Simultaneous use of auctions and posted prices

European Economic Review 78 (2015) 269–284

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European Economic Review journal homepage: www.elsevier.com/locate/eer

Simultaneous use of auctions and posted prices Patrick Hummel Google Inc., 1600 Amphitheatre Parkway, Mountain View, CA 94043, United States

a r t i c l e i n f o

abstract

Article history: Received 8 December 2014 Accepted 24 June 2015 Available online 4 July 2015

I consider a model in which several identical objects are sold simultaneously via an auction and a posted price mechanism. The model explains several empirical regularities regarding bidding behavior in eBay auctions such as the finding that some bidders bid multiple times over the course of the auction, and that bidders tend to bid with greater frequency near the end of the auction than the beginning. I also show that sellers prefer to simultaneously use auctions and posted prices than to use either mechanism individually. & 2015 Elsevier B.V. All rights reserved.

JEL classification: D44 Keywords: Auctions Posted prices

1. Introduction In modern times, prospective buyers of an object often have the opportunity to try to win the object in an online auction while the object is simultaneously available at a posted price. Major companies such as CompUSA, IBM, and Sam's Club have auctioned off objects on auction websites while simultaneously offering buyers the option of purchasing the same items at posted prices. Similarly, several dedicated eBay merchants simultaneously use auctions and posted prices in their businesses (Einav et al., 2013). There are two empirical regularities about bidding behavior in eBay auctions that are difficult to explain under standard models of auctions. First, bidders bid more frequently near the end of the auction than they do at the beginning of the auction (Ariely et al., 2005; Bajari and Hortaçsu, 2003; Ockenfels and Roth, 2006; Roth and Ockenfels, 2002; Steiglitz, 2007; Wilcox, 2000). This finding remains true even after one controls for the tendency of some bidders to engage in a practice known as “sniping”, in which a bidder submits a bid at the absolute last minute of the auction.1 Second, it is common for bidders to bid multiple times in eBay auctions. For instance, Bajari and Hortaçsu (2003) note that the average bidder makes two proxy bids in eBay auctions, and Ockenfels and Roth (2006) find that the average number of bids made by a bidder is 1.89. This paper proposes a model of simultaneous auctions and posted prices that accounts for all of the empirical regularities described above. I consider a model in which identical objects are sold simultaneously via an auction and a posted price mechanism. Buyers may purchase at the posted price at any time, and if they do, they immediately obtain an object at that price. However, if a buyer enters the auction, then the buyer can never hope to obtain an object at the auction until the

E-mail address: [email protected] For example, Bajari and Hortaçsu (2003) note that while 25% of winning bids on eBay are submitted in the final eight minutes, over half of all bids are submitted after 90% of the auction duration has passed, indicating that the bidders bid more frequently in the last 10% of the auction even if one ignores the last-minute bidding. Ockenfels and Roth (2006) also note that the rate at which bidders bid in the last hour exceeds the rate at which they bid throughout the rest of the auction on Amazon where sniping is not possible because placing a last-minute bid automatically extends the length of the auction. 1

http://dx.doi.org/10.1016/j.euroecorev.2015.06.006 0014-2921/& 2015 Elsevier B.V. All rights reserved.

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auction ends. Throughout I assume that buyers never demand more than one object, and buyers who value an object for more than its price would prefer to receive an object as quickly as possible. I consider two possible ways that the seller running the auction may choose to operate the auction. In the first scenario, the seller running the auction never reveals any information about any bids that have taken place until the end of the auction. In the second scenario, the seller always reveals the current value of the price bidders in the auction would have to pay if there is no more bidding before the end of the auction. I assume throughout that buyers who are potentially interested in buying an object arrive according to a Poisson process, and any buyers who arrive have values for an object that are independent and identically distributed draws from some distribution. Regardless of whether the seller reveals information about bids that have taken place, the equilibrium strategies for the buyers have the same overall form. Individuals whose values for an object are lower than the reserve price never attempt to buy an object at either the auction or the posted price. Prospective buyers with values between the auction reserve price and the posted price all enter the auction and bid their values. And buyers who have values for an object greater than the posted price sometimes choose to bid in the auction but sometimes choose to buy at the posted price. These buyers are more willing to bid in the auction if they have lower values for an object (because such a buyer would not be able to achieve much profit at the posted price) or if they arrive closer to the end of the auction (since then there is less cost to waiting until the end of the auction). The equilibria in these settings generate predictions that are consistent with the two empirical observations described in the second paragraph of this manuscript. Buyers will bid more frequently near the end of the auction, and if the seller reveals information about the current second-highest bid, then buyers often have an incentive to place multiple bids over the course of the auction. The first of these results holds because more buyers with values greater than the posted price will enter the auction near the end of the auction, and the second of these results holds because buyers with values greater than the posted price are willing to bid more near the end of the auction than the beginning, so they may be willing to revise their bids near the end of the auction if another bidder outbid their initial bid. Since high valuation buyers purchase at the posted price, but low valuation buyers purchase at the auction, simultaneously running auctions and posted prices effectively provides a way to price discriminate between high and low-value buyers. I show in the paper that regardless of whether the seller reveals information about bids that have taken place, the seller can achieve greater revenue by simultaneously selling objects via auctions and posted prices than by only using auctions or only posted prices. I also argue that this result will hold even if the seller may offer multiple shipping options or make use of sequential auctions. There has been relatively little work that analyzes what happens when buyers can simultaneously buy an object via an auction or a posted price. The only other such papers that I am aware of are Caldentey and Vulcano (2007), Etzion and Moore (2013), Etzion et al. (2006), and Sun (2008). My paper differs from these papers in several ways. My paper is the first to analyze equilibrium strategies in a setting in which the seller running the auction reveals information about bids that have taken place over the course of the auction.2 As a result, I am also able to show how the equilibrium strategies in such a setting are consistent with empirical bidding behavior in eBay auctions. Finally, in the framework I consider I am able to present analytic results that indicate that a seller can always do better by simultaneously selling objects via auctions and posted prices, but these papers do not.3 Celis et al. (2014) further present an analysis of a randomized mechanism that they call “buy-it-now or take-a-chance” in which bidders have the option of first buying an object at a posted price, and if nobody buys the object at a posted price, the object is then sold at random to one of the top d bidders. My paper differs from this work in that I explicitly consider an underlying dynamic model in which potential buyers arrive over time, and as a result I am able to account for empirical regularities in eBay auctions that relate to the dynamic nature of this auction. In addition to the above papers, there has also been work that compares the use of auctions with the use of posted prices in a variety of settings (Hammond, 2010, 2013; Julien et al., 2002; Kultti, 1999; Wang, 1993, 1998; Vakrat and Seidmann, 1999; van Ryzin and Vulcano, 2004; Vulcano et al., 2002; Zeithammer and Liu, 2008), work on optimal selling mechanisms when buyers arrive dynamically (Board and Skrzypacz, 2014; Vulcano et al., 2002), and work on models of eBay auctions and related markets (Ackerberg et al., 2006; Ambrus et al., 2014; Bajari and Hortaçsu, 2003; Chen et al., 2013; Hidvegi et al., 2006; Peters and Severinov, 2006; Ockenfels and Roth, 2006; Roth and Ockenfels, 2002). Finally, there has also been work on sequential search mechanisms when it is costly for buyers and sellers to interact that illustrates that it can be optimal to first see if there is an agent willing to complete a transaction at a particular price and then complete the transaction with the agent willing to accept the most favorable price possible if no agent accepted the initial offer (Ehrman and Peters, 1994; McAfee and McMillan, 1988). However, these papers do not analyze models where buyers must simultaneously consider whether to buy an object at an auction or a posted price. As such, my paper analyzes a very different model than these papers.

2 The only previous paper that considers a model in which the seller running the auction reveals information about bids that have taken place over the course of the auction is Etzion and Moore (2013), but this paper does not derive any analytic results about equilibrium strategies and instead solely focuses on conducting simulations under various heuristics bidders might adopt under such mechanisms. 3 This result also contrasts with Riley and Zeckhauser (1983), which finds circumstances under which always committing to a posted price is optimal.

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2. The model A seller sells a set of homogeneous objects simultaneously via a second-price auction and a posted price mechanism. There is a single object available at the auction and there are an infinite number of objects available for sale at the posted price. Throughout I let r denote the reserve price at the auction and p denote the posted price at which objects are being sold. The assumption that there are an infinite number of objects available for sale at the posted price is used to model situations in which the seller always has enough objects available to meet any buyer demand (or can produce or acquire more as needed), and is realistic in the many scenarios where merchants offer a seemingly limitless supply of objects for sale.4 The auction begins at time t¼0 and ends at time t ¼T. During this time, potential buyers arrive according to a homogeneous Poisson process with arrival rate λ. Each buyer who arrives has a value for an object that is an independent and identically distributed draw from a continuous cumulative distribution function F with corresponding continuous density f and support equal to ½v; v, where v 4 v Z0. Any time after a buyer arrives, the buyer may either buy an object at the posted price or place a bid in the auction. If a t buyer obtains an object at time period t by paying a price β, then the buyer obtains a utility of δ ðv  β Þ for the game, where v 5 denotes the buyer's value for an object, and δ A ð0; 1Þ denotes the buyer's discount factor. Throughout I also assume that a buyer's valuation for a second object beyond the first object is 0 and if a buyer never buys an object, the buyer obtains zero utility. A seller's total payoff for the game is equal to the total revenue that the seller was able to bring in from each of the objects that the seller sold. After the auction is over, a buyer still has the option of buying at the posted price p. I consider two different ways that a seller may run the auction. In the first scenario, the seller never gives any information about what bids the bidders have placed during the course of the auction, and the seller only reveals the price the winner of the auction must pay when the auction ends at time t¼T. In the second scenario, the seller always publicly reveals the price the winner of the auction would have to pay if no more bids were placed, and any time after a buyer has arrived, the buyer may either place a new bid in the auction or buy at the posted price.6 This formulation implicitly allows buyers who arrive to wait before bidding in the auction or buying at the posted price, though this possibility will never arise in equilibrium.

