. To simplify the exposition I work with two separate potential bidder populations, distinct only by their preference for positive or no reserve auctions.30 As mentioned

29This is a minor simplification: results go through unchanged conditional on the website’s high- level filters if bidders are thought to consider only a subset of wines when entering the platform. 30Results are identical when potential bidders choose between the two types of listings. In that case, potential bidders enter zero and positive reserve auctions to the point of being indifferent

15 earlier, bidders have zero continuation values as listing inspection cost are associated with each separate listing. Hence, losing bidders are indiﬀerent between exiting the market or playing the game again and entering in another listing.31

Notation. The platform fee structure f = {cB, cS, eB, eS, eR} contains respectively the buyer premium and seller commission (both are shares of the transaction price), buyer entry fee, listing fee, and reserve price fee. Entry cost (opportunity cost of o o time) for potential bidders in no-reserve (eB,r=0) and positive reserve price (eB,r>0) o B B S auctions are allowed to diﬀer, and for sellers are denoted by eS. Nr=0, Nr>0, N ,

Tr=0, Tr>0 denote the number of: potential bidders for no reserve auctions, potential bidders for positive reserve auctions, potential sellers, and listings (sellers) in no and positive reserve auctions. N B and N S denote the sets of potential bidders and sellers, B B B N = Nr=0 + Nr>0 and T = Tr=0 + Tr>0. Random variables are denoted in upper case and their realizations in lower case.

FV0 and FV denote the valuation distributions for potential sellers and bidders.

The empirical analysis controls for auction-level observables so V0 and V should be interpreted as conditional valuations, and the model assumes no unobserved auction- level heterogeneity. I interchangeably refer to V0 as a seller’s (marginal) cost. The population distributions are allowed to diﬀer, and satisfy:

Assumption Two-sided IPV. All i = {1, ..., N B} potential bidders independently S draw values vi from V ∼ FV and all k = {1, ..., N } potential sellers independently draw values v0k from V0 ∼ FV0 such that: 0 B B S vi ⊥ vi0 ∀i 6= i ∈ N and vi ⊥ v0k ∀i ∈ N and ∀k ∈ N , and FV and FV0 satisfy regularity conditions: fV (x) supp(V )=[v, v¯], supp(V0)=[v0, v¯0], FV is absolutely continuous, and increases 1−FV (x) in x ∀x ∈ [v, v¯] (Increasing Failure Rate)

Most importantly, this assumption states that conditional on the vector of observed product attributes, variation in values across buyers and sellers is of a purely idiosyncratic -private values- nature. In addition, the idiosyncratic variation is inde-

between the two. Just as in my model they would enter into the two types of listings to the point of depleting all expected surplus, as per the zero proﬁt entry condition. 31Empirically, I cannot distinguish between bidders who enter the platform for the ﬁrst time and losing bidders from other listings as bidder identities are unobserved.

16 pendent.32 Valuation distributions, allocation mechanism, population sizes, and all cost (fees and entry cost) are common knowledge.

2.2 Equilibrium strategies

I next solve for equilibrium strategies by backwards induction. I restrict attention to symmetric Bayesian-Nash equilibria in weakly undominated strategies requiring that strategies are best responses given competitors’ strategies and that beliefs are consistent with those strategies in equilibrium.33 Omitted proofs are provided in the appendix.

2.2.1 Auction stage

Conditional on entry decisions and the sorting of bidders over listings, the idiosyncratic- good auction platform is made up of independent second-price sealed bid auctions. I therefore derive standard reserve pricing (as in Riley and Samuelson(1981)) and bidding (as in Vickrey(1961)) strategies, up to the impact of buyer premium and seller commission. Lemma 1. A bidder with valuation v bids:

v b∗(v, f) ≡ (1) 1 + cB

Proof. This follows directly from Vickrey(1961): bidding more may result in negative utility and bidding less decreases the probability of winning without aﬀecting the transaction price.

Auctions without reserve price attract more bidders, but the beneﬁt of setting a positive reserve price increases in the seller’s value. Combined with a positive reserve price fee, the set of sellers that sets a zero reserve price is determined by a threshold- R crossing problem (as in Jehiel and Lamy(2015)). In what follows, v0 indicates the no-reserve screening value:

32 Independence and continuity are needed for identification of FV but can be omitted on the seller side. IFR guarantees uniqueness of the optimal reserve price and that listing-level bidder surplus decreases in the number of bidders (lemma 3). The IPV assumption abstracts from affiliation in players’ cost or values conditional on auction observables; structural analysis of a common values model is not feasible (as per the negative identification results in Athey and Haile(2002)). 33I exclude a no-trade equilibrium where no seller enters and therefore no bidder enters. Excluding this equilibrium would also be achieved by restricting attention to subgame-perfect BNE.

17 R 34 Lemma 2. A seller with valuation v0 ≥ v0 sets a reserve price solving:

∗ ∗ v0 1 − FV ((1 + cB)r (v0, f)) r (v0, f) = + ∗ (2) 1 − cS (1 + cB)fV ((1 + cB)r (v0, f))

Note that, if cS = cB = 0, the optimal reserve price is identical to the Riley and Samuelson(1981) public reserve price in auctions with a ﬁxed number of bidders. ∗ Because r (v0, f) is secret, it does not aﬀect the number of bidders in the seller’s listing. This is true for any reserve price strategy of competing sellers, and generally the entry equilibrium results are therefore valid as long as r∗ is monotonically increasing in v0. The optimal reserve price is increasing in cS and (given IFR) decreasing in cB. In what follows, I denote a buyer premium-adjusted optimal reserve price byr ˜: ( (1 + c )r∗(v , f) for v > vR r˜ = B 0 0 0 R 0 for v0 ≤ v0

2.2.2 Entry stage

∗ Any entry equilibrium of this game consists of two bidder entry probabilities (pr>0 ∗ and pr=0) as potential bidders learn values after entering (as in the random bidder entry model of Levin and Smith(1994)) and a seller entry threshold as sellers know their values before listing (as in the selective bidder entry model of Samuelson(1985)). I will simply refer to entry probabilities and threshold as strategies. Specific to the two-sidedness of the auction platform is that the bidder entry probability for positive reserve price listings depends on the seller entry threshold. The reason is that the reserve price distribution that bidders expect to draw from upon entry increases in a first order stochastic dominance sense when higher-value sellers populate the platform. This effect does not exist for bidder entry into zero reserve listings, although the game generates positive indirect network effects so that

34 R v0 is taken to be exogenous. Endogenizing it complicates estimation of the game, while early R analysis suggested little impact. Theoretically, endogenizing v0 merely strengthens the importance of seller selection for bidder entry in r > 0 auctions: as bidder entry into r > 0 auctions becomes R more attractive, the number of bidders per r > 0 listing increases, v0 would adjust downwards, which in turn results in a stochastically lower reserve price distribution and hence encourage additional R bidder entry into r > 0 auctions. Endogenizing v0 would be especially interesting to evaluate changes in the height of the (ﬂat) reserve price fee or the introduction of alternative policies such as a proportional fee. A more detailed analysis of the reserve price choice is left for future research, and might provide additional insight into unresolved puzzles regarding the use of secret reserve prices (see e.g. Jehiel and Lamy(2015) and references in Hasker and Sickles(2010)).

18 the entry probability in both cases depends on the number of listings. Results are derived under a large population approximation, which guarantees empirical tractability of the game and has the added beneﬁt that the assumption that players know the exact population sizes is relaxed.35

Assumption. The population of potential bidders is large relative to the number of B B ∗ ∗ bidders on the platform: (Nr>0,Nr=0) → ∞ and (pr>0, pr=0) → 0. Without the assumption, for r ∈ {r = 0, r > 0} the number of bid- B ders per listing given Nr potential bidders and Tr listings would be distributed B B Binomial( Nr p , Nr p (1 − p )), for bidder entry strategy p . With the assumption, Tr r Tr r r r the distribution is approximately Poisson, fully characterized by its mean:

B ∗ Lemma 3. For r ∈ {r = 0, r > 0}, with with large Nr and small pr, the number of bidders per listing has a probability mass function approximated by:

exp(−λ )λk f (k; λ ) = r r , ∀k ∈ +, (3) Nr r k! Z

The equilibrium λr is endogenous to the fee structure and in positive reserve auctions also depends on seller selection. In what follows, let πb(n, f, v0) be the expected listing-level bidder surplus in an auction with n − 1 competing bidders, fee structure f, when the seller has a value of v0 (unknown to bidders, to be taken an expectation of), and πs(n, f, v0) the expected listing-level seller surplus in such an auction. I slightly abuse notation to let πb(n, f, 0) denote expected bidder surplus in a listing without a reserve price. The entry equilibrium relies on the following properties:

Lemma 4. Listing-level expected surplus for bidders, πb(n, f, v0), decreases in n and v0, and for sellers, πs(n, f, v0), increases (decreases) in n (v0).

In the remainder of this section I ﬁrst show that any candidate seller entry ∗ threshold (v ˜0) maps to an equilibrium λr>0(f, v˜0), and then I use that mapping to ∗ ∗ solve for v0(f). It turns out that because λr>0(f, v˜0) is strictly decreasing inv ˜0 sellers’ best response entry thresholds satisfy the single-crossing property, so that the entry game reduces to a single agent discrete choice problem with a unique

35This approximation does not drive equilibrium existence and uniqueness; the appendix presents results for the original game including complete speciﬁcations of listing-level payoﬀs.

19 equilibrium.

Bidder entry. The bidder entry equilibrium is characterized by the λr>0 (λr=0) that solves potential bidders’ zero proﬁt condition in positive (zero) reserve price auctions.

In the case of r > 0, Πb,r>0(f, v˜0; λr>0) denotes potential bidders’ expected surplus from entering the platform. Besides fees and opportunity cost of time, it includes listing-level surplus πb(n, f, v0) in expectation over: 1) seller-values V0 given candidate 36 thresholdv ˜0, and 2) the Poisson-distributed number of competing bidders:

Πb,r>0(f, v˜0; λr>0) = Z R o E[πb(n + 1, f, v0)|V0 ∈ [v0 , v˜0]]fN,r>0(n; λr>0) − eB − eB,r>0 (4)

In the zero reserve price case, Πb,r=0(f; λr=0) does not depend on seller values: Z o Πb,r=0(f; λr=0) = πb(n + 1, f, 0)fN,r=0(n; λr=0) − eB − eB,r=0 (5)

∗ Lemma 5. For any candidate seller entry threshold v˜0, a unique equilibrium λr>0 solves potential bidders’ zero proﬁt condition in positive reserve auctions:

∗ λ (f, v˜ ) ≡ arg + {Π (f, v˜ ; λ ) = 0} (6) r>0 0 λr>0∈R b,r>0 0 r>0

∗ And a unique equilibrium λr=0 solves:

∗ λ (f) ≡ arg + {Π (f; λ ) = 0} (7) r=0 λr=0∈R b,r=0 r=0

Proof. Listing-level surpluses πb(n, f, v0) and πb(n, f, 0) strictly decrease in n (lemma

4) and fN,r>0(n; λ) increases in a ﬁrst-order stochastic dominance sense in λ. Entry decisions are conditional onv ˜0 or independent of it so the result also follows from Levin and Smith(1994), Ginsburgh, Legros and Sahuguet(2010).

Note that the above holds for any realization of (random variable) Tr>0 givenv ˜, and also for any Tr=0. Central for my analysis of the two-sided entry equilibrium is the following, more striking, result.

36 Πb,r>0(f, v˜0; λr>0) is independent of Tr>0, as conditional on the number of competing bidders in a listing, the number of other listings on the platform does not aﬀect bidder surplus.

20 ∗ Lemma 6. The equilibrium fN,r>0(n; λr>0(f, v˜0)) decreases in the ﬁrst-order stochas-

tic dominance sense in v˜0.

Proof. Candidate seller entry thresholdv ˜0 aﬀects Πb,r>0(f, v˜0; λr>0) only through the

distribution of reserve prices in those listings. A higherv ˜0 draws in sellers with

higher values that set higher reserve prices (lemma2), resulting in lower πb(n, f, v0) ∗ (lemma4). The zero proﬁt condition in (6) therefore dictates that λr>0(f, v˜0) strictly

decreases inv ˜0.

∗ Seller entry. The seller entry equilibrium is characterized by the v0 that solves ∗ the zero proﬁt entry condition for the marginal seller. Let Πs(f, v0; λr>0(f, v˜0), v˜0) R S denote expected surplus for a seller with valuation v0 > v0 when N − 1 competing 37 sellers enter the platform if and only if their valuation is less than thresholdv ˜0. It involves: 1) their listing-level expected surplus, and 2) an expectation over the

number of bidders per listing givenv ˜0 and bidders’ equilibrium best-response to this threshold: Z ∗ ∗ o Πs(f, v0; λr>0(f, v˜0), v˜0) = πs(n, f, v0)fN,r>0(n; λr>0(f, v˜0), v˜0) − eS − eS (8)

Lemma 7. A unique equilibrium seller entry threshold solves the marginal seller’s zero proﬁt condition:

∗ ∗ v (f) ≡ arg {Πs(f, v˜0; λ (f, v˜0), v˜0) = 0} (9) 0 v˜0s.t.FV0 (v ˜0)∈(0,1) r>0

∗ with λ (f, v˜ ) ≡ arg + {Π (f, v˜ ; λ ) = 0} as defined in (6). r>0 0 λr>0∈R b,r>0 0 r>0 Proof. The proof requires three parts. First, sellers have a unique best response for ∗ any competingv ˜0, because Πs(f, v0; λr>0(f, v˜0), v˜0) strictly decreases in their own v0. ∗ Second, given that 1) λr>0(f, v˜0) is strictly decreasing inv ˜0 (lemma6), and 2) entry of competing sellers does not affect seller surplus in other ways, the best response function is strictly decreasing in competing sellers entry threshold. Third, symmetry ∗ then delivers a unique equilibrium threshold, v0(f), that is the fixed point in seller value space solving (9) i.e., making the marginal seller indifferent between entering and staying out.

37 ∗ Using fN,r>0(n; λr>0(f, v˜0)) avoids introducing additional notation to capture that sellers care about competing bidders +1. This is without loss: the two distributions are identical by the envi- ronmental equivalence property of the Poisson distribution (Myerson(1998)).

21 Lemma’s5-7 describe the equilibrium of the game, which can be summarized as:

Proposition 1. For any fee structure f, the entry equilibrium of the auction platform game subject to the large population approximation is characterized by the triple of ∗ ∗ ∗ ∗ (v0(f), λr>0(f, v0(f)), λr=0(f)) that uniquely solve zero proﬁt conditions of potential bidders and the marginal seller.

The appendix contains a graphic description of the entry equilibrium for further intuition about its uniqueness property, and discusses model extensions. In general, ∗ any platform model that results in fN,r>0(n; pr>0, v˜0) FOSD decreasing inv ˜0 as in lemma6 delivers a unique two-sided entry equilibrium. Note that indirect network eﬀects are non-linear and deﬁned by how much ex-

pected surplus from entering the platform (Πb,r>0,Πb,r=0, and Πs) changes if additional users on the other side enter, before equilibrium adjustments. They create a clear trade-oﬀ for the platform that will be explored further in counterfactual simulations. Lower fees increase the number of listings and boost the sales volume, but higher fees populate the platform with lower reserve price listings and more bidders per listing. On top of that, with sunk entry cost and a seller listing fee, the platform needs to determine how to best allocate fees between the two sides.

Πb,r>0,Πb,r=0, and Πs depend crucially on the two value distributions and on latent entry cost. For example, if sellers are relatively homogeneous, reﬂected by low

dispersion in values drawn from FV0 , the seller selection channel is less important. In that case the benefit from adding additional listings might outweigh the cost of attracting higher-value sellers for the platform. A primary task for the remainder of the paper is therefore to recover the model primitives that pin down the magnitude of network effects and drive how changes in fees affect outcomes of interest.

3 Empirical strategies to recover model primitives

My equilibrium analysis highlights that seller selection and its impact on bidder entry in positive reserve auctions is of central importance for the empirical results, and that for any fee structure we can solve for a unique equilibrium characterized by ∗ ∗ ∗ ∗ (λr=0, λr>0(v0), v0) given model primitives. Treating the no-reserve screening value R v0 as structural avoids having to solve for another equilibrium threshold value. It also R implies that the part of the support of V0 < v0 is irrelevant in counterfactuals that

22 restrict attention to the case where at least one seller would find it profitable to set a positive reserve price. The screening value is simply recovered as the lowest implied seller values in positive reserve price auctions. The rest of this section discusses how R FV0 (v0)−FV0 (v0 ) o o o R FV , FV0|v0≥v = R , and latent entry cost (eS, er>0, er=0) are identified and 0 FV0 (v0 ) estimated. Key endogenous variables to do so are the number of actual bidders (A), the second-highest bid (B), and the reserve price (R). Random variable A departs from the allocated number of bidders N in auctions with a positive reserve price, to allow for (secret) reserve prices to censor bidders. Exogenous observables are denoted by X = {f, Z}, with Z the rich set of auction-level observables that account for auction-level heterogeneity.

3.1 Nonparametric identiﬁcation

Athey and Haile(2002, Theorem 1) prove identiﬁcation of FV in an ascending auction model that places identical restrictions on this distribution up to the presence of binding reserve prices. Identiﬁcation of FV therefore follows in my setting from observing

N and the empirical distribution FB in auctions without a reserve price. It relies on the equilibrium bidding strategy, and inverting the function mapping the resulting distribution of the second-highest value to its parent distribution. Speciﬁc assumptions underlying the result are the absence of unobserved heterogeneity (conditional on X) and that all interested bidders have the opportunity to place a bid.38

Given identiﬁcation of FV , each reserve price maps to that seller’s value by re- arranging the equilibrium reserve price strategy in (2):

1 − FV (r(1 + cB)) v0(r) = (1 − cS) r − (10) (1 + cB)fV (r(1 + cB))

38This is in line with the literature standard on analysis of ascending auction data. My data does not meet the stronger requirements to apply new identification methods for a bidding model with unobserved heterogeneity, which rely for instance on exogenous shifters in bidder participation (Hernández,Quint and Turansick(2020)) or the observation of multiple bid order statistics (e.g. Freyberger and Larsen(2017), Luo and Xiao(2020)). Roberts(2013) uses variation in reserve prices to control for unobserved heterogeneity but require sellers to be homogeneous. My rich set of auction observables furthermore explain a remarkably large share of the variation in second-highest bids, as documented below, minimizing the potential impact of unobserved heterogeneity. Furthermore, while Hickman et al.(2017), Platt(2017), and Bodoh-Creed et al.(2020) address intra-auction dynamics, my abstraction from this feature has relatively small effects on counterfactual estimates given that I explicitly model the entry decision of ex-ante identical potential bidders. Further details are provided in the appendix when discussing estimation.

23 where v0(r) denotes the seller value implied by reserve price r assuming equilibrium 39 play. The distribution of v0(r) is equal to the distribution of seller values conditional R ∗ on entering and setting a positive reserve price, so that ∀v ∈ [v0 , v0]:

F R (v) V0≥v0 Fv0(r)(v) = ∗ (11) F R (v ) V0≥v0 0

∗ ∗ Without identifying variation in v0 and unless v0 =v ¯0 and all potential sellers enter,

the population distribution F R is not nonparametrically identiﬁed on the part of V0≥v0 ∗ its support exceeding v0. However, the right-truncated distribution of potential seller values is the foundation for any counterfactual that reduces expected seller surplus, including unilateral fee increases. Counterfactuals show that this is the relevant part of the support in the BW context.

Entry cost are identiﬁed from the three zero proﬁt conditions, as Πb,r>0,Πb,r=0,

and Πs given equilibrium play are revealed in the data up to and strictly decreasing in entry cost. This does rely on knowing the equilibrium entry strategies, which ∗ are recovered as follows. v0 is revealed as the maximum of seller values implied by ∗ ∗ (10), λr=0 is equal to the mean A in zero reserve auctions, and λr>0 as the value

that maximizes the likelihood of the observed B and A given FV and the Poisson distribution of the number of potential bidders per listing. Additional details are provided in the estimation section. To summarize, results are based on the premise that the data is generated by equilibrium strategies of one iteration of the two-sided entry game. The distribution of bidder values is identified from variation in second-highest bids. The distribution of seller values is identified on the relevant part of its support by variation in reserve prices, but parametric restrictions are needed to extend identification to higher values. Seller entry cost are simply recovered as the amount that makes the marginal seller indifferent between entering and staying out, and bidder entry cost are identified as the amounts that, given bidder and seller values, justify the observed levels of participation.

39Note that it is not strictly necessary that sellers play the optimal Riley and Samuelson(1981) reserve price strategy: the identiﬁcation result applies to any known strategy. For uniqueness of the two-sided entry equilibrium I only strictly require that v0(r) is monotonically increasing in r.

24 3.2 Estimation method

Strategies to estimate model primitives closely follow the presented identiﬁcation

arguments. To extrapolate beyond the support on which F R is identiﬁed, and to V0≥v0 estimate FV independent of the number of bidders, I parameterize the latent value distributions and focus on estimating their ﬁnite-dimensional parameters. To account for auction-level heterogeneity, potential bidder and seller values are taken to satisfy ˜ ˜ ˜ R the following single-index structure: ln(V ) = g(Z) + V , and ln(V0|V0 ≥ v0 ) = g(Z) +

V0, with (V,V0, Z) mutually independent, the g(Z) term interpretable as the wine’s

quality, and V and V0 the idiosyncratic taste components. Estimation of bidder parameters by MLE is straightforward and in line with previous analysis of ascending auction data. To estimate the value distribution across auctions I ﬁrst homogenize the second-highest bid (as in Haile, Hong and Shum(2003)). ˜ ˜ For all bidders i: ln(Vi) = g(Z) + Vi, so that: ln(V(n−1:n)) = g(Z) + V(n−1:n). With cB = 0 in the data and given equilibrium play, qualityg ˆ(Z) is estimated by regressing the log of the second-highest bid on auction characteristics in auctions with r = 0 and with more than one bidder. The residual plus intercept deliver homogenized / 40 residualized values Vn−1:n, forming the basis of the likelihood function.

The empirical CDF’s of V (for diﬀerent number of bidders) and V0 (on the observed part of the support) show departures from symmetry, with bidder tastes being slightly left-skewed seller tastes quite signiﬁcantly so. Taste distributions are there- 2 2 fore parameterized as: V ∼ GGD(µb, σb , κb) and V0 ∼ GGD(µs, σs , κs), with GGD the

40 Speciﬁcally, let Tr0 denote the set of listings with a zero reserve price and h(.|nt, zt; θb) the density of homogenized transaction prices given the number of bidders nt. For all auctions with a zero reserve price, and with cB = 0 in the data, it is simply the probability that the homogenized second-highest bid bt is the second-highest among nt draws from FV . Hence ∀t ∈ Tr0:

nt−2 h(bt|nt, zt; θb) = nt(nt − 1)FV (bt; θb) [1 − FV (bt; θb)]FV (bt; θb) (12)

The log likelihood of bidder parameters given data is speciﬁed as: X L(θb; {nt, zt, bt}t∈Tr0 ) = ln((h(bt|nt, zt; θb))) (13)

t∈Tr0

25 Generalized Gaussian Distribution.41 It allows for additional flexibility relative to the Normal distribution, with values of κ > 0 (κ < 0) introducing skewness to the left (right). 2 Recovering the parameters of the seller taste distribution θs = (µs, σs , κs) is more ∗ complex as they depend on v0 that itself is a function of θs. A second issue stems ∗ from v0 being the solution to a fixed point problem with a nested threshold-crossing problem (equation9), making full maximum likelihood estimation (computationally) ˆ0 infeasible. To address this, I first obtain an initial estimate θs by maximum con- ˆ0 ˆ centrated likelihood estimation, then solve for the entry equilibrium given θs and θb, and re-estimate seller parameters using the resulting seller entry threshold. Specifi- cally, I exploit the mapping of equilibrium reserve prices to homogenized seller values (equation 10):

!! 1 − F (ln(˜r ) − g(ˆz ); θˆ ) vˆ = ln (1 − c ) r − V t t b − gˆ(z ), (14) 0t S t ˆ ˆ t (1 + cB)FV (ln(˜rt) − g(zt); θb) withr ˜t = rt(1 + cB) denoting the buyer premium-adjusted reserve price andv ˆ0t the implied conditional seller value in auction t. The density of implied seller values given ∗ ∗ entry threshold v0, rt, and zt (h(ˆv0t |v0, rt, zt; θs)) equals ∀t ∈ Tr>0:

F R (ˆv0t; θs) ∗ V0≥v0 h(ˆv0t|v0, rt, zt; θs) = ∗ (15) F R (v ; θs) V0≥v0 0

R As mentioned before, the no reserve screening value is simply estimated asv ˆ0 = ˆ0 min({vˆ0,t}t∈Tr>0 ). The initial θs maximize the resulting likelihood function concen-

41The GGD(µ, σ2, κ) has PDF:

φ(y) f(x; µ, σ2, κ) = ,with φ(.) the standard normal PDF and σ2 − κ(x − µ) x − µ 1 κ(x − µ) y = {κ = 0} + − ln(1 − ) {κ 6= 0} σ2 I κ σ2 I

26 42 trated atυ ˆTr>0 = max({vˆ0,t}t∈Tr>0 ):

X ∗ L(θs; {vˆ0,t, rt, zt}t∈Tr>0 , υˆTr>0 ) = ln(h(ˆv0t|v0 =υ ˆTr>0 , rt, zt; θs)) (16) t∈Tr>0 ˆ 0 θs = arg max L(θs; {vˆ0,t, rt, zt}t∈Tr>0 , υˆTr>0 ) (17)

∗ ˆ ˆ 0 The next steps are to compute the entry equilibrium to recover v0 given (θb, θs ) and ˆ re-estimate θs accordingly. The appendix provides details about the computation of the entry equilibrium. This algorithm resembles the Aguirregabiria and Mira(2002) nested pseudo likelihood (NPL) estimator, albeit with a nested concentrated likelihood estimator derived from the optimal reserve price strategy to recover structural parameters. NPL is more widely used as a solution to solving parameters involving ﬁxed point characterizations in the estimation of discrete choice entry games.43 o o o Entry coste ˆB,r>0,e ˆB,r=0, ande ˆS are estimated as the values that equal expected ˆ ˆ surplus from entering at the estimated (θb, θs) and given the computed entry equilibrium at estimated parameters. It is based on numerical approximations of expected surplus (4) and (5) for potential bidders and (8) for the marginal seller. Finally, to account for potential unexplained variation in the entry process, I allow for an additional share (p0,r>0 ≥ 0) of listings to attract no bidders. It is estimated by maximimizing the likelihood of the observed joint distribution of number of actual bidders (A) and the second-highest bid, given a (generalized) Poisson distribution of the number of bidders per listing (N). The empirical distribution of N in zero reserve auctions show that no such ﬂexibility is needed there. The appendix provides additional details.