3. Equilibrium strategies for buyers Throughout this section I assume that the reserve price in the auction is nonnegative and less than the posted price (r o p). If r op did not hold, then a buyer would necessarily have to pay at least as much to obtain an object in an auction as to buy the object at the posted price even though the buyer would be able to obtain the object more quickly if the buyer bought the object at the posted price. Thus any buyers who decided to try to buy an object would buy at the posted price. To ensure that there is at least some chance that some buyers will bid in the auction, I assume throughout that r o p. I begin by illustrating the types of strategies that buyers would use in equilibrium if sellers never reveal any information about bids that have taken place. First note that if a buyer has a value v o r, then the buyer will never try to either buy at the posted price or to bid in the auction because the buyer can never obtain a positive payoff from buying an object. Also note that if a buyer has a value v A ½r; p, then the buyer would never have an incentive to buy at the posted price, but the buyer would have an incentive to enter the auction. Furthermore, if such a buyer enters the auction, the buyer has a weakly dominant strategy of always bidding his or her value. Thus if a buyer has a value v A ½r; p, then the buyer follows a strategy of entering the auction and bidding his or her value at the auction. Now consider a buyer who has a value v 4 p. Note that if this buyer enters the auction, then the buyer has a weakly dominant strategy of bidding b¼p. If the buyer enters the auction and does not win an object at the auction, then the buyer T will choose to buy an object at the posted price as soon as the auction is over, and the buyer will obtain a payoff of δ ðv pÞ. By contrast, if the buyer wins an object at the auction and pays a price b for the object, then the buyer obtains a payoff of δT ðv bÞ. Thus the buyer is willing to pay any amount b satisfying δT ðv  bÞ Z δT ðv pÞ or b rp in the auction, and if a buyer with value v 4 p enters the auction, then the buyer has a weakly dominant strategy of bidding b¼p. Now I derive the payoff a buyer with value v 4 p obtains from entering the auction. To do this, let π denote the probability a buyer with value v 4 p believes that at least one other buyer with value v 4 p will enter the auction and let a denote what such a buyer regards as the expected value of the second-highest bid in the auction conditional on the buyer entering the auction and no other buyers with values v 4p entering the auction. Note that if the buyer enters the auction and at least one other buyer with value v 4 p enters the auction, then the buyer T obtains a payoff of δ ðv  pÞ because the buyer either wins an object at price p in the auction or the buyer loses the auction and then buys the object at price p via the posted price mechanism. And if the buyer enters the auction and no other buyers T with values v 4 p enter the auction, then the buyer obtains an expected payoff of δ ðv  aÞ because the buyer wins the 4

The results in this paper also hold regardless of whether the seller's goods come from an endowment or they are produced. t  tn Another reasonable modeling assumption would be to assume that a buyer who arrives at time t n obtains utility δ ðv  βÞ from buying an object for price β at time t. Making this alternative assumption would not affect any of the analysis in the paper. 6 While my model allows a buyer to buy at the posted price even if the buyer has the highest bid at that point, this possibility will not arise in equilibrium, so my results extend to a model in which this is prohibited. 5

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auction and pays a price of a in expectation. Combining these results, we see that if a buyer with value v 4 p enters the T auction, the buyer obtains an expected payoff of δ ½π ðv  pÞ þ ð1  π Þðv  aÞ. However, if the buyer instead buys at the posted price at time t when the buyer first arrives, the buyer obtains a payoff of δt ðv pÞ. From this it follows that if a buyer with value v 4p first arrives at time t, then this buyer has an incentive to enter T t t T the auction if and only if δ ½π ðv pÞ þ ð1  π Þðv  aÞ Z δ ðv  pÞ or δ r π þ ð1  π Þðv  aÞ=ðv pÞ. Now since a o p, the righthand side of this equation is decreasing in v. And the left-hand side of this equation is decreasing in t. This indicates that buyers with values v 4p are more likely to be willing to enter the auction if they have relatively lower valuations for an object or if they first arrive when a greater amount of time has elapsed in the auction. I now use these preliminaries to derive the following equilibrium result: Proposition 1. Suppose the seller running the auction does not reveal information about any bids that have been placed before the end of the auction. Then there exists a unique symmetric equilibrium characterized by a set of values fvn ðtÞg for t o T with the following properties: (a) No buyer with value v or enters the auction or buys at the posted price. (b) All buyers with values v A ½r; p enter the auction and bid their value. (c) A buyer with value v 4 p who arrives at time t oT enters the auction if v rvn ðtÞ and buys at the posted price at time t otherwise. (d) A buyer with value v 4 p who arrives at time t¼T enters the auction. (e) All buyers with values v 4p who enter the auction bid b¼p. (f) If a buyer enters the auction and does not win an object at the auction, then the buyer buys at the posted price at time T if and only if the buyer's value v is greater than p. (g) vn ðtÞ is strictly increasing in t.

We have already seen the mathematical reasons for why buyers would follow strategies of the form given in Proposition 1, so here I just try to explain the economic intuition behind Proposition 1. In addition to indicating that buyers with values v o r or v A ½r; p will follow their dominant strategies, this proposition also indicates that buyers with values v 4 p are more likely to bid in the auction if there is little time left in the auction or if they have smaller values for an object. The intuition for these results are as follows: When there is little time left in the auction, a buyer would only have to delay purchasing at the posted price for a small amount of time in order to try to win the object at a cheaper price in the auction, so it makes sense for a buyer to do so. By contrast, if a buyer would have to wait a long time for the auction to end, that buyer has more of an incentive to simply purchase at the posted price because attempting to win an object at an auction carries a much larger cost of delay. Also, when a buyer has a value that is only slightly greater than the posted price, the buyer will not be able to achieve much profit by purchasing at the posted price, so the buyer has a strong incentive to enter the auction. By contrast, if a buyer's value is significantly greater than the posted price, then the buyer will achieve large profits even if the buyer purchases at the posted price, so it makes sense for the buyer to try to achieve these profits as quickly as possible and simply purchase at the posted price. The result that potential buyers will bid with greater frequency near the end of the auction than at the beginning of the auction is consistent with empirical observations that a disproportionate percentage of bids in eBay auctions tend to be received near the end of the auction (Ariely et al., 2005; Bajari and Hortaçsu, 2003; Ockenfels and Roth, 2006; Roth and Ockenfels, 2002; Steiglitz, 2007; Wilcox, 2000). While the observation that a disproportionate percentage of bids in eBay auctions tend to be received near the end of the auction is inconsistent with eBay's recommendations that bidders simply bid their values for an object as soon as they arrive at the auction, and is also difficult to explain in a standard model of auctions, this prediction arises naturally in the present framework where potential buyers have an outside option of purchasing at a posted price. Proposition 1 indicates that there can be uncertainty as to whether a buyer with value v 4p will enter the auction, as buyers with relatively low values will choose to enter the auction, but buyers with higher values for an object might not choose to enter the auction. Given that only some of the buyers with values v 4 p will choose to enter the auction, it is interesting to ask how the probability that a buyer with value v 4 p will choose to enter the auction varies with the parameters of the game. I answer this question in Corollary 1: Corollary 1. Suppose the seller running the auction does not reveal information about any bids that have been placed before the end of the auction. Then the probability a buyer with value v 4 p enters the auction, unconditional on the time the buyer arrives and the precise realization of the buyer's value, is always nonincreasing in r and λ and strictly decreasing in r and λ for values of r and λ such that there is a positive probability that a buyer with value v 4p would choose not to enter the auction (i.e. vn ðtÞ is decreasing in r and λ). To understand this result, note that when the reserve price is relatively lower, a buyer who enters the auction will not have to pay as much for an object on average, so a buyer's expected payoff from entering the auction is relatively greater.

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Similarly, when the rate at which buyers arrive to purchase an object, λ, decreases, there will typically be less competition in the auction, and buyers who enter the auction are likely to obtain a relatively greater payoff. Thus decreases in the reserve price, r, and the rate at which buyers arrive, λ, mean that buyers with values v 4 p will be relatively more willing to enter the auction. Having addressed how buyers behave in the scenario in which the seller running the auction does not reveal any information about what bids have been placed over the course of the auction, I now illustrate the types of strategies that buyers use if the seller running the auction reveals information about the current price the winner would have to pay if no more bids were placed in the auction. In such a setting, the buyers follow strategies whereby they bid in the auction if and only if the amount they would have to bid in order to have the highest bid in the auction is sufficiently small. Formally, we have the following result:

Proposition 2. Suppose the seller running the auction reveals the value of the current price the winner would have to pay every time a new bid is placed. Then there exists a unique symmetric equilibrium for the buyers characterized by cutoffs wn ðv; tÞ with the following properties: (a) No buyer with value v or enters the auction or buys at the posted price. (b) All buyers with values v A ½r; p enter the auction and bid their value if their value is less than the current second-highest bid, and do not enter the auction or buy at the posted price otherwise. (c) A buyer with value v 4 p who has arrived at time t and not previously purchased at the posted price enters the auction and bids wn ðv; tÞ if this is higher than the current second-highest bid, and buys at the posted price otherwise. The buyer then immediately buys at the posted price if wn ðv; tÞ was not the highest bid in the auction. (d) wn ðv; TÞ ¼ p for all v 4 p. (e) wn ðv; tÞ is strictly decreasing in v for all v 4 p and t o T. (f) wn ðv; tÞ is strictly increasing in t.