42 ∗ ∗ υˆTr>0 is a consistent estimate of v0 withυ ˆTr>0 → v0 as Tr>0 → ∞ at the true population parameters, by the law of large numbers, asymptotically over multiple iterations of the game. 43The applicability of the algorithm is not surprising given similarity of the seller entry decision to a discrete choice entry game. Roberts and Sweeting(2010) previously applied NPL to an auction setting. Importantly, Pesendorfer and Schmidt-Dengler(2010), Kasahara and Shimotsu(2012) and Egesdal, Lai and Su(2015) provide conditions under which NPL does (not) converge to the true equilibrium. A best-response stable equilibrium is a suﬃcient condition for the algorithm to converge and this is certainly guaranteed by my game reducing to a single agent (marginal seller) discrete choice problem with unique equilibrium. As any number of iterations results in a consistent estimator ˆ θs, I solve for the equilibrium only once.

27 Table 5: Estimated structural parameters

2 2 Taste parameters µˆb σˆb κˆb µˆs σˆs κˆs 3.389 1.020 0.084 3.453 0.851 0.260 Main sample (0.033) (0.003) (0.003) (0.037) (0.006) (0.007) high-end sample 5.303 0.313 -0.615 5.570 0.477 -0.477 (0.041) (0.004) (0.011) (0.042) (0.011) (0.020) o o o R Entry parameters eˆB,r>0 eˆB,r=0 eˆS pˆ0,r>0 vˆ0 6.562 6.865 11.081 0.045 0.165 Main sample (0.172) (0.186) (0.288) (0.001) (0.043) high-end sample 16.954 17.984 14.764 0.152 4.860 (0.733) (0.795) (0.978) (0.006) (0.045) Estimates based on 2770 observations in the main sample and 598 observations in the high-end sample. Standard errors based on 100 nonparametric bootstrap repetitions are reported in parenthesis.

4 Estimation results

Estimates from the homogenization step show that the coefficients of key variables are as expected. Prices are higher for bottles sold by case, and conditional on this case effect the price is lower the more bottles are included in the lot. Specialized temperature-controlled warehouses and special format bottles (e.g. magnums) are sold with a premium. All fill levels that are not the best deliver (weakly) lower prices.44 It is important to point out that the rich set of auction observables obtained through web scraping explains a remarkably large share of total price variation: the adjusted R-squared is 0.491 for the main sample and 0.917 for the smaller high-end sample. This compares favorably even with how much price variation can be explained by observables in auction data with more homogeneous goods.45 It is especially encouraging given the notorious difficulty to address unobserved heterogeneity in ascending auctions (Hernándezet al.(2020)).

Table5 reports the rest of the estimated structural parameters. Estimation of θs

excludes the 9.2 (5.1 in high-end sample) percent of sellers for whichv ˆ0t is estimated

to be negative. Moreover, bothu ˆ0,t andg ˆ(Z) are trimmed at their 1st-99th percentiles

44Estimation results are reported in the appendix. Included variables relate to: type of wine, region of origin, number of bottles, special format bottles, the auction month, storage in a temperature- controlled warehouse, delivery cost/conditions, returns and insurance, payment options, seller ratings, and ullage. I also include variables obtained with textual analysis of the seller’s description. 45For instance, Bodoh-Creed, Boehnke and Hickman(2017) state that the 0-15% of variation with simple OLS regressions is representative of low predictive power in the literature, and they increase the ﬁt to on average 48% with elaborate random forest estimation and machine learning techniques.

28 (a) FV (b) Bn−1:n (c) FV0

(d) r > 0 (e) fN,r=0 (f) fV0 , high-end

Figure 2: Model fit to minimize the impact of outliers. The model fits the data well as illustrated by the various plots in figure2. Plots (b) and (d) of figure2 include draws of estimated quality to simulate second-highest bids and reserve prices, which are out-of-sample predictions for the reserve price sample. The second-highest bid is moreover simulated in expectation over the number of bidders per listing. As another measure of model fit I compute the mean absolute deviation between observed and predicted second-highest bids separately for n = {2, 3, .., 10} bidders: mean absolute deviations are small between 0.016-0.742 pounds and there is no clear pattern by number of bidders. Two-sample Kolmogorov-Smirnov tests furthermore cannot reject the null that observed and predicted reserve prices are drawn from the same population distribution (p-value 0.2). Plot (e) displays the good fit of the assumed Poisson distribution with the esti- ˆ∗ mated λr=0, versus its empirical distribution. It is of particular interest that the data does not reveal any overdispersion relative to the Poisson distribution. This indicates that while preferences for high-level characteristics (filters) might vary across the population of potential bidders, the uniform sorting over listings assumed in estimation

29 captures first order effects of entry behavior in the BW data. A chi-square goodness of fit test formally fails to reject that N is generated by a Poisson distribution (p-value 0.2). The high-end sample does contain some underdispersion and the test barely fails to reject the null (p-value 0.06).46 Estimated listing inspection cost are significantly higher in the high-end sample. But relative to the second-highest bid, estimated entry cost are higher in the main sample: 9 percent versus 5 percent in the high-end sample. Estimates do in both cases correspond to the idea that listing inspection cost are significant in this idiosyncratic goods environment. Setting no reserve price attracts on average 1.2 additional bidders. It makes intuitive sense that this participation differential is larger in the high-end sample (1.9). The probability of being the sole entrant and winning the more expensive wine for the 1 pound opening bid is more valuable. o o Another source of model validation comes from comparinge ˆB,r=0 withe ˆB,r>0. While they are allowed to be different, there is no reason to suspect that it is significantly more time-intensive to inspect listings with or without a reserve price if the reserve price does itself not reveal any information about the quality of the item. o o Indeed,e ˆB,r=0 ande ˆB,r>0 are statistically indistinguishable both the main and high- o o end sample. Recall thate ˆB,r=0 ande ˆB,r>0 are computed in two different cuts of the data as the values that justify the observed participation levels (e.g. that respectively ˆ ˆ set Πb,r=0(.) = 0 and Πr>0(.) = 0), based on λr=0 and λr>0 that are estimated with different methods. This suggests that the parsimonious model provides a plausible description of behavior and payoffs on this platform.

5 Counterfactuals

This section uses model estimates to simulate the auction platform game in counterfactual scenario’s. Each variation on the fee structure requires solving for a new entry equilibrium and auction outcomes. But ﬁrst I would like to emphasize the centrality of “commission index” α = cB +cS . 1+cB Ginsburgh et al.(2010) show that expected platform revenue (and bidder and seller

46This could be explained by (some) bidders entering auctions with a lower standing price, rather than randomly. Counterfactual simulations abstract from such behavior insofar as there are departures from the uniform matching assumption in the high-end sample.

30 (a) Platform revenue (b) Sales volume

Figure 3: Illustrating the commission index and revenue-volume trade-oﬀ

The game is estimated on a grid of: cB × cS (cB = {−0.3, −0.2, −0.1, 0, 0.1}, cS = {−0.1, 0, 0.1, 0.2, 0.3}) and interpolated linearly. Values are normalised by baseline levels and are based on parameter estimates from the main sample.

47 surplus) are independent of (cB, cS) as long as α remains constant. Hence only the commission index and flat fees matter for the platform revenue-maximization problem. Plot a of figure3 confirms that simulated counterfactual platform revenue levels line up perfectly with the commission-index level lines (in orange). However, this is as much as theory can tell us: the combinations of cB and cS that keep platform revenue and user surplus constant. In what follows, I use model estimates to address the two key indeterminacy’s of two-sided markets: 1) what fee structures improve platform profitability, and 2) how do fee changes affect user welfare? A second remark is that the platform faces a trade-off between maximizing revenues and the volume of sales. Intuitively, increasing fees lowers the sales volume but increases the share of that volume paid to the platform. In the case of commissions, it is important to note that this holds even when keeping α constant. For example, increasing cB and decreasing cS such that α is unchanged would lower the volume because bidders scale down their bids (lemma1) while at the same time the reserve price and sale probability are unaffected (as shown by Ginsburgh et al.(2010)). 48 Plot

47Ginsburgh et al.(2010) do not model seller entry or heterogeneity, but their result applies here ∗ since the marginal seller’s expected surplus (and hence v0 ) remains constant as well. 48 Note that platform revenue = volume×(cB +cS) + income from (eS, eB, eR). Even when keeping

31 b of ﬁgure3 illustrates this point: simulated volume levels decrease when moving up

(higher cB) along the commission index level lines. Similarly, increasing the listing fee generates more revenue but depresses the sales volume by lowering the number of listings on the platform. Fee structures that increase platform revenue at the ex- pense of reducing volume (by much) are considered unattractive, and to avoid adding structure on platform growth dynamics, I report a non-parametric volume constraint alongside platform revenues.49

5.1 Seller selection and indirect network eﬀects

The impact of fee changes depend on entry elasticities of potential bidders and sellers and hence on network effects generated by user interactions on the platform. I first simulate homogenized auctions according to equilibrium strategies and estimated parameters, while altering either the number of bidders (M) or sellers (T , incorporating selection) on the platform. I then estimate expected bidder and seller surplus. Results are reported in table6 for various M and T , and are all based on the main sample with r > 0 for illustration. To put a value on the usual definition of indirect network effects: adding one additional bidder to the platform increases expected surplus of the marginal seller by 0.0022 pounds, and this effect is twice as large as the reverse. One benefit of the structural analysis is that it relaxes the assumption that these network effects are constant. In fact, results show significant heterogeneity. Lower-value sellers benefit more: the indirect network effect is 50 percent larger for the median potential seller than the marginal one (at the 85th percentile). Increasing the number of sellers also has a weaker effect on bidders than reducing it. These results are driven by the estimated bidder and seller taste parameters, which impact the importance of the seller selection channel. For example, little dispersion in seller tastes / reserve prices increases the indirect network effect of attracting additional sellers. Estimates indicate that taste distributions are such that seller selection plays a significant role on the BW platform. When adding 100 listings, the fact that these are higher-reserve listings reduces the positive indirect network effect

cB +cS entry constant, and hence , the sales volume decreases in cB. 1+cB 49The trade-off is crucial in any scenario where the volume of sales affects future revenues, for instance through word of mouth or brand awareness. See also Evans and Schmalensee(2010) who explain why startups focus on network growth in early years using a platform model with myopic users, no switching cost, and significant indirect network effects.

32 Table 6: Estimated indirect network eﬀects

Exogenous change number sellers (T): -100 -40 -20 -10 +10 +20 +40 +100 Effect on Πb -0.11 -0.056 -0.028 -0.014 0.013 0.025 0.045 0.080 Exogenous change number bidders (M): -100 -40 -20 -10 +10 +20 +40 +100 Effect on Πs (marginal seller) -0.206 -0.082 -0.041 -0.021 0.021 0.041 0.082 0.205 Effect on Πs (median seller) -0.377 -0.151 -0.075 -0.038 0.038 0.075 0.150 0.376 Simulations based on r > 0 homogenized auctions in the main sample, where M + 100 corresponds to M + 1.25% bidders and T + 100 corresponds to T + 5.34% sellers. on bidders by 62 percent. Similarly, the reduction in bidder surplus is 47 percent less when removing 100 listings relative to the no-selection benchmark.

5.2 Welfare impacts

Being able to quantify welfare effects of fee changes in a two-sided market is of imme- diate policy relevance. While it is widely understood that both sides of the market are affected by price changes on either side, the difficulty to quantify network effects has been a bottleneck for applying antitrust policy to two-sided markets.50 To illustrate a key aspect of two-sided markets with seller (listing) heterogeneity, I first simulate the effect on sellers of increasing the listing fee by 1 pound. In a model that ignores entry, expected surplus for all sellers on the platform would decrease by 1 pound and no other user groups would be affected. Instead, when recomputing the equilibrium with two-sided entry, the expected surplus for sellers who remain on the platform decreases by less than 1 pound. The higher listing fee excludes some of the highest-value sellers from the platform, increasing expected surplus for potential bidders and driving up the number of bidders per listing. One could describe this negative network effect of seller entry on other sellers (e.g. an own-side externality) brought about by a positive indirect network effect resulting from entry of additional bidders as a “lemons effect” after Akerlof(1970), as it is the exclusion of high-reserve setting sellers from the platform that drives it.51

50See e.g. Bomse and Westrich(2005), Tracer(2011), Evans and Schmalensee(2013). For example, in this eBay case, sellers claiming that eBay charged supracompetitive fees were denied a class action suit due to the absence of a method to quantify damages in the presence of network eﬀects. Furthermore, the landmark 2018 Ohio vs Amex Supreme Court decision required plaintiﬀs (merchants) to provide evidence that anti-steering rules negatively impact consumers as well. 51However, the market does not unravel as seller entry is not characterized by adverse selection given that the expected surplus from entering the platform decreases in the seller’s value draw.

33 (a) Only increasing listing fee (b) Adding bidder entry subsidy

Figure 4: Lemons eﬀect: heterogeneous change in expected seller surplus when increasing listing fee by one pound.

∗ Estimated eﬀects plotted by decile of FV0|V0≥v˜0 , for sellers who are infra-marginal (e.g. with v0 ∈ [v ˜0, v0 ]) both at baseline and in the counterfactual.

Figure4 furthermore shows that the magnitude of the lemons effect is inversely related to the infra-marginal seller’s value draw. Increasing the listing fee by 1 pound reduces expected seller surplus by 11-23 percentage points less than without taking two-sided entry into account. The effect increases with the degree of seller heterogeneity in the market. To illustrate, the figure includes results simulated after increasing 2 the variance of the distribution of seller values (σs ) by 10 percent (“added seller heterogeneity”). Fully accounting for welfare impacts on sellers, those that set no reserve price simply experience the full 1 pound loss in surplus, and for the 2 percent of sellers who are pushed out of the market but would otherwise set a positive reserve price the expected surplus must be lower in the counterfactual scenario. These results are especially interesting as they support the special circumstance in two-sided markets that (some) users could be better off when paying higher fees. Plot b of figure4 furthermore demonstrates that the network effects on BW can be exploited to make all sellers (weakly) better off despite paying a 1 pound higher listing fee, by using the proceeds to subsidize bidder entry. The budget-neutral amount of bidder entry subsidy is computed to deplete all additional revenues raised from the ∗ ∗ higher listing fee. This makes the marginal entrant with V0 = v0 indifferent as v0 remains roughly the same at the estimated magnitudes. Infra-marginal sellers with

34 Table 7: Antitrust damages of doubling commission index (cS + 0.102)

Total damage Incidence on Hammer price Buyer damage Seller damage (1000s pounds) sellers (%) (% change) (% post-hammer) (% post-hammer) Benchmark pro-rata 100.0 0.0 0.0 10.2 Benchmark elastic sellers 100.0 0.0 0.0 10.2 Simulated impacts: No entry 13.4 91.1 -4.1 1.0 10.5 No seller entry 16.8 76.3 -6.0 3.9 12.5 Full two-sided entry 17.1 60.7 -0.9 7.5 11.6 Estimates for the non-hypothetical case are estimated in auctions with r > 0 in the main sample. To stay close to antitrust applications, damages are computed as a share of the counterfactual expected hammer price (expected sale probability multiplied by the expected hammer (transaction) price conditional on a sale). Buyer and seller damages are computed in expectations for groups of buyers and sellers, with a buyer being the in expectation winning bidder, including in unsold listings.

R ∗ V0 ∈ [˜v0 , v0] are better off: in the main sample their expected surplus increases by up to 1.2 pounds. Even sellers setting no reserve price are better off as, at the estimated model primitives and magnitudes, subsidy-induced entry of additional bidders into auctions with r = 0 outweighs the higher listing fee. Due to the zero profit entry condition, bidders are not affected and as the number of listings remains constant also the total surplus for bidders as a group remains unaffected. No intervention by a social planner is needed to bring about these benefits: the fee change is estimated to increase platform profits by 1 percent and sales volume by 4 percent, driven by a higher sale probability and transaction prices.52

Antitrust damages. The incidence of a (potentially anticompetitive) change in fees crucially depends on assumptions made about entry and whether or not sellers set a reserve price. For instance, the idea that winning bidders are not affected by changes in either buyer or seller commission (as argued in e.g. McAfee(1993), Ashenfelter and Graddy(2005), and Marks(2009)) is correct only in a market without entry and with fully elastic sellers. A different paradigm was adopted in the 2001 commission fixing case of Sotheby’s and Christie’s: in the absence of a method to evaluate the incidence of commission increases, pro-rata damages were deemed appropriate and most of the

52In terms of practical implementation, the platform could invest in lowering listing inspection cost by standardizing listings more or by introducing an estimated quality index. A negative bidder entry fee is infeasible (if it costs users less to enter the platform and collect it), but a voucher to reduce the transaction price for winning bidders would also stimulate entry. In a similar vein, the next section discusses a negative buyer commission to encourage bidder entry.

35 $512 million settlement went to winning bidders.53 With this rule of thumb, damages to buyers (sellers) are equal to the overcharge in buyer (seller) commission as a share of the realized hammer price. A more extreme solution is adopted in a 2010 case where sellers claiming that eBay charged supracompetitive fees were denied a class action suit, due to the absence of a methodology to empirically quantify damages in the presence of network effects.54 Clearly, a major benefit of my structural approach is that it allows for the estimation of realistic welfare impacts of any fee change. I demonstrate this by simulating the effects of doubling the commission index, by increasing the seller commission from 0.102 to 0.204. Results are reported in table7 and furthermore illustrate biases in simpler models without (seller) entry.55 While in both benchmark cases the incidence of the seller commission increase fall for 100 percent on sellers, this number is substantially lower in simulations. At the estimated model primitives, the incidence on sellers drops to 91 percent when only accounting for the fact that they set reserve prices (see the “no entry” row in table 7). When also endogenizing bidder entry but keeping the set of listings constant (the “no seller entry” row), the incidence on sellers drops further to 76 percent. Fewer bidders enter because reserve prices are higher and the additional loss in surplus is driven by excluding some that would become highest bidders. In the full two-sided entry equilibrium also the number of listings decreases, although additional bidders attracted by the more favorable reserve price distribution on the platform undo’s part of the reduction in surplus. In my application, the elastic seller benchmark does a reasonable job in approxi- mating the damages to an average seller as a percentage of the counterfactual hammer price. However, instead of suffering no welfare loss at all, damages to buyers are estimated to be substantial at 7.5 percent of the average hammer price. Of course the structural model also facilitates simulation of welfare impacts of multiple fee changes simultaneously, and allows for more detailed break-downs by user subgroups.

53See In re Auction Houses Antitrust Litigation and Ashenfelter and Graddy(2005). 54See In re eBay Seller Antitrust Litigation. 55Damages are computed as the reduction in expected surplus resulting from the increase in commission, for groups of (expected) buyers and sellers on the platform, and per-user as percentage of the expected counterfactual hammer price. For an equivalent increase in the commission index brought about by increasing the buyer commission to 0.1281, total damages and the incidence on sellers is the same, but as the hammer price decreases by more the estimated percentage damages are larger. These results are provided in the appendix.

36 5.3 Platform revenue

In two-sided markets, it is profitable to subsidize entry of users on the side that generates stronger positive externalities on the other side, who can then be charged a higher price (Rochet and Tirole(2006)). As documented above, bidders generate stronger indirect network effects than sellers and this partly driven by the fact that additional sellers attracted to the platform set higher reserve prices. This is not lost on platform management, who up to a non-negativity constraint have set the optimal lowest level of buyer commission cB = 0 and bidder entry fee eB = 0. The previous section discussed benefits of subsidizing bidder entry in the form of lowering listing inspection cost or giving cash back to winning bidders. Here I consider a negative buyer commission, which is merely a discount on successful sales. While charging negative commissions would certainly be innovative in the auction platform world, it is similar to the (temporary) discount vouchers periodically offered at eBay or the cash-back policies of certain credit cards. To study the impact of fee changes on the composition of listings on the platform, other than related to seller heterogeneity, results in this section include homogenized auctions based on parameter estimates from the high-end sample. Also in this richer setting, a self-imposed non-negativity constraint on the buyer commission is binding. Figure5 plot a) illustrates this point: platform revenues cannot increase by changing the allocation of commissions to buyers and sellers, unless it subsidizes buyers with a winning bidder discount. In fact, estimates reveal that volume-constrained revenues can increase by over 40 percent when combining a negative cB with a larger increase in cS. The latter is needed to finance the winning bidder discount. It simultaneously increases the commission index to bring about benefits from the selection of lower- value sellers.

In the space of unit versus percentage seller fees, at cB = 0 the volume and revenue levels run in parallel in both the main and high-end samples. Hence any global improvement involves a buyer discount to relax the volume constraint and/or relies on compositional changes from changing the share of high-end listings on the platform. To illustrate, with a 10 percent buyer discount the platform achieves a 30 percent revenue increase (without reducing volume) when either increasing cS to 0.23, or by increasing the listing fee to about 19 pounds (ﬁgure5 plot b). The latter policy is especially attractive for a platform interested in establishing itself in the higher- end market, with the share of high-end listings increasing by 20 percent. However,

37 (a) (cB,cS) at baseline eS (b) (eS,cS) at cB = −0.1

Figure 5: Platform revenue at alternative fee structures

Contour plots of simulated platform revenues, normalised by baseline revenues. Grey vertical bar corresponds to baseline cS = {0.9 (high-end), 0.102 (main sample)}, horizontal bars indicate baseline cB = 0, eS = 2.1, all including 20% VAT. with the buyer discount costing more in high-end listings at estimated parameters, its proﬁt share decreases by 52 percent in the high unit fee scenario, while the high-end proﬁt share increases by 8.5 percent in the high percentage fee case. The model relies on the monopoly position of the platform, motivated by the fact that BW is the only large UK wine auction platform that uses the unvetted seller-managed listing format. An interesting direction for further research would be to model competition in fee structures between (auction) platforms. While that is beyond the scope of this paper, consider competition in only one fee. If a competing platform best-responds to a fee increase on BW by also increasing the fee, this would show up as a higher entry (opportunity) cost for the targeted user. In that case their true entry elasticity w.r.t. an increase in the fee on BW would be lower, and estimated revenue impacts from increasing fees on BW would be conservative.

6 Conclusions

This paper studies an auction platform with two-sided entry. In this setting, the selection of lower-value sellers, setting lower reserve prices, increases the value of

38 the platform to potential bidders. I present a structural model that captures user interactions in such a platform in order to study the welfare and revenue impacts of the platform’s fee structure. I also provide a computationally feasible estimation algorithm and show that relevant model primitives are nonparametrically identified with basic auction data. The model is based on descriptive evidence for seller selection and listing inspection cost and is shown to fit the data well. Counterfactual simulations highlight that the network effects generated by entry and user interactions are nonlinear, the selection of higher-value sellers depletes much of the indirect network effect on bidders, and the benefit of bidder entry is lower for higher-value sellers. What is termed a “lemons effect” clearly illustrates the role of seller selection in this two-sided market. The reduction in surplus of increasing the unit listing fee by one is, for sellers who remain on the platform, less than one as it causes some higher-value sellers (e.g. “lemons”) not to enter. As such, the platform becomes more attractive to potential bidders and this drives up transaction prices for remaining sellers. This effect increases with the degree of seller heterogeneity in the market. Pairing the listing fee increase with a budget-neutral bidder entry subsidy furthermore (weakly) increases expected surplus for all users on the platform. Platform revenues can increase significantly when combining a bidder discount (negative buyer commission) with higher seller fees. Results furthermore account for compositional effects beyond the distribution of seller values, using model estimates from a sample of higher-end wines. Increasing the unit listing fee rather than the percentage seller commission results in a platform with relatively more higher-end listings but a lower profit share from those listings. A key contribution is that the paper brings a two-sided market perspective to the empirical ascending auction literature by modelling endogenous seller entry and how it interacts with bidder entry. Listing independence is key to solve the two-sided entry equilibrium, and delivers the tractability that stationarity restrictions bring to dynamic (auction) models. Further research is needed to extend the auction platform game to other settings, such as those exhibiting congestion or dynamic incentives, in order to study welfare impacts of mechanism design changes more broadly. While my empirical results are based on a specific platform and are narrow in that sense, the paper provides handles to make progress on applying antitrust policy to two-sided markets. The structural model applies most readily to other peer-to-peer platforms with listing heterogeneity.

39 Acknowledgements

I am indebted to Andrew Chesher, Phil Haile, Lars Nesheim, Adam Rosen and Aureo´ de Paula for fruitful discussions and continued guidance. I also thank Dan Acker- berg, Larry Ausubel, Matt Backus, Dirk Bergemann, Thomas Chaney, Natalie Cox, Martin Cripps, Matt Gentry, Emel Filiz-Ozbay, Ken Hendricks, Tom Hoe, Hyejin Ku, Laurent Lamy, Kevin Lang, Guy Laroque, Brad Larsen, Thierry Mayer, Konrad Mierendorff, Rob Porter, Imran Rasul, José-Antonio Esp´ın-Sánchez, Andrew Sweet- ing, Michela Tincani, Jean Tirole, Frank Verboven, Daniel Vincent, Quang Vuong, Martin Weidner, seminar participants, and anonymous reviewers for helpful com- ments. All errors are my own. I am grateful for financial support from the Economic and Social Research Council (PhD Studentship).

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44 For Online Publication

A.1 Additional details entry equilibrium

This supplementary material provides further intuition behind the entry equilibrium and presents listing-level payoﬀ functions. It also shows that the large population approximation is merely adopted for computational feasibility and does not drive the results. In what follows,r ˜ denotes the optimal reserve price increased with buyer ∗ premium,r ˜ = (1 + cB)r (v0, f). Before knowing their valuation, the expected bidder surplus in a listing with n bidders equals:

1 π (n, f, r) ≡ [V − max(V , r˜)|V ≥ r˜][1 − F (˜r)], (A.1) b nE (n:n) (n−1:n) (n:n) V(n:n) with the last term denoting the sale probability and the max(.) term the transaction price including buyer premium. For auctions without a reserve price:

1 π (n, f, 0) ≡ [V − V ] (A.2) b nE (n:n) (n−1:n)

Expected surplus for a seller with valuation v0 in a listing with n bidders:

πs(n, f, v0) ≡ (E[max(Vn−1:n, r˜)|Vn:n ≥ r˜](1 − cS) − v0) [1 − FV(n:n) (˜r)] (A.3)

Consider the case where the number of listings Tr>0 is known to potential bidders. Tr>0 Letv ˜0 denote a candidate seller entry threshold and Πb,r>0(f, v˜0; p) potential bidders’ expected surplus from entering the platform as a function of their entry probability p:

N B,r>0−1 T,r>0 X R T,r>0 o Πb,r>0 (f, v˜0; p) = E[πb(n + 1, f, v0)|V0 ∈ [v0 , v˜0]fN,r>0(n; p) − eB − eB,r>0, n=0 (A.4)

It takes the expectation of πb(n, f, v0) (equation A.1 with optimal r as in equation 2) over: i) possible seller values given sellers’ entry threshold and ii) the number of competing bidders given their entry probability. I present this alternative model here to show more clearly that, in equilibrium, f T,r>0 is independent of the realization Nr>0 of Tr>0 which implies that it must also be independent of the expectation over Tr>0

given thresholdv ˜0.