A buyer's behavior when the seller reveals the value of the second-highest bid every time a new bid is placed shares many similarities with the buyers' behavior when the seller does not reveal this information. Potential buyers with small values for an object ðv o rÞ never buy an object and buyers with moderate values for an object ðv A ½r; pÞ always enter the auction rather than paying the more expensive posted price to buy an object immediately. Finally, buyers with large values for an object ðv 4 pÞ may either choose to bid in the auction or to buy at the posted price depending in part on the buyer's value for an object and the time remaining until the auction ends. Such buyers are more willing to bid in the auction if there is little time left in the auction or if they have smaller values for an object in the sense that the maximum amount they are willing to bid at the auction is larger when there is less time remaining in the auction and when they have smaller values for an object. The intuition for this last part is identical to the intuition behind the corresponding result in Proposition 1. It is worth noting that buyers who arrive will either bid in the auction or purchase an object at the posted price as soon as they arrive rather than waiting. A buyer who bids in the auction can always choose to buy at the posted price later if the competing bids in the auction become too high, so bidding in the auction does not restrict the seller's options. However, if a buyer makes a bid in the auction that is higher than the current second-highest bid, then the buyer can discover if this bid is high enough to be the high bid in the auction, and immediately purchase at the posted price if another bidder has already made a bid in the auction that is higher than this bidder's willingness to pay. This option is not available to the buyer if the buyer waits before bidding in the auction. Thus buyers always prefer to bid in the auction as soon as they arrive than to wait until later to bid. The equilibrium in Proposition 2 also has the feature that if a buyer with value v 4 p stays in the auction for an extended period of time, then the buyer will necessarily choose to increase his or her bid over the course of the auction since the value of wn ðv; tÞ is strictly increasing in t for v 4 p. Thus we would expect to see some bidders place multiple bids over the course of the auction in the setting considered in Proposition 2. The possibility that bidders place multiple bids over the course of an auction is consistent with empirical evidence on eBay auctions, as Ockenfels and Roth (2006) and Wilcox (2000) note that it is quite common for bidders to place multiple bids in eBay auctions, and Steiglitz (2007) notes further anecdotes regarding bidders placing multiple bids on eBay. Nonetheless, this is a strategy that is difficult to explain under more standard models of auctions and it is also inconsistent with eBay's recommendations that buyers simply bid the maximum amount they are willing to pay for an object as soon as they arrive at the auction. Finally, I comment on one other aspect of the equilibrium in Proposition 2. In the equilibrium in Proposition 2, a bidder with value v 4 p will continuously revise his bid over time even if he has not been outbid. It is worth noting, however, that there is an equivalent equilibrium in which a bidder with value v 4 p only revises his bid over time if the bidder has previously been outbid. Revising one's bid in the auction only has an effect on the individual's payoff if the bidder has previously been outbid, so making this change to a bidder's strategy would have no effect on the payoffs of any of the potential buyers in Proposition 2. If bidding were infinitesimally costly, then the equilibrium in which bidders with values v 4 p only revised their bids if they were outbid would become the unique equilibrium.

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4. Revenue effects This section illustrates that a seller can obtain greater expected revenue by simultaneously selling objects via an auction and a posted price mechanism than by only using one of these mechanisms. This provides a possible explanation for why we might see so many major companies auction off objects on websites while simultaneously offering buyers the option to purchase the same items for sale at a posted price. First I prove the following result: Proposition 3. A seller obtains a strictly larger expected payoff by simultaneously selling an object via an auction and also selling objects via a posted price mechanism than by only selling objects via a posted price mechanism. This result holds regardless of whether the seller reveals information about the second-highest bid that has been placed in the auction. To understand the intuition behind this result, suppose that when the seller only sells items via a posted price mechanism that the seller chooses to sell the items at a posted price of p. If the seller instead simultaneously sells objects via an auction and a posted price mechanism, then the seller can use a reserve price of p at the auction and simultaneously sell the objects with a slightly higher posted price via the posted price mechanism. In this case, almost all buyers whose values for the object exceed the new posted price will choose to simply buy at the posted price rather than wait around to try to win an object at the auction since the smallest possible price at the auction is nearly as large as the posted price. Thus it is unlikely that more than one potential buyer will bid in the auction, and the seller will be able to sell an object to all potential buyers with values v Z p who arrive. Thus if the seller simultaneously sells objects via an auction and a posted price mechanism, the seller will effectively be able to sell objects for higher prices to buyers who have high valuations for an object while still selling objects to buyers who have moderate valuations for an object at the same price as before. Since simultaneously selling objects via a posted price mechanism and an auction effectively enables the seller to price discriminate between buyers who have a high valuation for an object and buyers who have more moderate valuations for an object, simultaneously using these two mechanisms leads to greater seller profit than only selling objects via a posted price mechanism. This result holds regardless of whether the seller reveals information about individual bids over the course of the auction. Proposition 3 illustrates that a seller can do better by simultaneously selling objects via a posted price mechanism and an auction than by only selling objects via a posted price mechanism. To illustrate that a seller can do better by simultaneously using these two mechanisms than by using one separately, it is also necessary to illustrate that the seller can do better by simultaneously selling objects via an auction and a posted price mechanism than by only selling objects via an auction. This is done in Proposition 4: Proposition 4. A seller obtains a strictly larger expected payoff by simultaneously selling an object via an auction and also selling objects via a posted price mechanism than by only selling an object via an auction. This result holds regardless of whether the seller reveals information about the second-highest bid that has been placed in the auction. To understand the intuition behind this result, suppose that the seller would auction off an object using a reserve price of r if the seller were only selling via an auction, and consider what would happen if the seller instead sold an object at an auction with a reserve price of r while simultaneously selling the objects at a significantly higher posted price. If this posted price is high enough, almost all buyers who have values for an object greater than the reserve price will choose to enter the auction. And if it happens to be the case that buyers only try to buy at the auction, then the decision to also use the posted price mechanism has no effect on revenue since the final auction price will be the same as it was before. The only circumstance under which a buyer will choose not to buy at the auction is if an individual is one of the very few buyers who has a value for an object greater than the posted price and the seller ends up selling the item being sold in an auction at a price very close to the posted price. But under this circumstance the seller is better off simultaneously selling objects via an auction and the posted price mechanism. In this scenario, the seller sells an object at a very high price in the auction and also sells additional objects at a posted price in the posted price mechanism. By contrast, if the seller were only selling an object in an auction, the seller would still sell this object at a high price, but the seller would forgo the extra revenue from the posted price mechanism. Thus the seller does better by simultaneously selling objects via an auction and a posted price mechanism than by only selling via an auction. This result again holds regardless of whether the seller reveals information about individual bids over the course of the auction.

5. Extensions 5.1. Comparing revenue across information structures The previous section illustrated that simultaneously using auctions and posted prices results in greater revenue for the auctioneer than using either mechanism separately. This result holds regardless of whether the seller reveals information about the price the current high bidder in the auction would have to pay if no other bidders enter the auction before the auction finishes. Should the seller reveal information about the price the current high bidder in the auction would have to pay if no other bidders enter the auction before the auction finishes?

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Unfortunately there is no general result as to whether it is beneficial for the seller to reveal information about the price the current high bidder in the auction would have to pay if no other bidders bid in the auction. The reason for this has to do with the fact that there are some circumstances in which revealing this information will benefit the seller, but there are other circumstances where this will hurt the seller's revenue. I elaborate below. If the seller does not reveal information about the second-highest bid in the auction, there are two strategies that a buyer with value greater than the posted price might follow. Either the buyer will enter the auction and stay in the auction until the end, or the buyer will buy at the posted price immediately. If the seller does reveal this information, then buyers with values greater than the posted price may still follow either of these strategies in equilibrium. But there is also one other strategy they might follow that would not be available to a buyer if the seller did not reveal this information. A buyer might enter the auction with a bid that is higher than the current second-highest bid, learn that his bid is still lower than the highest bid in the auction, and then immediately purchase an object at the posted price. Whether the possibility that a buyer might follow this strategy will be beneficial to the seller will then depend on the types of buyers that arrive. If exactly two buyers arrive and both of these buyers have values greater than the posted price and enter the auction, then this possibility will hurt the seller. If the seller did not reveal information about the secondhighest bid in the auction, then both of these buyers would make a bid in the auction equal to the posted price, and both of these buyers would ultimately purchase the object for a price equal to the posted price. By contrast, if the seller revealed information about the second-highest bid, then one of these buyers would enter the auction with a bid strictly less than the posted price, learn that this bid is lower than the highest bid in the auction, and then immediately purchase an object at the posted price. As a result of this, the buyer who wins the auction would pay strictly less than the posted price, and the seller would earn less total revenue than in the case where the seller did not reveal information about the second-highest bid in the auction. Now consider a setting in which there are again exactly two buyers that arrive but one of these buyers has a value between the reserve price in the auction and the posted price, and the other buyer has a value greater than the posted price but still enters the auction. In this case, the possibility that a buyer might enter the auction with a bid that is higher than the current second-highest bid, learn that his bid is still lower than the highest bid in the auction, and then immediately purchase an object at the posted price could help the seller. If the seller did not reveal information about the second-highest bid in the auction, then this buyer would enter the auction with a bid equal to the posted price, win the auction, and the seller would not sell an item to the lower-value seller. By contrast, if the seller does reveal this information, this same buyer might initially enter the auction with a lower bid, realize that this is not the current high bid, and then immediately buy at the posted price. In this case, the seller would also sell an item to the lower-value seller in the auction and make more money as a result of revealing the information about the second-highest bid. The discussion in the previous two paragraphs illustrates that whether revealing information about the second-highest bid benefits the seller depends crucially on which types of buyers arrive. If multiple buyers with values greater than the posted price arrive, then revealing this information may harm the seller, and if only one such buyer arrives, then revealing this information may be beneficial. Because the relative likelihoods of these events and the sizes of the payoff differences arising under these circumstances depend crucially on the parameters of the model, there is no general result as to whether revealing information about the current price the high bidder in the auction would have to pay if no other bidders arrive would benefit the seller. While there is no general result as to whether revealing information about the second-highest bid benefits the seller, it is worth noting that there is good reason for the seller to prefer to reveal the second-highest bid rather than the highest bid. To see this, suppose exactly one bidder has entered the auction. Also suppose that a second bidder with value v 4 p arrives, and this bidder is not willing to bid as much in the auction as the first bidder. In this case, if the seller only reveals the secondhighest bid, the bidder with value v 4 p who just arrived may nonetheless make a bid w in the auction to see if there is some other bidder in the auction who has bid more than his willingness to pay, and then immediately purchase at the posted price upon realizing that a bid of w will not be sufficient to have the high bid in the auction. However, if the seller reveals the highest bid in the auction, then the bidder with value v 4p will realize that the high bid in the auction is greater than the maximum bid he is willing to make in the auction, even without bidding in the auction himself. Thus in this case, the bidder with value v 4p may just purchase at the posted price right away without first making a bid in the auction and increasing the price that the high bidder in the auction will have to pay. Since this could lead to lower revenue for the seller, the seller is better off only revealing the current value of the second-highest bid, rather than revealing the value of the highest bid.

5.2. Auctions with multiple objects Throughout the analysis so far, I have assumed that when a seller runs an auction that the seller sells no more than one object via the auction. This assumption is consistent with practice, as eBay typically does not allow sellers to sell more than one object in a single auction. However, it is interesting to ask how the results of the paper would be affected if the seller could sell q Z 1 objects via a single auction simultaneously by using a uniform price auction in which the q highest bidders in the auction paid the (qþ1)th-highest bid. Would the possibility that a seller sells multiple units in an auction affect any of the substantive conclusions of the paper? And if a seller were allowed to either sell one object or multiple objects at an auction, would the seller want to avail himself of the option to sell multiple objects at the auction?

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I first note that the possibility that the seller may sell multiple objects via a single auction does not change any of the results in the previous two sections. In a working paper version of this manuscript (Hummel, 2014), I note that an equilibrium satisfying the qualitative properties Propositions 1 and 2 exists when a seller simultaneously sells q objects via a uniform price auction and also sells objects via a posted price mechanism. Furthermore, the results in Propositions 3 and 4 also extend to a setting in which the seller can sell q objects via a uniform price auction. In particular, simultaneously selling q objects via an auction and also selling objects via a posted price mechanism dominates either just using the posted price mechanism or just selling q objects via an auction. Given the option of selling multiple objects via an auction rather than just one object, would the seller then want to sell multiple objects via an auction? The answer to this question will depend crucially on the parameters of the game. If the reserve price is zero and the posted price is substantial, then the seller may very well prefer to sell no more than one object via an auction because if the seller sells q 4 1 objects via an auction, then it may be quite likely that no more than q bidders enter the auction and the seller does not make any profit from the auction. However, if the seller only sells one object via the auction, then the seller may be more likely to make positive profit from the auction. However, if the seller follows the strategy for setting reserve prices and posted prices given in the proof of Proposition 3 that ensures that the simultaneous use of auctions and posted prices yields greater revenue than only using posted prices, then the seller may very well wish to sell multiple objects via auction. In particular, let Q 41 denote the maximum number of objects that a seller is allowed to sell in a single auction, and suppose a seller follows a strategy of setting a reserve price that is equal to the optimal posted price if the seller only used the posted price mechanism and then also setting a posted price that is slightly larger than the reserve price for the posted price mechanism, as in the strategy I have considered in my proof of Proposition 3. In this case, I am able to prove the following result: Proposition 5. Suppose the seller sets a reserve price r in the auction that equals the optimal posted price if the seller only used the posted price mechanism. Also suppose the seller sets a posted price p ¼ r þ ϵ for some sufficiently small ϵ 4 0. Then the seller will prefer to sell Q objects via the auction, where Q denotes the maximum number of objects that a seller is allowed to sell in a single auction. This result does not guarantee that the seller will always prefer to sell multiple objects via the auction, as the result only applies for a particular strategy of setting the posted price and the auction reserve price. Nonetheless it does illustrate that there are important situations in which the seller would want to simultaneously offer as many objects for sale in a single auction as the seller could, as the seller may wish to do this for a particular strategy in which the simultaneous use of auctions and posted prices results in greater payoffs than solely using posted prices. 5.3. Different shipping options Thus far I have focused attention on a setting in which a buyer receives an object immediately after purchasing the object. While this assumption is standard, in reality there is normally a delay from the time that a buyer purchases an object on eBay and the time the buyer receives the object due to the time it takes the object to be shipped. Moreover, on eBay a seller may offer multiple shipping options by allowing the buyer to pay an extra fee for expedited shipping. How would the possibility that the seller may offer multiple shipping options affect the results? One intuition might be that offering different shipping options could eliminate the benefits that can be achieved by simultaneously using auctions and posted prices. When a seller offers multiple shipping options, this effectively gives the seller a way to price discriminate amongst different buyers since the seller will be able to charge higher-value buyers an extra fee for expedited shipping. Since offering multiple shipping options enables the seller to price discriminate even without simultaneously using auctions and posted prices, one might conjecture that no further benefits from using the mixed-auction format can be achieved if the seller offers multiple shipping options. However, even when the seller can offer multiple shipping options, it will still be beneficial for the seller to simultaneously make use of auctions and posted prices. To illustrate this, I consider a model in which, regardless of whether a seller uses auctions, posted prices, or both selling mechanisms, a seller may offer two shipping options, standard shipping and expedited shipping. If a buyer purchases an object at time t using standard shipping, then the buyer receives the object at time t þ t s for some t s 4 0, but if the buyer purchases the object using expedited shipping, then the buyer receives the object at time t þt e for some t e A ½0; t s Þ. However, if the buyer purchases the object using expedited shipping, then the buyer must pay an extra cost s 4 0 to receive the item more quickly, and the seller must pay an extra cost c A ð0; s to ship the item more rapidly. In this setting, if a buyer with value v for an object purchases an object for a price p at time t and selects standard t þ ts

shipping, then the buyer obtains a payoff of δ t þ te

obtains a payoff of δ

v  δ p. If the same buyer instead selects expedited shipping, then the buyer t

v  δ ðp þ sÞ. Thus a buyer prefers to purchase using expedited shipping if and only if δt þ te v  δt ðp þ sÞ 4 δt þ ts v  δt p, which holds if and only if δt ðδte  δts Þv 4 δt s or v 4s=ðδte  δts Þ. t

A consequence of this is that whether a buyer chooses to pay for expedited shipping or just use standard shipping is independent of the price the buyer pays for the object. Thus if a seller simultaneously makes use of auctions and posted prices, the fact that a seller offers both of these shipping options does not change the qualitative nature of the equilibrium bidding strategies in Section 3. The most substantial difference that arises as a result of the different shipping options is that

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the maximum amount a buyer with value v for an object would be willing to pay for an object is maxfδ s v; δ e v  sg since t

t

t þ ts

a buyer obtains a non-negative payoff from purchasing an object at a price p if and only if maxfδ t þ te

δ p; δ t

v

v  δ ðp þsÞg Z0, which holds if and only if p r maxfδ v; δ v  sg. Thus bidders with values less than the posted t

ts

te

price will make bids of maxfδ s v; δ e v sg rather than v in the auction and bidders with values greater than the posted price will use a slightly different cutoff in deciding whether to bid in the auction or buy at the posted price. But the overall nature of the equilibrium strategies Propositions 1 and 2 will not be affected by the possibility of multiple shipping options; the t

t

only difference is that buyers will act as if their values are maxfδ s v; δ e v  sg rather than v in deciding both whether and how to bid in the auction. But since the qualitative nature of the equilibria characterized Propositions 1 and 2 will not be affected by the possibility of multiple shipping options, similar arguments to those used to Propositions 3 and 4 can also be used to prove these results when the buyer offers multiple shipping options. Thus we have the following result: t