1 Bidding in one listing at a time, the entry problem for potential bidders is then o equivalent to one in which they consider entry into a listing, as entry cost eB,r>0 are associated with each listing. Components of equation (A.4) are:

R E[πb(n + 1, f, v0)|V0 ∈ [v0 , v˜0]] = (A.5) Z v˜0 R πb(n + 1, f, v0)fV0|V0∈[v ,v˜0 ](v0)dv0 R 0 v0 B,r>0 N − 1 p p B,r>0 f T,r>0(n; p) = ( )n(1 − )N −1−n (A.6) N,r>0 n T T

T,r>0 B,r>0 where fN,r>0(n; p) denotes the Binomial probability that n out of N −1 competing potential bidders arrive in the same listing as the potential bidder who considers

entering the platform. Unpacking further, the seller’s v0 matter through its impact

on the reserve price that bidders face, in expectation over all v0’s such that the seller enters and sets a positive reserve price:

f R (v0) V0≥v0 f R (v0) = (A.7) V0|V0∈[v0 ,v˜0] F R (v ˜0) V0≥v0

R Only right-truncation at entry thresholdv ˜0 is made explicit as the screening value v0 is taken as given.

πb(n+1, f, v0) is strictly decreasing in n (lemma 3). So the bidder entry problem is equivalent to the Levin and Smith(1994) entry model, which assumes that expected bidder surplus decreases in n. The equilibrium bidder entry probability p∗T,r>0 solves zero proﬁt condition:

∗T,r>0 T,r>0 p (Tr>0, f, v˜0) ≡ argp∈(0,1) Πb (f, v˜0; p) = 0 (A.8)

In this equilibrium the number of (competing) bidders per listing follows a Bino- ∗T,r>0 ∗T,r>0 mial distribution with mean (N B,r>0 − 1) p and variance (N B,r>0 − 1) p (1 − Tr>0 Tr>0 ∗T,r>0 p ).56 Furthermore, a no-trade entry equilibrium at p = 0 that trivially solves Tr>0 (A.8) always exists, and it is excluded from the analysis based on the empirical observation that bidders currently the positive trade equilibrium. ∗T,r>0 A key property is that p is independent of Tr>0: bidders only derive pos-

itive surplus from the listing that they are matched to. Tr>0 does not aﬀect

56 See the “Omitted proofs” section of this appendix. The variance of Nr>0 would be larger when S also taking the expectation over Tr>0 given N andv ˜0.

2 BR v¯0i (.)

BR ∗ v¯0i (v ¯0−i, λr>0(v ¯0−i)) ∗ v0(λ)

∗ ∗ v0(λ )

BR ∗ v¯0i (v ¯0−i, λ > λr>0(v ¯0−i)) v¯ ∗ ∗ ∗ 0−i v0(λ ) v0(λ)

Figure A.1: Graphic representation of unique entry equilibrium result

R E[πb(n + 1, f, v0)|V0 ∈ [v0 , v˜0]]. The zero profit condition guarantees that in equilibrium a change in T causes p∗T,r>0 to adjust to keep f T,r>0(.) constant. The same r>0 Nr>0 reasoning applies when Tr>0 is the stochastic outcome of the seller’s entry threshold:v ˜0 only affects the equilibrium mean number of bidders per listing through R E[πb(n + 1, f, v0)|V0 ∈ [v0 , v˜0]] and not through its effect on the distribution of Tr>0.

A.2 Equilibrium uniqueness model extensions

Figure A.1 shows graphically why the entry equilibrium is unique in this model. It depicts the best-response function (entry threshold) of seller i when competing sellers BR ∗ adopt thresholdv ˜0−i. The solid line,v ¯0i (v ¯0−i, λr>0(v ¯0−i)), shows what happens on the equilibrium path. As described in lemma7, this function is downward-sloping: a

higher competing seller entry threshold decreases expected seller surplus for any v0, BR decreasing the threshold v0i for which seller i breaks-even. This results in a unique (symmetric) seller entry threshold where the best-response function intersects the 45 degree line. The challenge in two-sided markets is what happens oﬀ the equilibrium path. Sim- ply put, multiple equilibria exist when, if one side adopts a non-equilibrium entry strategy, this strategy is sustainable due to the best-response of users on the other BR ∗ side. The dashed line in ﬁgure A.1,v ¯0i (v ¯0−i, λ > λr>0(v ¯0−i)), represents seller i’s

3 best response threshold when bidders enter more numerously than their equilibrium ∗ strategy (e.g. λ > λ (v ¯0−i)). Expected seller surplus is higher for any v0 than in ∗ equilibrium, resulting in a higher seller entry equilibrium v0(λ). However, this is not an equilibrium in the two-sided entry game as it violates bidders zero-proﬁt condition: with expected bidder surplus strictly decreasing inv ˜0 (detailed in lemma5), ∗ ∗ ∗ λ > λ can only be sustained by somev ˜0 < v0(λ ). In turn, the latter leaves money ∗ on the table for sellers with values ∈ [v ˜0, v0(λ)] and is therefore also excluded as an equilibrium. As uniqueness of the entry equilibrium is a welcome feature of the presented two-sided market model, the following discusses in a high-level manner which type of model extension satisﬁes this property as well. Consider a generic monopoly platform servicing trade between users of Type 1 with users of Type 2. Users are potentially heterogeneous; possessing some “quality” drawn from a unidimensional distribution.

n1 n1 Let Π1( , n1, n2, q1, q2) and Π2( , n1, n2, q1, q2) denote expected surplus from trading n2 n2 on the platform, respectively for Type 1 and Type 2 users, as a function of how many

n1 57 of them are on the platform (n1, n2, ) and their known average qualities (q1, q2). n2 When quality qi is included this captures selection of Type i users in the usual sense: ∂qi < 0, e.g. resulting from threshold-crossing entry strategies. ∂ni Based on the theoretical analysis above, a suﬃcient condition for the two-sided market to have a unique entry equilibrium:

n1 n1 dΠ2( , n1, n2, q1, q2) dΠ1( , n1, n2, q1, q2) n2 n2 (i) < 0, and (ii) |n1,q1 < 0 (A.9) dn2 dn2 where the total derivative dΠ2(.) includes the equilibrium entry response of Type 1 dn2 dΠ1(.) users, while |n ,q captures how n2 affects expected surplus for Type 1 users dn2 1 1 before a response of n1 and q1. The first part of the condition guarantees the single crossing property on the equilibrium path, and the second part excludes profitable deviations off the equilibrium path. For example:

n1 1. In the presented idiosyncratic-good auction platform: Π1( , q2) for bidders n2 n1 and Π2( ) for sellers. It satisﬁes (A.9) as expected seller surplus is indepen- n2 n1 dent of n2 conditional on and the latter decreases in n2 (i) due to selection of n2 57This is more general than what the theoretical two-sided market literature usually works with. The canonical two-sided market model in Rochet and Tirole(2006) for instance assumes that there are no own-side network eﬀects and that users value all users on the other side equally so that n1 n1 Π1( , n1, n2, q1, q2) = Π1(n2) and Π2( , n1, n2, q1, q2) = Π2(n1). n2 n2

4 higher-value sellers generating higher reserve prices on the platform and therefore reducing expected bidder surplus (ii).

2. An extension to (1) where buyers look for speciﬁc items, so that having more sellers increases the match quality/probability (a scale eﬀect). Captured by

n1 n1 Π1( , n2, q2) and Π2( , n2), this satisﬁes (A.9) as long as the match-quality n2 n2 beneﬁt of additional listings does not outweigh the higher reserve price cost to potential bidders.

3. Adding congestion cost to (2) makes it easier to satisfy (A.9) as increasing n2 n1 then reduces Π1 (ii), making the bidder response to lower starker and sellers’ n2 best-response function steeper (i).

4. An extension to (1) with competing sellers, where competition gets ﬁercer when ∗ more sellers enter (e.g. r decreasing in n2 while increasing in v0). Captured by n1 n1 Π1( , n2, q2) and Π2( , n2), this merely results in a steeper seller best response n2 n2 function, satisfying (A.9).

n1 n1 5. An extension to (1) with bidder selection: Π1( , q1, q2) and Π2( , q1). In- n2 n2 creasing n2 will lead to more bidder entry, since increasing the dispersion of the distribution of bidder values on the platform increases expected winning bidder surplus in ascending auctions. Hence, (A.9) requires that the resulting partial eﬀect of increasing n1 is outweighed by the negative seller selection eﬀect on n2 the equilibrium n1 (condition ii). Condition i then follows, as facing lower value n2 bidders only adds to the negative impact of more listings on Π2.

Note that this analysis extends to platforms with posted prices as the sales volume (if not the price) remains endogenous to the number of buyers.

A.3 Omitted proofs

Optimal reserve price

Proof lemma 2. I derive the reserve price related to homogenized values V0 and deﬁne x hat and check notation as:x ˆ = x(1 + cB) andx ˇ = . Let R denote expected 1+cB revenue for a seller with valuation v0 when setting reserve price r in an auction with

5 n bidders:

n n−1 R = v0FV (ˆr) + (1 − cS)rnFV (ˆr) [1 − FV (ˆr)]+ (A.10) Z v¯ n−2 T (1 − cS) xnˇ (n − 1)FV (x) [1 − FV (x)]fV (x)dx rˆ

The three terms in the above equation for R cover three cases: i) no sale takes place, ii) a sale takes place but the second-highest bid is less than the reserve price and iii) the sale takes place and the second-highest bid exceeds the reserve. Maximizing R with respect to r:

∂R = v nF (ˆr)n−1f (ˆr)(1 + c ) + (1 − c )nF (ˆr)n−1[1 − F (ˆr)] (A.11) ∂r 0 V V B S V V n−2 +(1 − cS)rn(n − 1)FV (ˆr) fV (ˆr)(1 + cB)[1 − FV (ˆr)] n−1 −(1 − cS)rnFV (ˆr) fV (ˆr)(1 + cB) n−2 −(1 − cS)rn(n − 1)FV (ˆr) [1 − FV (ˆr)]fV (ˆr)(1 + cB)

The second and last line cancel out. Re-arranging delivers the optimal reserve price ∗ r (v0, f) which solves:

∗ v0 1 − FV (r(1 + cB)) r (v0, f) ≡ {r = + } (A.12) 1 − cS (1 + cB)fV (r(1 + cB))

∗ r (v0, f) is unique ∀(v0, f) given the IFR property of FV , increasing in v0, and independent of n.

Poisson decomposition property for number of bidders per listing

Proof lemma 3. The proof concerns the statement that when N B potential bidders enter a platform with T listings with probability p, the distribution of the number of N B p bidders per listing is approximately Poisson with mean T . Let M denote the total number of bidders on the platform, distributed Binomial(N Bp, N Bp(1 − p)). The limiting distribution of M when the population of potential bidders N B → ∞ and associated p → 0 s.t. N Bp remains constant is Poisson(λ = N Bp). Bidders on the 1 platform sort over T listings, entering each listing with probability q = T . Due to the stochastic number of bidders on the platform, the probability that m bidders get allocated in listing t and n enter into other listings also includes the probability that

6 m + n bidders enter the platform.

exp(−λ)λ(m+n) (m + n)! f (m, n) = (q)m(1 − q)(n) (A.13) Nt,N−t (m + n)! m!n!

This joint distribution function can be manipulated to conclude that:

∞ X exp(−λq)(λq)m exp(−λ(1 − q))(λ(1 − q))n exp(−λq)(λq)m f (m) = = Nt m! n! m! n=0

This is referred to as the decomposition property of the Poisson distribution in Myerson(1998). Novel here is the stochastic nature of M; the above shows that M

does not need to be independent of T . The t subscript can be dropped from fNt as the distribution is identical for all listings t = {1, .., T }.

Binomial decomposition property. Alternatively, we can show that N ∼ B p B p p Binom(N T ,N T (1 − T )) and apply the large sample approximation afterwards. Including the expectation over the number of bidders on the platform, M, the probability mass function of the number of bidders per listing, ∀n ∈ Z≥:

N B X P [N = n] = P [N = n|m]P [M = m] = (A.14) m=0 | {z } EM [P [N=n|m]] B N B n X N B m 1 1 pm(1 − p)N −m (1 − )m−n m n T T m=0

and 0 otherwise. Using the law of iterated expectations (E[N] = EM [E[N|m]]) and iterated variance (V ar(N) = EM [V ar(N|M = m) + V ar(EM [N|M = m])): 1 1 1 1 [V ar(N|M = m)] = [m (1 − )] = (N B)p (1 − ) (A.15) EM EM T T T T 1 1 = N Bp − (N B)p( )2 T T

M 1 1 V ar( [N|M = m]) = V ar( ) = ( )2V ar(M) = ( )2N Bp(1 − p) (A.16) EM T T T 1 1 = −( )2N Bp2 + ( )2N Bp T T

7 Re-arranging shows that the mean and variance of N are:

m [M] N Bp [N] = [ ] = E = (A.17) E EM T T T p p V ar(N) = N B (1 − ) (A.18) T T This proves that given the large population assumption, a success probability of en- p tering in listing t (for any t ∈ {1, .., T }) equal to T results in fN being approximately N B p Poisson with mean T . Listing-level properties

∂πb(n,f,v0) Proof lemma 4. ∂n < 0 (deﬁned in (A.1)) because FV satisﬁes the increasing failure rate (IFR) property. Li(2005) prove that a monotonically nondecreasing

failure rate implies decreasing spacings so that E[V(n+1:n+1) − V(n:n+1)] − E[V(n:n) − V(n−1:n)] ≤ 0. This holds without a reserve price or fees and since both are independent R of n, and the inequality is strict in the IFR case. For v0 ≤ v0 sellers set no reserve price πb(n,f,v0) so πb(n, f, v0) = πb(n, f, 0) is independent of v0, and otherwise ≤ 0 since the v0 ∂πs(n,f,v0) optimal reserve increases in v0 (lemma 2). On the seller side, ∂n > 0 (deﬁned in ∗ (A.3)) as r (v0, f) is independent of n (lemma 2) and FV(n:n) is stochastically increasing in n. It is clear from (A.3) that ∂πs(n,f,v0) ≤ 0 and intuitively: higher seller values ∂v0 reduce gains from trade.

A.4 Additional details estimation algorithm

This section provides details about the estimation of structural parameters not in- o o o cluded in the main text. This regardse ˆB,r>0,e ˆB,r=0,e ˆS, andp ˆ0,r>0. The estimated entry cost (opportunity cost of time) solve the relevant zero proﬁt ˆ ˆ R conditions, given estimated parameters (θb, θs,v ˆ0 ,p ˆ0,r>0) and given the entry equilib- ˆ rium at those parameters. As estimating θs itself requires one iteration of solving for ˆ0 the entry equilibrium given initial parameters θs , the estimation algorithm proceeds R ˆ0 as follows. First, based onv ˆ0 andυ ˆTr>0 , estimate θs by maximum concentrated likeli- o,0 hood as described in the main text. Then, solve for initial entry cost estimates (ˆeB,r>0, o,0 o,0 ∗ ˆ 0 ˆ eˆB,r=0,e ˆS ) as detailed below. Given these initial estimates solve for v0(f; θs , θb) and ˆ update seller parameters to θs as described. Finally, use these values to re-estimate the entry cost.

8 o,0 o,0 Fore ˆB,r>0 ande ˆB,r=0, the initial estimator is the same as the ﬁnal estimator ˆ although the latter is based on the updated θs. They are estimated as the value of the entry cost that sets respectively Πb,r>0(.) and Πb,r=0(.) equal to 0 as dictated by the zero proﬁt entry condition. This clearly depends on the relevant distribution of ˆ∗ ˆ∗ the number of bidders per listing, and hence λr>0,p0,r ˆ=0, and λr=0. In auctions with ∗ no reserve price, the mean observed N is a consistent estimator of λr=0:

ˆ∗ 1 X λr=0 = nt (A.19) |Tr=0| t∈Tr=0

ˆ∗ ˆ∗ Note that λr=0 and λr=0 are only obtained to estimate entry cost and they are not ˆ∗ treated as structural parameters. I now turn to the estimation of λr>0. In positive reserve prices a diﬃculty is that only the actual number of bidders A is observed, which might be less than the number of bidders in the listing N. In the BW data the reserve price is secret, but the platform provides some information about it (“reserve not met”, “reserve almost met”, or “” if the standing price exceeds the reserve). If the reserve price were observed (and the only reason for bidders not ∗ submitting a bid), a consistent estimate of λr>0 equals the value that maximizes the

likelihood of the homogenized second-highest bids bt and number of actual bidders

at in positive reserve auctions given estimated bidder valuation parameters and ho-

mogenized reserve prices rt. In particular, the joint density of (bt, at) if the number

of potential bidders nt would be known, withr ˜t = rt(1 + cB), ∀t ∈ Tr>0:

ˆ ˆ nt h(bt, at|nt, rt, zt, θb) = {FV (˜rt; θb) }I{at = 0} (A.20) ˆ nt−1 ˆ {ntFV (˜rt; θb) [1 − FV (˜rt; θb)]}I{at = 1} n t ˆ nt−at ˆ at { FV (˜rt; θb) [1 − FV (˜rt; θb)] nt − at ˜ ˆ at−2 ˜ ˆ ˜ ˆ at(at − 1)FV (bt; θb) [1 − FV (bt; θb)]FV (bt; θb)}I{at ≥ 2}

ˆ Note that h(bt, at|nt, rt, θb) = 0 when nt = 0. The first line covers the probability that all nt bidders draw a valuation below the reserve price, the second line the probability that one out of nt draw a valuation exceedingr ˜ while the others don’t, and the final two lines capture the probability that at out of nt draw a valuation exceeding the reserve and that the second-highest out of them draws a conditional value equal to ˜ bt = bt(1 + cB). Without observing nt, a feasible specification takes the expectation

9 ∗ 58 over realizations of random variable N ∼ generalized P ois(λr>0, p0,r>0). Using the more ﬂexible two-parameter Poisson distribution allows for an unspeciﬁed reason for observing no bids, in addition to all bids falling below the reserve price. This is the ˆ∗ basis of the likelihood function that (λr>0, pˆ0,r>0) maximizes:

ˆ ∗ g(bt, at|rt, zt, θb; λr>0, p0,r>0) = (A.22) ∞ X ˆ ∗ h(bt, at|nt = k, rt, zt, θb)fNr>0|Nr>0≥A(k; λr>0, p0,r>0) k=at ∗ X ˆ ∗ L(λr>0, p0,r>0; {bt, at, rt, zt}t∈Tr>0 ) = ln(g(bt, at|rt, zt, θb; λr>0, p0,r>0)) (A.23) t∈Tr>0 ˆ∗ ∗ (λr>0, pˆ0,r>0) = arg max L(λr>0, p0,r>0; {bt, at, rt, zt}t∈Tr>0 ) (A.24)

The estimator does not require interpretation of losing bids. While the resulting estimator does capture the censoring of bidders to some extent, is not a full treatment 59 ˆ of intra-auction dynamics. Also the estimated θb are based on the assumption that the second-highest bid equates to the second-highest out of N = A values in no- reserve auctions. It is worth emphasizing that the eﬀect of this abstraction is limited in my model with endogenous two-sided entry. To see why, consider the case where ∗ the true λr>0 is larger than estimated due to some bidders entering after the standing bid exceeds their valuation. This implies that the true FV would be stochastically dominated by the estimated distribution, as the transaction price is really the second- highest out of more draws from FV . The truee ˆB,r>0 would also have to be lower than estimated, as the per-bidder expected surplus from entering the platform is lower. Combined, counterfactual simulations based on these alternative primitives would deliver similar results. Without changing the fee structure, simulating entry decisions of lower-value potential bidders facing lower entry cost results in the exact same outcomes. As such, the abstraction from intra-auction dynamics is internally consistent although due to non-linearities in the system the direction of the eﬀect of

58The generalized Poisson distribution has PDF:

exp(−λ )λk f (k; λ , p ) = (1 − p ) r r + p {k = 0} (A.21) Nr>0 r>0 0,r>0 0,r>0 k! 0,r>0I

which reduces to a standard Poisson distribution for p0,r>0 = 0. 59Hickman et al.(2017) (for the case of non-binding reserve prices) and Bodoh-Creed et al.(2020) (for binding reserve prices) provide more comprehensive models to account for intra-auction dynamics in ascending auctions. My empirical setting is in between these cases, with the platform revealing some information about the secret reserve price, and the algorithm proposed by Platt(2017) based on a Poisson arrival process would apply if p0,r>0 = 0.

10 the abstraction when changing fees cannot be signed ex-ante. o,0 o,0 The above describes how initial valuese ˆB,r=0 ande ˆB,r>0 are estimated. The initial o,0 valuee ˆS needs to be estimated differently, as pinning it down as the value that sets the surplus for a marginal seller with v0 =υ ˆTr>0 equal to 0 guarantees that the ∗ updated v0 will always equalυ ˆTr>0 . Instead, asυ ˆTr>0 is the sample maximum of a ∗ noisy first stage estimator and therefore likely overestimates the true v0, the initial o,0 o,0 o,0 eˆS is set to max(êB,r>0, eˆB,r=0). After simulating the entry equilibrium (described ˆ o o o in the next section) at initial estimates, and re-estimating θS,e ˆB,r>0,e ˆB,r=0, ande ˆS are updated as the values that solve zero profit conditions at the equilibrium solution. ˆ R ∗ˆ ∗ˆ Note that θb,v ˆ0 ,p0,r> ˆ 0, λr>0, and λr=0 are never updated in the estimation algorithm.

A.5 Numerical approximation equilibrium

Solving for the entry equilibrium involves hard to compute (triple) integrals. This section details the numerical approximations relied on for computational feasibility. The equilibrium is computed for homogenized auctions based on conditional value distributions. The notation also does not make explicit that these distributions are in fact the estimated conditional value distributions. I also use shorthand notation ∗ r˜ = (1 + cB)r (v0, f) and omit the sample size n in order statistics. The goal is to approximate for a given fee structure and set of parameter estimates the entry ∗ ∗ ∗ ∗ equilibrium (λr>0(f, v0), λr=0(f), and v0(f)) as deﬁned in (6), (7), and (9). It requires computing the expected surplus from entering the platform for bidders and sellers as a function of λ andv ˜0, and then solving for the values that satisfy the zero proﬁt entry conditions.

To compute Πb,r>0(f, v˜0; λ) we need to obtain πb(n, f, v0) deﬁned in (A.1) in expectation over v0 and n, minus entry cost. Also writing out the expectation operator in πb(.) we obtain:

11 max(n)−1 " v˜ # Z 0 f R (v0) X V0|V0≥v0 Πb,r>0(f, v˜0; λ) = πb(n + 1, f, v0) dv0 × (A.25) R F R (v ˜0) n=0 v0 V0|V0≥v0 o fN,r>0(n; λ) − eB − eB,r>0 1 Z v¯ Z vn πb(n, f, v0) = vn − max(˜r, vn−1dFVn−1|Vn=vn (vn−1))dFVn (vn) (A.26) n r˜ v Z vn n−1 FVn (vn) = nFV (x) fV (x)dx (A.27) v Z vn n−2 (n − 1)FV (y) fV (y) FVn−1|Vn=vn (vn−1) = n−1 dy (A.28) v FV (vn)

∗ and fN,r>0(n; λ) deﬁned in (3). This is then suﬃcient to compute λr>0 for any value

ofv ˜0:

∗ λr>0(v ˜0) ≡ argλ {Πb,r>0(f, v˜0; λ) = 0} (A.29)

∗ With Πb,r>0(f, v˜0; λ) strictly decreasing in λ, λr>0 solves a threshold-crossing condition ∗ that is nested in the ﬁxed point problem that deﬁnes v0(f). Moreover, the triple integral makes Πb(.) costly to compute for any candidatev ˜0. For auctions with a zero ∗ reserve price, λr=0 is similarly computed as a threshold-crossing problem based on

Πb,r=0 which is simply:

max(n)−1 X o Πb,r=0(f, λ) = πb(n + 1, f, 0)fN,r=0(n; λ) − eB − eB,r=0 (A.30) n=0 with πb(n + 1, f, 0) deﬁned in (A.2).

To compute Πs(f, v˜0; λ) we need to obtain πs(n, f, v0) deﬁned in (A.3) in expec-

12 tation over the number of bidders, minus entry cost:

B Nr>0 ∗ X ∗ o Πs(f, v0; λ (v ˜0)) = πs(n, f, v0)fN,r>0(n, λ (v ˜0)) − eS − eS (A.31) n=0 1 Z v¯ πs(n, f, v0) = (max(r, vn−1dFVn−1|Vn≥r˜(vn−1))× (A.32) 1 + cB v

(1 − cS) − v0)[1 − FV(n) (˜r)] (A.33) Z v¯

FVn−1|Vn≥r˜(vn−1) = FVn−1|Vn=x(vn−1)dFVn (x) (A.34) r˜

∗ This is then suﬃcient to compute v0(f) for any fee structure and given potential ∗ bidders’ best response characterized by λr>0(f, v˜0):

∗ ∗ v0 ≡ argv˜0 {Πs(f, v˜0; λr>0(v ˜0)) = 0} (A.35)

Given high computational cost of implementing these functions literally, I choose to rely on numerical approximation to speed things up. I implement the following pseudo-code to compute the entry equilibrium, where object names in bold facilitate easy replication with access to my computer code.

• Initiating probability vectors for the simulation of bidder and seller values with importance sampling. Simulate 250 values from Unif(0, 1) and collect in vector v probs (making sure that 1e−4 and 1 − 1e−4 are lower bounds on extremum probabilities). Initiate a finer grid v probs fine by sampling 25000 values from Unif(0, 1) with identical minimum extremum values. Simulate 500 values from Unif(0, 1) and collect in vector v0 probs fine (making sure that 1e−4 and 1 − 1e−4 are lower bounds on extremum probabilities). Sample a coarser grid for seller values by drawing without replacement 48 values from v0 probs fine and add the extremum values, call this vector v0 probs. Set max(n) = 15 (pick a sensible number based on estimated λ’s). Never change these values.