t

Remark 1. The result that the simultaneous use of auctions and posted prices should be preferred to using either auctions or posted prices individually is robust to the possibility that the seller offers multiple shipping options. 5.4. Sequential auctions One other assumption I have made in the analysis so far is that there is only a single auction, and bidders optimize without taking into account the possibility that there may be other auctions for the object in the future. While this is a natural starting point for analysis, in some settings, if a bidder fails to win an object in an auction, that bidder may have other opportunities to win the object in other auctions in the future. Moreover, a seller may wish to continue to sell objects via an auction after the seller's initial auction is complete, thereby resulting in the seller using an infinite sequence of auctions rather than a single auction. The possibility that sellers may sell objects using sequential auctions has been analyzed by Said (2011, 2012) in another setting in which new buyers may arrive over time. How would the possibility that the seller sells objects using sequential auctions affect the results? When sellers make use of sequential auctions, bidders may follow significantly different strategies than the strategies they would use in a standard one-shot auction. As noted in Said (2011), in such auctions bidders have an incentive to shade their bids down by their continuation value, which reflects the option value of participating in future auctions. This same possibility will also arise when the seller is concurrently selling the same object using sequential auctions and a posted price mechanism. Since bidders with values between the reserve price and the posted price will be able to achieve a positive expected payoff by not winning the current auction, and instead participating in a future auction, such bidders will not be willing to bid up to their value, and will instead bid less than their value. Thus, unlike in the equilibria Propositions 1 and 2, if a seller is simultaneously selling objects using sequential auctions and posted prices, bidders with values between the reserve price and the posted price will not necessarily bid their values. While a full characterization of the equilibrium when the seller simultaneously uses sequential auctions and posted prices is beyond the scope of this paper, there is good reason to believe that simultaneously using sequential auctions and posted prices will still dominate only using sequential auctions. Consider an extension of the model in Said (2012) in which there is a sequence of single-object auctions, each of which takes place during the time interval ½kT; ðk þ 1ÞT for k ¼ 0; 1; …; 1, and buyers arrive stochastically according to a Poisson process. As in Said (2012), I assume that at the end of each auction, there is some exogenous probability that a losing bidder does not enter future auctions. However, if the seller is simultaneously using a posted price mechanism, I allow for the possibility that a losing bidder who does not enter future auctions can immediately buy at the posted price just after losing an auction. The proof of Proposition 4 proceeds by analyzing a scenario in which one simultaneously runs an auction and a posted price mechanism using a posted price that is so large that a bidder would generally attempt to enter the auction first and only purchase at the posted price if the bidder failed to win an object in an auction. Using a posted price in such a fashion increases the number of objects the bidder is able to sell if a bidder who is willing to pay the posted price fails to win an object at the auction, while having no adverse effect on revenue otherwise. The same reasoning used to prove Proposition 4 should apply equally well to a setting in which a posted price mechanism is used in conjunction with sequential auctions. If a bidder who is willing to pay the posted price fails to win an auction, and would not enter future auctions for an exogenous reason, then by offering a posted price mechanism one will succeed in selling an object to a buyer who would not have ever attempted to buy an object again otherwise, thereby increasing the seller's revenue. Thus the fact that sellers may make use of sequential auctions should not overturn the conclusion that the simultaneous use of auctions and posted prices dominates using only auctions.

6. Empirical evidence This section discusses some of the empirical evidence surrounding the use of mixed-auction formats. The results of my analysis would suggest several predictions that could be tested empirically by analyzing eBay auctions, so it is worthwhile to analyze whether these predictions are consistent with empirical evidence.

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First, given that it is more profitable for a seller to simultaneously sell objects using auctions and posted prices, we should expect many sellers on eBay to frequently choose to simultaneously sell the same object via auctions and posted prices. This is indeed the case empirically. For instance, Einav et al. (2013) note in their empirical study of eBay auctions that “an overwhelming feature of the data is that sellers very commonly use both auctions and posted prices. Moreover, even if sellers favor one sale format, they commonly continue to use both over time, without converging to just one.” Another feature we should expect to see if bidders are following the equilibrium strategies I have given is that the final prices in auctions tend to be lower than the posted prices that are simultaneously available. In the model I have given, bidders never have an incentive to make a bid in the auction that is greater than the posted price, so under these strategies we would expect to see that sales prices in auctions are lower than the corresponding posted prices. This again appears to typically hold for most product categories on eBay. The broadest analysis I am aware of that compares final prices in auctions to posted prices is the analysis in Einav et al. (2015). Einav et al. (2015) pursues an empirical analysis of eBay markets that analyzes thousands of products and hundreds of thousands of sales, in contrast to most prior studies which had focused on a small set of products (such as Bajari and Hortaçsu (2003) and Lucking-Reiley (1999)). Einav et al. (2015) find that “auction prices, on average, are significantly below equivalent posted prices”. In particular, they find that over half of the auction sales they observe occur at a discount of 13 percent or more relative to the posted price. Furthermore, less than 20 percent of auctions resulted in higher sales prices than the equivalent posted prices, and most of these involved very small overpayments. Thus the prediction that prices in auctions tend to be lower than posted prices appears to be broadly consistent with empirical evidence from eBay, although there may be particular cases in which auctions prices have exceeded the corresponding posted prices.7 In addition to finding that auctions prices fall below posted prices, we should also expect to see that auction prices will typically be greater if the object is not simultaneously available for sale through a posted price. When the same object is simultaneously being sold via an auction and a posted price mechanism, we have seen in the equilibrium analysis Propositions 1 and 2 that some bidders who would normally enter the auction in the absence of the posted price will instead simply buy at the posted price, and some bidders who do enter the auction will make lower bids than they would have in the absence of the posted price. Thus the simultaneous presence of a posted price mechanism should depress auction prices relative to posted prices. This again appears to be consistent with evidence from eBay. Einav et al. (2013) note that the average discount in auction prices relative to posted prices tends to be larger when the object is simultaneously available for sale via a posted price on eBay than when it is only available through an auction at that time. Finally, given the results in Propositions 3 and 4, we should find that sellers who simultaneously sell objects using auctions and posted prices on eBay will make more money in expectation than they would if they only used either mechanism individually. Although I am not aware of any empirical evidence that addresses this exact question, some existing evidence on eBay auctions that speaks to a closely related question suggests that this may be the case. Various papers such as Ackerberg et al. (2006), Anderson et al. (2008), and Einav et al. (2015) have analyzed eBay auctions with a buy-it-now price in which a buyer can preempt the auction and immediately purchase the object at the posted price. These auctions with buy-it-now prices have a similar flavor to the type of mixed auction and posted price mechanism I have analyzed in this paper. Ackerberg et al. (2006), Anderson et al. (2008), and Einav et al. (2015) all find evidence that using auctions with buy-it-now prices can result in larger revenues than simply running a standard auction, which Einav et al. (2015) attribute in part due to being able to “discriminate between impatient but possibly high value buyers, and bargain hunters who are willing to wait and bid in the auction”, much as in the equilibrium in my model. While the results in my paper seem broadly consistent with empirical evidence, it is worth noting that the explanation my model offers for the simultaneous use of auctions and posted prices need not be the only explanation. For instance, Einav et al. (2013, 2015) emphasize seller experimentation as an explanation for the fact that so many sellers on eBay simultaneously make use of auctions and posted prices. While seller experimentation is undoubtedly important in online marketplaces, this possibility is not inconsistent with the idea that some sellers may simultaneously use auctions and posted prices to maximize their profits. This point is noted by Einav et al. (2015) who write that “it is unlikely that every price or auction design change is motivated purely by the desire to experiment” as the concurrent listing of auctions and posted prices may “be part of a consumer segmentation strategy”. Furthermore, Einav et al. (2013) also explicitly note that price discrimination through the simultaneous use of auctions and posted prices could be effective in a model with deal shoppers and convenience buyers who are behavioral types.8 7. Conclusion This paper has presented a model in which buyers who arrive stochastically can purchase objects either via an auction or a posted price. In this framework, there is an equilibrium in which buyers with low valuations never attempt to purchase an object, buyers with moderate valuations always enter the auction, and buyers with higher valuations than the posted price 7 In particular, Ariely and Simonson (2003) found that auction prices frequently exceeded posted prices in their analysis of 500 auctions for CDs and DVDs on a large Internet auction site, and Malmendier and Lee (2011) found that auction prices exceeded posted prices 42 percent of the time in their analysis of 167 eBay auctions for the board game, Cashflow 101, that took place from February to September 2004. 8 Bauner (2015) also explains the coexistence of auctions and posted prices with a model of behavioral types in which there are auction lovers who prefer to participate in auctions and normal types.

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bid in the auction if their valuations are not too high or there is little time until the auction ends, but immediately buy at the posted price otherwise. This type of equilibrium exists regardless of whether the seller reveals information about bids that have taken place over the course of the auction. The equilibrium in my model can explain several empirical regularities regarding bidding behavior in online auctions that are difficult to understand when one considers the auction in a vacuum. In eBay auctions it is common for prospective buyers to bid multiple times over the course of the auction and for buyers to bid near the end of the auction more often than they bid earlier in the auction. While these strategies contradict eBay's advice that buyers should simply bid the amount they value an object as soon as they arrive, they arise naturally in the setting where auctions and posted prices are used simultaneously. Here prospective buyers enter and bid in the auction more often when they arrive later and they also generally have an incentive to change their bids over time since they are willing to bid more as the end of the auction nears. The results in the paper also provide a possible explanation for why so many firms have chosen to sell objects via an auction while simultaneously making the same objects available for purchase at a posted price. Such a strategy enables a firm to effectively price discriminate between high-valuation buyers who want to pay more at a posted price in order to obtain an object quickly and low-valuation buyers who prefer to wait until the end of an auction in order to have a chance of buying an object at a lower price. For this reason, a firm obtains larger expected profits by simultaneously using auctions and posted prices than by using either mechanism separately. Finally, I suggest one possibility for future research. I have shown that the seller can do better by simultaneously using auctions and posted prices than by using either mechanism individually. However, there is still the open question of what the optimal selling mechanism would be in this setting. Perhaps a seller would want to simultaneously use auctions and posted prices with a posted price that varies over time. Further research could reveal whether a seller can do better by varying the posted price over time.