− −9 ˆ • Importance sampling of Vn:n and Vn−1:n|Vn:n. Setv ¯ = FV 1(1 − 1e ; θb) and v = 0. Code the distributions in (A.27) and (A.28). For each n = 1, .., 15, simulate 250 values from the two distributions. For the highest valuation, solve for F −1 (v probs; θˆ ), separately for each n, resulting in matrix Vn:n b h mat of dimension [250 × 15]. For the second-highest valuation, solve for F −1 (v probs; θˆ ), where for each entry j in v probs v equals the Vn−1:n|Vn:n=vn b n

13 jth entry in h mat from the relevant n column. Doing this separately for each n > 1 results in matrix sh mat of dimension [250 × 15] with the ﬁrst column made up of zeros.

• Linear interpolation of h mat and sh mat on finer grid using v probs fine, separately for each n column. This results in two matrices of dimension [25000 × 15], h mat fine and sh mat fine.

• Calculating optimal reserve price for grid of v0’s. Importance sampling of V0: solve for F −1(v0 probs; θˆ ) and store in vector v0 vec of dimension [50 × 1]. V0 s ˆ ∗ Given also θb, compute optimal r (v0 vec) and store in vector r vec.

• Compute listing-level bidder and seller surplus for v0-n combinations. Initiate matrices of

v0 mat, n mat, and r mat with values of v0 in the first dimension and n in the second dimension (so n mat and r mat are constant in the first dimension and v0 mat is constant in the second dimension). These three matrices are of dimension [50×15]. For each entry, use the pre-calculated matrices h mat fine and sh mat fine to approximate listing-level surplus with monte carlo simulations, separately for bidders in auctions with positive and no reserve prices (the latter being a vector) and for sellers in auctions with a positive and with no re-

serve prices (both being matrices). For example, consider a (v0, 2) combination

with v0idx being the index of v0 in the 2nd column of v0 mat. πb(2, f, v0) is approximated as the mean of the second column of h mat ﬁne including only

all values exceeding r mat(v0idx, 2) × (1 + cB), minus the mean of the same

entries in sh mat ﬁne or minus r mat(v0idx, 2)×(1+cB) if that is higher, and ˆ 2 multiplied by the sale probability (1 − FV (log((1 + cB)r mat(v0idx, 2)); θb) ), all divided by two.

• Linear interpolation of listing-level surplus on v0 probs ﬁne. This results in listing-level surplus matrices of dimensions [25000 × 15] for bidders in positive reserve price auctions (pib posr mat), for sellers in positive reserve price auctions (pis posr mat), and for sellers in no reserve price auctions (pis nor mat). For bidders in auctions with no reserve price (pib nor vec) we obtain a vector of dimension [1 × 15] as their listing-level surplus is independent of the seller’s value. Also pre-calculate a vector of probabilities that −1 V0 = v0 using F R (v0 probs) and interpolate on the ﬁner v0 grid, resulting V0|V0≥v0 in pdf v0 mat.

14 ˆ • Repeat the ﬁve previous steps only once for each new θs or fee structure.

With the pre calculated listing-level surplus matrices as functions of v0 and n, ∗ the computation of v0 as a ﬁxed point problem with a nested threshold-crossing ∗ problem to ﬁnd λ for each candidatev ˜0 is fast and straightforward.

• Coding equation (A.31) with nested in it equation (A.29). Make sure that for

every candidatev ˜0, the entries of pdf v0 mat that function as weights of the f R (v0) V0|V0≥v0 ∗ listing-level bidder surplus (the in (A.25)) sum to one. The λ (v ˜0) in F R (v ˜0) V0|V0≥v0 2 (A.29) is obtained as the root of (Πb(f, v˜0; λ)) . I use Matlab’s fzero function with tolerance levels for the function and parameter of 1e−6, which delivers stable results. Then I pass (A.31) to a non-linear solver to find the fixed point, again using fzero root finding with the same tolerance levels.

Contraction mapping. Relevant for the NPL-like estimation method, the follow- ∗ ing argumentation shows that v0 is characterized by a contraction mapping. Let j −j j Πs(v0, v0 ) denote the expected surplus for seller with valuation v0 when entering the platform and setting a reserve price, with competing sellers’ entry threshold only af- ∗ −j fecting Πs through its eﬀect on the the equilibrium mean number of bidders λ (v0 ). 0 −j The fee structure and other exogenous inputs are omitted from notation. Let v0(v0 ) −j 0 −j denote the seller’s best response to threshold v0 ; to enter i.f.f v0 ≤ v0(v0 ). A nec- ∗ essary and suﬃcient condition for v0 being characterized by a contraction mapping is −j ∗ that there are no other values of v0 6= v0 that deliver zero surplus for the marginal 0 −j −j seller so that v0(v0 ) = v0 . We need to consider three cases:

−j ∗ ∗ −j ∗ ∗ ∗ −j • Case of v0 > v0: λ (v0 ) < λ (v0) which means that Πs(v0, v0 ) < 0. Since Πs j 0 −j −j ∗ is decreasing in the seller’s v0, the resulting v0(v0 ) < v0 < v0. We conclude −j −j that Πs(v0 , v0 ) is not an equilibrium.

−j ∗ ∗ −j ∗ ∗ ∗ −j • Case of v0 < v0: λ (v0 ) > λ (v0) which means that Πs(v0, v0 ) > 0. With Πs j 0 −j −j ∗ decreasing in the seller’s v0, the resulting v0(v0 ) > v0 > v0. Also in this case, −j −j Πs(v0 , v0 ) is not an equilibrium.

−j ∗ • The final case is the unique fixed point in seller value space, where v0 = v0. ∗ ∗ −j 0 −j −j ∗ By definition of v0,Πs(v0, v0 ) = 0 so that v0(v0 ) = v0 = v0.

This proves that Equation A.35 is a contraction mapping.

15 A.6 Reserve price approximation

Reserve prices are approximated as the average between the highest standing price for which the reserve price is not met and the lowest for which it is met. If all bids would be recorded in real time, this approximation would be accurate up to half a bidding increment due to the proxy bidding system. To relieve traffic pressure on the site I only track bids on 30-minute intervals. The reserve price approximation could be more than half a bidding increment off if the bids are not placed at regular intervals. As a compromise with constant high website traffic a separate dataset is collected that accesses open listings at 30-second intervals for the duration of two weeks, to test the reserve price approximation in the main sample. My estimation method requires that the estimated distribution of reserve prices is consistent for its population counterpart. Equality of the distribution of approximated reserve prices in the main sample and the distribution of (approximated) reserve prices in the smaller high frequency sample is tested with a two sample nonparametric Kolmogorov-Smirnov test. To account for different listing compositions the empirical reserve price distributions are right-truncated at the 90th percentile of the high frequency reserve price sample. The null hypothesis is that the two right H M truncated reserve price distributions are the same. In particular, letting FR and FR respectively denote the empirical distribution of right truncated approximated reserve prices in the high frequency (H) and main (M) sample, the Kolmogorov-Smirnov test statistic is defined as: H M Dh,m = sup |FR (x) − FR (x)|, (A.36) x with supx the supremum function over x values and h and m respectively denoting the relevant number of observations in the high frequency and main samples, which are 330 and 596 (only for sold lots). With Dh,m = 0.059, the null cannot be rejected p h+m at the 5 pervent level (Dh,m > 1.36 ( hm ), the p-value = 0.4406).

16 A.7 Additional tables and ﬁgures

Table A.1: Independent listings: regression analysis

Dependent var: bidders / listing bids / bidder transaction price reserve price coef. s.e. coef. s.e. coef. s.e. coef. s.e. Product: any wine 30 days 0.0002 (0.0002) 0.0002 (0.0002) -0.020 (0.032) 0.002 (0.032) 7 days 0.001 (0.001) 0.0003 (0.001) -0.093 (0.095) -0.121 (0.108) 2 days 0.001 (0.001) 0.001 (0.001) -0.135 (0.138) -0.368 (0.167) Product: type (red) 30 days 0.002 (0.002) 0.001 (0.001) 0.214 (0.251) 0.350 (0.265) 7 days 0.016 (0.005) 0.005 (0.004) 0.923 (0.639) 0.203 (0.823) 2 days 0.009 (0.006) 0.004 (0.005) 0.385 (0.834) -1.426 (1.046) Product: region (Bordeaux) 30 days 0.001 (0.001) 0.0002 (0.0004) -0.042 (0.075) 0.044 (0.081) 7 days 0.004 (0.002) 0.0003 (0.001) -0.112 (0.227) -0.241 (0.279) 2 days 0.004 (0.002) 0.001 (0.002) -0.301 (0.328) -0.964 (0.421) Product: region x type 30 days 0.002 (0.002) 0.00001 (0.002) 0.428 (0.370) 0.255 (0.391) 7 days 0.015 (0.007) 0.004 (0.006) 3.038 (1.020) 0.894 (1.234) 2 days -0.004 (0.010) -0.002 (0.008) 2.151 (1.463) -2.242 (1.617) Product: region x type x vintage 30 days -0.010 (0.007) -0.008 (0.006) 1.537 (1.164) 0.332 (1.037) 7 days -0.026 (0.017) -0.011 (0.015) 16.553 (2.890) 6.649 (2.466) 2 days -0.049 (0.021) -0.013 (0.018) 17.627 (3.841) 6.152 (2.992) Product: subregion (Margaux) x type x vintage 30 days -0.005 (0.004) 0.0002 (0.004) 1.143 (0.689) 0.108 (0.633) 7 days 0.004 (0.012) -0.001 (0.010) 4.249 (1.748) -0.330 (1.867) 2 days -0.010 (0.014) 0.003 (0.012) 3.438 (2.082) -2.514 (2.096) Observations 3,500 3,500 2,235 2,351 Sample all all sold lots reserve auctions Results from 72 separate OLS regressions of how the number of competing listings affects the four outcome variables (columns). Competing listings defined as offering the same product in the same market, using 6 different product definitions and a market being all listings ending within a 30 day, 7 day, or 2 day rolling window of the listing. Regressions condition on product fixed effects.

17 Table A.2: Heckman selection model estimates

FIRST STAGE Dependent variable: List (1) (2) Market variables: # Potential sellers, x1000 −3.248 −3.252 (1.288) (1.330) # Potential sellers r > 0, x1000 0.071 −0.395 (1.841) (1.900) Potential seller variables: Share of markets entered 4.314 4.267 (0.282) (0.302) # Listings other markets, x100 −1.427 −1.624 (0.333) (0.376) (# Listings other markets, x100)2 0.335 0.372 (0.079) (0.087) Month account created (lower=older) −0.033 0.016 (0.011) (0.014) # New members when joined −0.001 (0.001) # New potential sellers when joined 0.022 (0.005) # Ratings, x10.000 4.977 (3.464) (# Ratings, x10.000)2 −6.648 (11.79) Has negative ratings 0.037 (0.099) Share of ratings positive 0.139 (0.089) Share of ratings neutral −0.122 (0.397) Constant 67.252 −31.831 (21.510) (28.918)

SECOND STAGE Dependent variable: residˆ r = 0 (1) (2) (3) (4) Inverse Mills Ratio −64.646 −57.205 0.083 0.104 (14.027) (14.046) (0.019) (0.016) Observations 3,500 3,500 3,500 3,500 F Statistic 21.241 16.587 17.926 18.196 Includes Z (auction descriptors) No No Yes Yes First stage model (1) (2) (1) (2)

First stage based on all potential seller-month combinations, for all potential sellers who listed at least once during the sample period. List = 1 if seller has at least one listing in the month (columns 1 and 2). r = 0 is a dummy equal to 1 if the seller has at least one listing with r = 0. residˆ is the estimated reserve conditional on observables.

18 Table A.3: Suggestive evidence against bidder selection: hammer price, multiple product/market deﬁnitions, main sample with r = 0

(1) (2) (3) (4) (5) (6) (7) Nr. bidders 10.130 10.742 10.710 10.659 10.666 10.172 8.804 in auction (0.664) (0.611) (0.618) (0.613) (0.628) (0.690) (0.714) Nr. bidders -0.061 0.032 0.026 0.053 0.059 -0.097 0.378 product/market (0.072) (0.028) (0.038) (0.045) (0.101) (0.210) (0.186) Product FE: Yes Yes Yes Yes Yes Yes Yes Time trend: Yes Yes Yes Yes Yes Yes Yes Sample r = 0 r = 0 r = 0 r = 0 r = 0 r = 0 r = 0 Observations 989 989 989 989 989 989 989 Adjusted R2 0.365 0.239 0.294 0.269 0.317 0.365 0.346 Product/market speciﬁcations: (1): type×vintage×region, 4 weeks, (2)-(7) market: 2 day rolling window, (2) any wine, (3) type, (4) region, (5) region×type, (6) region×type×vintage, (7) subregion×type×vintage

Table A.4: Suggestive evidence against bidder selection: sale price (hammer price in sold auctions), various samples, basic product/market deﬁnition

(1) (2) (3) (4) (5) (6) (7) (8) (9) Nr. bidders 15.747 13.184 13.180 6.786 5.861 5.861 8.841 7.938 7.865 in auction (1.146) (1.170) (1.170) (0.432) (0.427) (0.428) (0.618) (0.642) (0.643) Nr. bidders −0.111 −0.144 −0.155 −0.068 −0.040 −0.034 0.001 −0.067 −0.033 product/market (0.076) (0.144) (0.148) (0.028) (0.050) (0.052) (0.034) (0.069) (0.071) Product FE: No Yes Yes No Yes Yes No Yes Yes Time trend: No No Yes No No Yes No No Yes Sample Full Full Full Main Main Main r = 0 r = 0 r = 0 Observations 2,235 2,235 2,235 1,874 1,874 1,874 985 985 985 Adjusted R2 0.078 0.274 0.274 0.117 0.350 0.349 0.179 0.330 0.332

Table A.5: Suggestive evidence against bidder selection: hammer price, multiple samples, basic product/market deﬁnition

(1) (2) (3) (4) (5) (6) (7) (8) (9) Nr. bidders −0.674 −3.486 −3.446 2.307 3.651 3.640 11.006 10.130 10.079 in auction (2.096) (2.426) (2.426) (0.430) (0.461) (0.461) (0.630) (0.664) (0.665) Nr. bidders 0.721 −0.175 −0.234 −0.011 0.074 0.084 −0.053 −0.061 −0.038 product/market (0.161) (0.329) (0.333) (0.031) (0.060) (0.061) (0.035) (0.072) (0.074) Product FE: No Yes Yes No Yes Yes No Yes Yes Time trend: No No Yes No No Yes No No Yes Sample Full Full Full Main Main Main r = 0 r = 0 r = 0 Observations 3,500 3,500 3,500 2,826 2,826 2,826 989 989 989 Adjusted R2 0.005 0.036 0.036 0.010 0.166 0.166 0.240 0.365 0.365

19 Table A.6: Homogenization estimates (main sample)

DEPENDENT VARIABLE: log(second-highest bid per bot- Point S.E. tle) estimate Number bottles -0.3386 0.0552 Number bottles2 0.0142 0.0035 Contains more than one bottle -0.2344 0.0801 Case of 6 0.4517 0.1244 Case of 12 0.6933 0.2299 Special format bottle 0.1228 0.0685 Stored in warehouse 0.3102 0.2072 Month auction ends -0.0136 0.0046 Textual description related to “en Primeur” 0.1765 0.0472 Textual description related to delivery / shipping -0.0086 0.0387 Textual description related to expert opinion 0.1766 0.0417 Number words in textual description 0.0011 0.0002 Delivery and payment options: Duty estimate -0.0171 0.0096 VAT estimate 0.0066 0.0154 Delivers to UK 0.0234 0.0489 Returns accepted -0.2120 0.1445 Payment by bank 0.3009 0.0866 Payment by Paypal -0.1221 0.0457 Payment by cheque 0.0589 0.0515 Payment in cash 0.0545 0.1115 Ships with Royal Mail 0.0494 0.0503 Ships with Parcel Force -0.1776 0.0493 Ships fast 0.3437 0.0684 Insurance included 0.1274 0.0433 Can be collected in person 0.0833 0.0446 Can only be collected in person -0.0183 0.1056 Lowest shipping cost quote 0.0094 0.0039 Seller ratings: Seller has ratings -0.1591 0.0540 Seller ratings -0.0008 0.0002 Seller ratings2 3.6e-07 6.9e-08 Fill level: Mid Shoulder (HS) -0.3737 0.2259 Into Neck (IN) -0.1298 0.1813 High Shoulder (HS) -0.3997 0.2250 Missing -0.0567 0.1863 Top Shoulder (TS) -0.4374 0.2189 Very Top Shoulder (VTS) -0.1768 0.2088 Base of Neck (BN) -0.2162 0.1883

20 Table A.6: ...continued (main sample)

Type: White -0.2047 0.0600 Sparkling 0.0884 0.1046 Assorted -0.0483 0.0685 Fortiﬁed 0.1365 0.1176 Rose -0.6587 0.3303 Region: Tuscany -0.2011 0.0741 Burgundy 0.0219 0.0665 Rhone -0.0204 0.0641 Champagne 0.2806 0.1254 Veneto -0.1849 0.1370 Provence -0.2191 0.2406 Alsace -0.2795 0.1754 France 0.1618 0.0690 Other -0.1275 0.0911 Piedmont/Lombardy -0.1658 0.0907 Rioja -0.3566 0.1404 South Australia -0.0900 0.0887 Douro -0.0175 0.1842 Mendoza -0.6049 0.2237 Bekaa Valley 0.0913 0.3533 Oporto -0.0255 0.1877 Scotland -0.0622 0.2595 Assorted 0.3425 0.0764 Australia -0.1144 0.1002 United States 0.3146 0.1940 Cognac 0.5515 0.3048 Spain -0.3686 0.2601 California 0.0663 0.1468 Portugal 0.3361 0.1859 Loire -0.6398 0.2653 Cuba -0.3927 0.3549 Italy -0.2613 0.1423 Oregon -0.7325 0.2616 South Africa -0.2169 0.3109 Ribera del Duero 0.1077 0.3155 Islay 0.4411 0.4889 South West France 0.1415 0.4845 Intercept 3.8629 0.2203 Observations 2005 R2 0.4911 Coeﬃcients and standard errors from OLS regression, based on auctions with r = 0 and at least two bidders.

21 Table A.7: Homogenization estimates (high-end sample)

DEPENDENT VARIABLE: log(second-highest bid per bot- Point S.E. tle) estimate Number bottles -0.2278 0.0277 Number bottles2 0.0048 0.0010 Contains one bottle 0.5018 0.0848 Case of 6 -0.1969 0.0904 Case of 12 0.0915 0.1407 Stored in warehouse -0.1647 0.1845 Special format bottle 0.1438 0.0932 Month auction ends -0.0039 0.0061 Textual description related to “en Primeur” 0.0358 0.0496 Textual description related to delivery / shipping -0.0017 0.0476 Textual description related to expert opinion -0.0475 0.0581 Number words in textual description 0.00018 0.0003 Delivery and payment options: Duty estimate -0.0027 0.0070 VAT estimate 0.0005 0.0038 Delivers to UK -0.0828 0.0611 Returns accepted -0.0099 0.1074 Payment by bank 0.0166 0.1192 Payment by Paypal -0.1554 0.0587 Payment by cheque -0.0279 0.0616 Payment in cash 0.2319 0.1365 Ships with Royal Mail 0.1109 0.0738 Ships with Parcel Force -0.1972 0.0873 Ships fast -0.1491 0.1062 Insurance included 0.0009 0.0507 Can be collected in person -0.0525 0.0556 Can only be collected in person -0.1262 0.1224 Lowest shipping cost quote 0.0033 0.0028 Seller ratings: Seller ratings -0.0004 0.0003 Number seller ratings -0.0806 0.0536 Number seller ratings2 2.0e-07 1.3e-07 Fill level: Into Neck (IN) 0.0400 0.0609 Base of Neck (BN) 0.0087 0.0971 Mid Shoulder (HS) -0.0988 0.1425 Top Shoulder (TS) 0.2645 0.2474 Very Top Shoulder (VTS) -0.0797 0.1106 High Shoulder (HS) -0.0365 0.1730 Low Shoulder (LS) or worse -0.2564 0.2355

22 Table A.7: ... continued (high-end sample)

Type: White -0.1875 0.1179 Red 0.0207 0.0875 Assorted 0.0519 0.0889 Fortiﬁed 0.1473 0.1766 Region: Rhone -0.1290 0.1268 Bordeaux 0.0616 0.0918 France -0.0820 0.0863 Champagne -0.1126 0.1121 Other -0.15120 0.1508 South Australia -0.1578 0.1211 United States -0.3858 0.2828 Tuscany -0.1961 0.1416 Spain -0.4803 0.3469 Scotland -0.1395 0.3675 Burgundy 0.0353 0.1098 Ribera del Duero 0.1770 0.2439 California -0.2141 0.1637 Piedmont/Lombardy -0.1210 0.1710 Portugal 0.0137 0.1786 Douro -0.1803 0.2469 Australia -0.1566 0.2075 Italy -0.5007 0.2743 Veneto -0.0922 0.3426 Alsace -0.3104 0.3530 Islay 0.1814 0.3745 Cognac -0.4997 0.3740 Oporto -0.2059 0.3667 Intercept 5.6440 0.2127 Observations 372 R2 0.9167

Coeﬃcients and standard errors from OLS regression, based on auctions with r = 0 and at least two bidders.

23 Table A.8: Antitrust damages of doubling commission index (cB + 0.1281)

Total damage Incidence on Hammer price Buyer damage Seller damage (1000s pounds) sellers (%) (% change) (% post-hammer) (% post-hammer) Benchmark pro-rata 0.0 0.0 12.81 0.0 Benchmark elastic sellers 100.0 -11.36 0.0 12.81 Simulated impacts: No entry 13.5 90.5 -15.0 1.2 11.8 No seller entry 16.5 76.9 -16.5 4.2 14.0 Full two-sided entry 17.0 61.0 -12.1 8.3 13.0 Estimates for the non-hypothetical case are estimated in auctions with r > 0 in the main sample. To stay close to antitrust applications, damages are computed as a share of the counterfactual expected hammer price (expected sale probability multiplied by the expected hammer (transaction) price conditional on a sale). Buyer and seller damages are computed in expectations for groups of buyers and sellers, with a buyer being the in expectation winning bidder, including in unsold listings. Increasing the buyer commission from 0 to 0.1281 brings about the doubling of the commission index, just as increasing the seller commission from 0.102 to 0.204 as done in Table7 in the main text.

Figure A.2: Ullage classiﬁcation and interpretation

Source: Christie’s. Numbers refer to auction house Christie’s interpretation of the ﬁll levels, which are for Bordeaux- style bottles: 1) Into Neck: level of young wines. Exceptionally good in wines over 10 years old. 2) Bottom Neck: perfectly good for any age of wine. Outstandingly good for a wine of 20 years in bottle, or longer. 3) Very Top- Shoulder. 4) Top-Shoulder. Normal for any claret 15 years or older. 5) Upper-Shoulder: slight natural reduction through the easing of the cork and evaporation through the cork and capsule. Usually no problem. Acceptable for any wine over 20 years old. Exceptional for pre-1950 wines. 6) Mid-Shoulder: probably some weakening of the cork and some risk. Not abnormal for wines 30/40 years of age. 7) Mid-Low-Shoulder: some risk. 8) Low-Shoulder: risky and usually only accepted for sale if wine or label exceptionally rare or interesting. For Burgundy-style bottles where the slope of the shoulder is impractical to describe such levels, whenever appropriate [due to the age of the wine] the level is measured in centimetres. The condition and drinkability of Burgundy is less aﬀected by ullage than Bordeaux. For example, a 5 to 7 cm. ullage in a 30 year old Burgundy can be considered normal or good for its age.

24 Table A.9: Additional details selected counterfactual fee simulations

Fee structure: BL (1) (2) (3) (4) (5) (6) (7) (8) cB 0 0 -0.1 -0.1 -0.1 -0.03 0.05 0 -0.1 cS (main) 0.102 0.1 0.2 0.25 0.05 0.08 0.102 0.152 0.102 cS (high-end) 0.09 0.1 0.2 0.25 0.05 0.08 0.09 0.14 0.09 cL 2.1 2.1 2.1 2.1 10 6.75 2.1 2.1 20 BL Percentage change relative to baseline (BL)

Platform revenue 40.5 2.3 10.1 25.5 -61.2 0.7 27.6 31.8 32.6 Volume 20.9 -1.1 7.9 1.3 26.7 4.7 -13.4 -10.6 -0.1 Platform composition changes: Transaction price (avg) 1.3 -0.5 9.6 8.7 10.8 -0.4 -8.7 -4.8 -6.2 Sale probability (avg) 0.8 -0.1 -1.1 -3.7 5.8 -0.5 -4.8 -5.4 -13.2 Seller entry prob (main) 0.8 0.2 -0.7 -1.6 -0.3 -4.3 -4.3 -4.9 -24.4 Seller entry prob (high) 0.9 -0.8 -1.6 -7.7 3.0 -0.7 -3.9 -4.5 -4.1 Share high-end listings 0.2 -0.8 -0.8 -5.2 2.7 3.1 0.3 0.3 21.2 Share high-end proﬁts 0.4 5.8 6.7 21.6 -157.5 -19.4 4.3 4.3 -51.9 Number bidders r > 0 (main) 4.3 0.1 0.1 -0.6 8.2 5.3 -0.7 -0.8 15.2 Number bidders r > 0 (high) 5.3 -1.7 -3.1 -12.7 19.4 6.9 -6.8 -8.2 18.4 Welfare impacts: Total seller surplus (m) 42.5 -0.4 -4.6 -8.8 17.8 -6.3 -17.3 -18.3 -39.9 Total seller surplus (h) 32.7 -6.5 -10.6 -41.6 74.5 14.4 -23.8 -27.1 30.9 Total winning bidder surplus (m) 66.1 -0.3 -1.2 -2.6 5.0 -0.9 -4.8 -5.2 -13.7 Total winning bidder surplus (h) 43.3 -2.7 -3.7 -15.8 18.1 4.4 -8.7 -9.8 10.5 Per-person winning bidder surplus (m) 23.3 -0.5 -0.5 -1.0 5.4 3.5 -0.6 -0.3 14.1 Per-person winning bidder surplus (h) 72.7 -1.9 -2.2 -8.7 14.7 5.1 -5.0 -5.5 15.2 Per-person seller surplus (m) 15.0 -0.6 -4.0 -7.2 18.2 -2.2 -13.6 -14 -20.5 Per-person seller surplus (h) 54.8 -5.7 -9.1 -36.7 69.5 15.2 -20.8 -23.7 36.4

Simulations based on homogenized auctions in the main and high-end samples. Reserve price fee ﬁxed at baseline level.

25 Figure A.3: Listing page example