Acknowledgements I thank Preston McAfee, Michael Schwarz, the anonymous associate editor, and the anonymous referees for helpful comments and discussions. Appendix Lemma 1. Let β denote the probability with which a buyer with value v 4 p for an object enters the auction unconditional on the time at which the buyer arrives and the precise realization of the buyer's value. Then π, the probability with which a buyer with value v 4 p believes that at least one other buyer with value v 4 p will enter the auction, and a, the expected value of the secondhighest bid conditional on a buyer with value v 4 p entering the auction and no other buyers with values v 4p entering the auction, can both be written exclusively as functions of β (for fixed F, λ, r, and p). Moreover, π ðβÞ and aðβÞ are both continuous and strictly increasing functions of β. Proof. The value of β uniquely determines the distribution of the random number of buyers with values v 4 p who enter the auction, and this distribution in turn uniquely determines the values of π and a. Thus π and a can both be written exclusively as functions of β, which I denote by π ðβÞ and aðβÞ. Moreover, π ðβÞ and aðβÞ are both continuous and strictly increasing functions of β. □ t T

¼ π ðβÞ þ ð1  π ðβÞÞðv aðβ ÞÞ=ðv pÞ. Lemma 2. For any t oT, there is a unique v A ðp; 1Þ that is a solution to the equation δ Moreover, if all other buyers are following a fixed strategy of the form given in Proposition 1 such that the probability a buyer with value v 4 p enters the auction is β, then an individual buyer's best response is to follow a strategy of the form given in Proposition 1 with vn ðtÞ ¼ vnβ ðtÞ, where vnβ ðtÞ denotes the solution to this equation as a function of t. tT

A ð1; 1Þ, limv-p π ðβÞ þð1  π ðβ ÞÞðv aðβÞÞ=ðv  pÞ ¼ 1, and limv-1 π ðβÞ þ Proof. Note that some such v exists because δ ð1  π ðβÞÞðv  aðβ ÞÞ=ðv pÞ ¼ 1. Thus by the intermediate value theorem, there exists some v A ðp; 1Þ satisfying

δt  T ¼ π ðβÞ þð1  π ðβÞÞðv  aðβÞÞ=ðv  pÞ. Moreover, any such v is unique because δt  T is independent of v and π ðβÞ þ ð1  π ðβÞÞðv  aðβÞÞ=ðv  pÞ is strictly decreasing in v for v A ðp; 1Þ. Since vnβ ðtÞ is defined to be the unique v that is a tT

solution to the equation δ and δ

t T

t T

¼ π ðβ Þ þ ð1  π ðβÞÞðv  aðβÞÞ=ðv  pÞ, we have δ

4 π ðβÞ þ ð1  π ðβÞÞðv  aðβÞÞ=ðv pÞ for v 4 vnβ ðtÞ

o π ðβ Þ þ ð1  π ðβÞÞðv  aðβÞÞ=ðv  pÞ for v o vnβ ðtÞ. Thus an individual buyer with value v 4 p who arrives at time t is

indifferent between entering the auction and not entering the auction if the buyer has value v ¼ vnβ ðtÞ, strictly prefers

entering the auction if v o vnβ ðtÞ, and strictly prefers buying at the posted price if v 4vnβ ðtÞ. From this it follows that if all other buyers are following a fixed strategy of the form given in Proposition 1 such that the probability that a buyer with value v 4 p enters the auction is β, then an individual buyer's best response is to follow a strategy of the form given in Proposition 1 with vn ðtÞ ¼ vnβ ðtÞ. □

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Lemma 3. Let hðβÞ denote the probability that a buyer with value v 4 p enters the auction when the buyer is following a strategy of the form given in the statement of Proposition 1 with vn ðtÞ ¼ vnβ ðtÞ unconditional on the time at which the buyer arrives and the n n n precise realization of the buyer's value. Then there is a unique β A ½0; 1 satisfying hðβ Þ ¼ β and the unique symmetric equilibrium is for buyers to follow a strategy of the form given in the statement of Proposition 1 with vn ðtÞ ¼ vnβn ðtÞ Proof. Note that hðβ Þ is a continuous and non-increasing function from ½0; 1 to ½0; 1 since the fact that π ðβ Þ and aðβÞ are

continuous and strictly increasing in β means that vnβ ðtÞ is continuous and strictly decreasing in β and hðβÞ is continuous and

non-increasing in β. Thus hðβÞ has a unique fixed point, and there exists a unique β A ½0; 1 such that hðβ Þ ¼ β . n

n

n

But if β A ½0; 1 satisfies hðβ Þ ¼ β and all other buyers are following a fixed strategy of the form given in Proposition 1 n

n

n

such that the probability a buyer with value v 4p enters the auction is β , then an individual buyer's best response is to follow a strategy of the form given in Proposition 1 with vn ðtÞ ¼ vnβn ðtÞ, a strategy in which this buyer enters the auction with n

probability β . But this means that if all other buyers are following a strategy of the form given in Proposition 1 with vn ðtÞ ¼ vnβn ðtÞ, then an individual buyer's best response is to also follow a strategy of the form given in Proposition 1 with n

vn ðtÞ ¼ vnβn ðtÞ. This indicates that there is a symmetric equilibrium of the form given in Proposition 1. Furthermore, since there is a unique β A ½0; 1 satisfying hðβ Þ ¼ β , this symmetric equilibrium is unique. □ n

n

n

Proof of Proposition 1. We know from Lemma 3 that there is a unique symmetric equilibrium of the form given in Proposition 1, so it only remains to show that vn ðtÞ is strictly increasing in t. To prove this, it suffices to show that if vn ðtÞ t T

¼ π þ ð1  π Þðv  aÞ=ðv  pÞ for any t, then vn ðtÞ is strictly increasing gives the value of v that is a solution to the equation δ in t. Now since a o p, the right-hand side of this equation is strictly decreasing in v. And the left-hand side of this equation is decreasing in t. From this it follows that the solution to this equation vn ðtÞ must be strictly increasing in t. □ Proof of Corollary 1. Write π ðβ; λ; r; pÞ, aðβ; λ; r; pÞ, and hðβ; λ; r; pÞ to make explicit the dependence of π ðβ Þ, aðβÞ, and hðβÞ on λ, r, and p. Note that π ðβ; λ; r; pÞ is strictly increasing in λ since it is more likely that some other buyer with value v 4p will enter the auction if

λ is larger. And aðβ; λ; r; pÞ is strictly increasing in λ because even if no other buyers with values v 4 p

enter the auction, it is still more likely for there to be more buyers in the auction when λ is larger, so aðβ; λ; r; pÞ is increasing in λ. Thus π ðβ; λ; r; pÞ þ ð1  π ðβ; λ; r; pÞÞðv aðβ ; λ; r; pÞÞ=ðv pÞ is strictly decreasing in t T

of v A ðp; 1Þ that is a solution to the equation δ

λ for any v 4 p and vnðβ;λ;r;pÞ ðtÞ, the value

¼ π ðβ; λ; r; pÞ þ ð1  π ðβ; λ; r; pÞÞðv  aðβ; λ; r; pÞÞ=ðv pÞ, is strictly decreasing

in λ. From this it follows that hðβ; λ; r; pÞ is nonincreasing in λ and strictly decreasing in λ if hðβ; λ; r; pÞ o1. A similar argument shows that hðβ; λ; r; pÞ is nonincreasing in r and strictly decreasing in r if hðβ; λ; r; pÞ o1.

λ and r, it follows that if βn ðλ; rÞ denote the unique βn A ½0; 1 satisfying n hðβ ; λ; r; pÞ ¼ β , then β ðλ; rÞ is nonincreasing in λ and r. Moreover, for values of λ such that β ðλ; rÞ o 1, we have n n hðβ ðλÞ; λ; r; pÞ o1, and hðβ; λ; r; pÞ is strictly decreasing in λ and r for values of β near β ðλ; rÞ, which implies that the unique n n n β A ½0; 1 satisfying hðβ ; λ; r; pÞ ¼ β must also be strictly decreasing in λ and r. Thus the probability a buyer with value v 4 p enters the auction is always nonincreasing in λ and r and strictly decreasing in λ and r for values of λ and r such that there is But since hðβ; λ; r; pÞ is nonincreasing in n

n

n

a positive probability that a buyer with value v 4 p would not enter the auction. □

Proof of Proposition 2. First note that a buyer with value v 4 p who enters the auction has a dominant strategy of making a bid w at any time t such that the buyer would be indifferent between being the high bidder in the auction at time t if the highest competing bidder in the auction also made a bid of w and purchasing at the posted price at time t.9 Thus in analyzing the appropriate values of wn ðv; tÞ in this bidding strategy, I find the values of wn ðv; tÞ such that a bidder with value v 4 p would be indifferent between being the high bidder in the auction if the highest competing bidder in the auction also made a bid of wn ðv; tÞ and purchasing at the posted price. First I show that wn ðv; TÞ ¼ p for all v 4 p. Note that if a buyer buys at the posted price at time t¼T, then the buyer obtains a payoff of δ ðv  pÞ. And if a buyer has the high bid in the auction and the highest competing bid is wn ðv; TÞ, then the buyer T

obtains a payoff of δ ðv  wn ðv; TÞÞ. Thus the buyer is indifferent between being the high bidder in the auction and T

immediately purchasing at the posted price if and only if δ ðv wn ðv; TÞÞ ¼ δ ðv  pÞ, which holds if and only if p ¼ wn ðv; TÞ. Thus wn ðv; TÞ ¼ p for all v 4 p. Now I show that wn ðv; tÞ is strictly increasing in t for all v 4p. To see this, suppose an individual with value v 4 p is indifferent between having the high bid in the auction at time t if the highest competing bidder in the auction made a bid of w and purchasing at the posted price at time t. Then if uðv; w; tÞ denotes the expected payoff a bidder with value v 4 p T

T

9 In a second-price auction, a bidder's dominant strategy is to make a bid w such that the bidder would be indifferent between winning and losing the auction if the highest competing bidder also made a bid of w. Here if a bidder does not have the high bid in the auction, the bidder's payoff equals the payoff the bidder would achieve by purchasing at the posted price. Thus a bidder's dominant strategy is to make a bid w such that the buyer would be indifferent between being the high bidder in the auction at time t if the highest competing bidder in the auction also made a bid of w and purchasing at the posted price at time t.

P. Hummel / European Economic Review 78 (2015) 269–284

281

obtains from being the high bidder in the auction at time t when the highest competing bid is w, we have t uðv; w; tÞ ¼ δ ðv  pÞ. Consider some time t 0 4 t. Note that if no other buyers bid more than w by time t 0 , then the individual's expected payoff for the game beginning at time t is uðv; w; t 0 Þ. And if at least one other buyer bids more than w by time t 0 , then the individual's expected payoff for the game beginning at time t 0 is some u0 ouðv; w; t 0 Þ. Thus if π t0 denotes the probability that no other buyers bid more than w by time t 0 , then uðv; w; tÞ ¼ π t0 uðv; w; t 0 Þ þ ð1  π t0 Þu0 . Since u0 ouðv; w; t 0 Þ, this in turn implies t t0 t0 that uðv; w; tÞ ouðv; w; t 0 Þ. And since uðv; w; tÞ ¼ δ ðv pÞ 4 δ ðv pÞ, this further implies that δ ðv  pÞ ouðv; w; t 0 Þ. Now uðv; w; t 0 Þ represents the expected payoff a bidder with value v 4 p obtains from being the high bidder in the auction t0

at time t when the highest competing bid is w, so uðv; w; t 0 Þ is decreasing in w. Thus in order for δ ðv  pÞ ¼ uðv; w0 ; t 0 Þ to hold, it must be the case that w0 4 w. This implies that for an individual with value v 4 p to be indifferent between having the high bid in the auction at time t 0 if the highest competing bidder in the auction made a bid of w0 and purchasing at the posted price at time t 0 , it must be the case that w0 4 w. Thus wn ðv; t 0 Þ 4 wn ðv; tÞ, meaning wn ðv; tÞ is strictly increasing in t for all v 4p. Now I show that wn ðv; tÞ is strictly decreasing in v for all v 4 p and t o T. To see this, suppose that the highest competing bid in the auction is w and an individual with value v 4 p enters the auction at time t with the high bid in the auction. Note that if there is some future date t 0 4t at which it becomes necessary for this individual to bid more than wn ðv; t 0 Þ in order to t0 have the high bid in the auction, then the individual buys at the posted price at time t 0 and obtains a payoff of δ ðv  pÞ. And if bids evolve in such a way that the individual stays in the auction until the end and ultimately buys an object in the auction ^ 4w, then the individual's payoff in the auction is δT ðv  wÞ. ^ But if no other new individuals enter the auction at some price w with bids higher than w, then the individual ultimately buys the object at a price w at the end of the auction and obtains a T payoff of δ ðv  wÞ. t0

Putting these cases together we see that with some probability π 1 A ð0; 1Þ the individual obtains a payoff of δ ðv pÞ for T ^ for some w ^ 4 w, and with some t 0 4 t, with some probability π 2 A ð0; 1Þ the individual obtains a payoff of δ ðv  wÞ probability 1  π 1  π 2 the individual's ultimate payoff from entering the auction at time t is δ ðv  wÞ. Thus an individual's expected payoff from entering the auction at time t if the highest competing bid in the auction is w is 0 0 ^ þ ð1  π 1  π 2 ÞδT ðv  wÞ, where the expectation E½δt  is taken over the random realization of π 1 E½δt ðv pÞ þ π 2 δT ðv E½wÞ ^ is taken over the t 0 given that the individual ultimately buys at the posted price at some time t 0 4 t and the expectation E½w ^ given that the individual ultimately buys an object in the auction at some price w ^ 4w. random realization of w T

But if an individual with value v 4 p buys at the posted price at time t, the individual obtains a payoff of δ ðv  pÞ. Thus the individual is indifferent between having to place a bid of w to have the highest bid in the auction at time t and buying at the t

t0

t0

^ þ ð1  π 1  π 2 Þδ ðv  wÞ ¼ δ ðv  pÞ or π 1 ðδ  E½δ Þp þ posted price at time t if and only if π 1 E½δ ðv  pÞ þ π 2 δ ðv  E½wÞ T

t0

T

t

t

^ þð1  π 1  π 2 Þðδ p  δ wÞ ¼ ½π 1 ðδ  E½δ Þ þð1  π 1 Þðδ  δ Þv. But the left-hand side of this equation is π 2 ðδ p  δ E½wÞ strictly decreasing in w and the right-hand side of this equation is strictly increasing in v. Thus the cutoff value of w that gives the bid an individual wishes to enter into the auction at time t, wn ðv; tÞ, is strictly decreasing in v. □ t

T

t

T

t

t

T

Proof of Proposition 3. Let pn denote the optimal posted price that the seller would use if the seller only used the posted price mechanism. Consider what would happen if the seller instead sold an object via an auction with a reserve price of pn while simultaneously selling all other objects via a posted price mechanism with price pn þ ϵ for some small ϵ 4 0. This change potentially benefits the seller in that the seller may be able to charge higher prices to buyers who ultimately buy an object. But the change potentially hurts the seller in that the seller may now fail to sell some objects that the seller would have sold before if more than one buyer enters the auction and at least one of these buyers has value v A ½pn ; pn þ ϵ. I thus seek to bound the relative costs and benefits from these two possibilities below. Note that if the seller sells objects using this strategy and a buyer with value v 4 pn þ ϵ arrives, the probability that this buyer enters the auction, unconditional on the buyer's precise value for an object and the precise time at which the buyer arrives, is OðϵÞ for the following reason: If a buyer with value v 4pn þ ϵ buys an object at the posted price at time t, then the buyer obtains a payoff of δ ðv  ðpn þ ϵÞÞ, but if a buyer with value v 4 pn þ ϵ buys an object at the auction, the buyer can t

obtain a payoff no greater than δ ðv  pn Þ. Thus a necessary condition for a buyer to want to enter the auction is that T

δ ðv p Þ Z δ ðv  ðp þ ϵÞÞ or δ ðv  pn Þ Zv ðpn þ ϵÞ or ðδT  t  1Þv Z ðδT  t  1Þpn  ϵ or v r pn þ ϵ=ð1  δT  t Þ. And since T t pn þ ϵ=ð1  δ Þ  ðpn þ ϵÞ ¼ OðϵÞ in the limit as ϵ-0, it follows that the probability a buyer with value v 4 pn þ ϵ for an object enters the auction is OðϵÞ as ϵ-0. From this it follows that as ϵ-0, the expected number of buyers with values v 4 pn þ ϵ who buy at the posted price is λTð1  Fðpn ÞÞ þ OðϵÞ. And since such buyers pay a price that is higher by an amount ϵ than the price they would pay if the T

n

t

n

T t

seller were not simultaneously selling objects via an auction, this indicates that this change benefits the seller by an amount λTð1  Fðpn ÞÞϵ þOðϵ2 Þ as a result of the higher prices the buyer is able to charge. Now the expected number of buyers with values v A ½pn ; pn þ ϵ who enter the auction is λTðFðpn þ ϵÞ Fðpn ÞÞ, meaning the probability that a buyer with value v A ½pn ; pn þ ϵ enters the auction is λTðFðpn þ ϵÞ Fðpn ÞÞ þ Oðϵ2 Þ. And since λTðFðpn þ ϵÞ  Fðpn ÞÞ þ Oðϵ2 Þ ¼ λT ϵf ðpn Þ þ Oðϵ2 Þ, it follows that the probability that a buyer with value v A ½pn ; pn þ ϵ enters the auction is λT ϵf ðpn Þ þ Oðϵ2 Þ.

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And the expected number of buyers with values v 4 pn þ ϵ who enter the auction is OðϵÞ. Putting these results together, we see that the expected number of buyers with values v A ½pn ; pn þ ϵ who enter the auction and fail to win an object at the auction is Oðϵ2 Þ. And for every buyer with value v A ½pn ; pn þ ϵ who enters the auction and fails to win an object, the seller loses a total of pn by simultaneously selling objects via an auction and a posted price mechanism. Thus if a seller simultaneously sells objects via an auction and a posted price mechanism, the seller loses no more than Oðϵ2 Þ revenue in expectation due to the seller's failure to sell some objects that the seller would have sold before. Combining this result with the fact that the seller gains λTð1 Fðpn ÞÞϵ þ Oðϵ2 Þ as a result of the higher prices the buyer is able to charge to individuals who buy objects shows that for sufficiently small ϵ 4 0, the seller obtains a strictly greater payoff by simultaneously running the auction and posted price mechanisms described than by only running a posted price mechanism with posted price pn . Thus a seller obtains a strictly larger expected payoff by simultaneously selling objects via an auction and a posted price mechanism than by only selling objects via a posted price mechanism. □ Proof of Proposition 4. Let r denote the optimal reserve price that the seller would use if the seller were only selling an object via an auction and the seller were not simultaneously selling objects via a posted price mechanism. Consider what would happen if the seller instead sold an object via an auction with a reserve price r while simultaneously selling objects via a posted price mechanism with posted price v  ϵ for some small ϵ 40. I first show that in the setting where the seller does not reveal any information about the bids that have been placed before the end of the auction, the seller would obtain strictly greater expected revenue for sufficiently small ϵ 4 0. In order to prove this, I first show that if ϵ 40 is sufficiently small, then all buyers who arrive will choose to enter the auction. To see this, note that if all buyers who arrive choose to enter the auction and π ðϵÞ denotes the probability that at least one other buyer with value v 4v  ϵ arrives, then limϵ-0 π ðϵÞ ¼ 0. Also note that if all buyers who arrive choose to enter the auction and aðϵÞ denotes the expected value of the second-highest bid in the auction conditional on a buyer with value v 4 v  ϵ entering the auction and no other buyers with values v 4 v  ϵ entering the auction, then as ϵ-0, aðϵÞ approaches the expected value of the second-highest bid in an auction conditional on a buyer with value v entering the auction and there being no posted price mechanism. Thus limϵ-0 aðϵÞ ¼ a for some a o v. But if all buyers who arrive choose to enter the auction, then a buyer with value v 4v  ϵ who arrives at time t prefers to tT enter the auction if and only if δ o π ðϵÞ þ ð1  π ðϵÞÞðv  aðϵÞÞ=ðv  ðv  ϵÞÞ. Now π ðϵÞ þ ð1  π ðϵÞÞðv aðϵÞÞ=ðv  ðv  ϵÞÞ is minimized for values v 4v  ϵ when v ¼ v. And since limϵ-0 π ðϵÞ ¼ 0 and limϵ-0 aðϵÞ ¼ a for some a o v, it follows that limϵ-0 π ðϵÞ þð1  π ðϵÞÞðv  aðϵÞÞ=ðv  ðv  ϵÞÞ ¼ 1 for v ¼ v. But this means that for sufficiently small ϵ 4 0, we have δt  T o π ðϵÞ þ ð1  π ðϵÞÞðv aðϵÞÞ=ðv  ðv  ϵÞÞ for all v 4v  ϵ. Thus if ϵ 40 is sufficiently small, all buyers who arrive will choose to enter the auction. Now suppose that ϵ 40 is sufficiently small that all buyers who arrive will choose to enter the auction. Note that if no more than one buyer who arrives by time T has value v Zv  ϵ for an object, then the seller sells an object to the highest bidder at the second-highest bid, which is the same bid regardless of whether the seller also uses the posted price mechanism. Thus if no more than one buyer who arrives by time T has value v Z v  ϵ for an object, the seller earns the same total amount of revenue regardless of whether the seller also uses the posted price mechanism. Now if at least two buyers who arrive by time T have values v Zv  ϵ for an object and the seller does not simultaneously sell objects via a posted price mechanism, then the seller will sell an object for a price between v  ϵ and v. And if at least two buyers who arrive by time T have values v Zv  ϵ for an object and the seller does simultaneously sell objects via a posted price mechanism with posted price v  ϵ, then the seller sells at least two objects for a price of v  ϵ. But for sufficiently small ϵ 4 0, this means that the seller obtains a strictly greater expected profit by simultaneously selling objects via a posted price mechanism with posted price v  ϵ. The result then follows if the seller running the auction does not reveal any information about the bids that have been placed before the end of the auction. Now I show that in the setting where the seller reveals information about the bids that have been placed, the seller would obtain strictly greater expected revenue for sufficiently small ϵ 40. To see this, first note that if exactly one buyer with value v 4 v  ϵ enters the auction and ultimately wins an object in the auction, then the object in the auction is sold for the same price regardless of whether the seller simultaneously uses the posted price mechanism. Thus the seller will sell the object in the auction for the same price regardless of whether the seller simultaneously uses the posted price mechanism in this case, and the simultaneous use of the posted price mechanism will have no effect on the seller's payoff. Now consider what happens if there is some buyer with value v 4 v  ϵ who ends up buying an object at the posted price instead of winning an object at the auction. Note that if a buyer with value v 4 v  ϵ for an object buys at the posted price at t time t, then the buyer obtains a payoff of δ ðv  ðv  ϵÞÞ. But if the buyer places the highest bid in the auction at time t and the second-highest bid as of time t is w, then there is a positive probability that no other buyers will arrive by time T and the T T buyer will obtain a payoff of δ ðv wÞ. And at a minimum the buyer can always obtain a payoff of δ ðv  ðv  ϵÞÞ by buying at the posted price at time T. Thus if π denotes the probability that no buyers arrive between time t and time T, then a buyer T T with value v 4 v  ϵ obtains an expected payoff greater than or equal to πδ ðv wÞ þð1  π Þδ ðv  ðv  ϵÞÞ by placing a bid of w if such a bid will give the buyer the highest bid in the auction at time t. From this it follows that a necessary condition for a buyer with value v 4 v  ϵ to want to buy at the posted price at time t is that

δt ðv  ðv  ϵÞÞ Z πδT ðv  wÞ þ ð1  π ÞδT ðv  ðv  ϵÞÞ 3

P. Hummel / European Economic Review 78 (2015) 269–284

283

δt  T ðv  ðv  ϵÞÞ Z π ðv wÞ þ ð1  π Þðv  ðv  ϵÞÞ 3 1h

π

i

δt  T  ð1  π Þ ðv  ðv  ϵÞÞ Z v  w 3

  1 t  T w Zv  1 þ δ  1 ðv  ðv  ϵÞÞ:

π

Now for values of v 4 v  ϵ, we have v ½1 þ ð1=π Þðδ

t T

T

t T

 1Þðv  ðv  ϵÞÞ Z v  ϵ ½1 þ ð1=π Þðδ

t T

 1Þϵ ¼ v  2ϵ ð1= π Þ

ðδ  1Þϵ Zv  2ϵ  ð1=π 0 Þðδ  1Þϵ, where π 0 denotes the probability that no buyers arrive between time 0 and time T. Thus if a buyer with value v 4v  ϵ chooses to buy at the posted price, then it necessary that the second-highest bid in the auction is at least v  zϵ for some constant z 40 that is independent of the time the buyer arrives. From this it follows that if a buyer with value v 4v  ϵ chooses to buy at the posted price, then the seller sells the object in the auction for a price of at least v  zϵ for some z. And the seller sells at least one object at the posted price of v  ϵ. But if the seller chose not to simultaneously sell objects via a posted price mechanism, then the seller would only sell one object for a price no greater than v. Thus for sufficiently small ϵ 4 0, the seller obtains greater expected revenue by simultaneously selling objects via a posted price mechanism than by only selling an object via an auction. □ Proof of Proposition 5. Let pn denote the optimal posted price that the seller would use if the seller were only selling objects via the posted price mechanism, and suppose the seller simultaneously runs an auction with a reserve price of pn and uses a posted price mechanism with a posted price of pn þ ϵ for some small ϵ 40. I seek to show that for any positive integer q o Q , the seller can obtain at least as much revenue in expectation by making q þ 1 objects available for sale in the auction than by only making q objects available. There are two ways that making q þ 1 objects available for sale via an auction rather than only q objects can influence the seller's payoff. First, if at least q þ 1 bidders enter the auction, then the seller may be able to sell more objects via the auction than the seller would have sold if the seller were only selling q objects via an auction. Second, sellers with values greater than the posted price may make different decisions about whether to enter the auction or to immediately purchase at the posted price. By the same reasoning in the proof of Proposition 3, the probability a randomly selected buyer enters the auction is ΘðϵÞ. Thus the probability at least q þ 1 bidders enter the auction is Θðϵq þ 1 Þ, so the seller obtains Θðϵq þ 1 Þ more revenue in expectation by making q þ 1 objects available for sale via the auction rather than q objects from the possibility that at least q þ 1 bidders enter the auction. Next I seek to show that the expected revenue losses due to the possibility that buyers with values greater than the posted price may make different decisions about whether to enter the auction are Oðϵq þ 2 Þ: To see this, first note that if there are q objects in an auction and a bidder with value greater than the posted price enters the auction, then the expected price in the auction is pn þ Θðϵq þ 1 Þ for the following reason: In order for the expected price to differ from pn , it is necessary for at least q other bidders to enter the auction. The probability at least q other bidders enter the auction is Θðϵq Þ. And conditional on q other bidders entering the auction, the final price in the auction will exceed pn by ΘðϵÞ. Thus if there are q objects in an auction and a bidder with value greater than the posted price enters the auction, then the expected price in the auction will exceed pn by Θðϵq þ 1 Þ. Now a buyer with value v Z pn þ ϵ will want to enter the auction if and only if δ ðv  cÞ Z δ ðv ðpn þ ϵÞÞ, where c is the T

t

expected price in the auction if this buyer enters the auction. This holds if and only if δ ðpn þ ϵÞ  δ c Z ðδ  δ Þv, which in t

T

t

T

turn holds if and only if v rðδ ðpn þ ϵÞ  δ cÞ=ðδ  δ Þ. And since the expected price in an auction with q objects if a bidder with value greater than the posted price enters the auction is pn þ Θðϵq þ 1 Þ, it follows that increasing the number of objects in the auction from q to q þ1 decreases this expected price by pn þ Θðϵq þ 1 Þ  ðpn þ Θðϵq þ 2 ÞÞ ¼ Θðϵq þ 1 Þ. And since increasing the number of objects in the auction from q to q þ 1 decreases the value of c by Θðϵq þ 1 Þ, this change also increases the probability a bidder with value v Zpn þ ϵ will enter the auction by Θðϵq þ 1 Þ. But conditional on such a bidder making a different decision about whether to enter the auction, the seller's expected revenue can only change by OðϵÞ: If fewer than q other bidders enter the auction, then the seller sells the same number of objects as before, and the only difference resulting from this change is that prices may change by OðϵÞ. And the probability that at least q other bidders enter the auction is Θðϵq Þ ¼ OðϵÞ. Thus conditional on a bidder with value v Z pn þ ϵ making a different decision about whether to enter the auction, the seller's expected revenue can only change by OðϵÞ. Combining this with the result in the previous paragraph shows that the expected revenue losses due to the possibility that buyers with values greater than the posted price may make different decisions about whether to enter the auction or to immediately purchase at the posted price are Oðϵq þ 2 Þ. Now we have seen that the seller's expected revenue gains from selling q þ1 objects via the auction rather than q objects from the possibility that at least q þ 1 bidders enter the auction are Θðϵq þ 1 Þ. Combining this with the result in the previous paragraph shows that the if the seller sells q þ 1 objects via the auction rather than q objects, then the seller's expected revenue increases. □ t

T

t

T

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