Optimal Reverse-Pricing Mechanisms

Transcription

1 Otimal Reverse-Pricing Mechanisms Martin Sann 1 Robert Zeithammer Gerald Häubl 3 February 19, 010 Abstract Reverse ricing is a market mechanism under which a consumer s bid for a roduct leads to a sale if the bid exceeds a hidden accetance threshold the seller has set in advance. The seller faces two key decisions in designing such a mechanism: First, he must decide where in the rocess to collect the revenue that is, whether to commit to a minimum marku above cost (and thus define the bid-accetance threshold given cost) and whether to set a fee for the consumer s right to bid. Second, the seller must decide whether to facilitate or hinder consumer learning about the current bid-accetance threshold. We analyze these decisions for a rofit-maximizing small intermediary retailer selling to consumers who can also urchase the roduct in an outside osted-rice market. The otimal revenue model is to charge a fee for the right to bid and then accet all bids above cost, rather than to set a ositive minimum marku above cost. Avoiding minimum markus in favor of a bidding fee is more rofitable because of increased efficiency arising from more entry by consumers and higher bids by the entrants. When consumers learn about the bid-accetance threshold before they enter the market, efficiency increases further, and generating revenue through a bidding fee can comensate the seller for his loss of information rent when the cometition from the outside osted-rice firm is relatively weak. Keywords: reverse ricing, name-your-own-rice, analytical modeling, e-commerce. 1 Martin Sann: Munich School of Management, Ludwig-Maximilians-University (LMU), Geschwister-Scholl- Platz 1, München, Germany, Phone: +49 (89) , Fax: +49 (89) , sann@sann.de Robert Zeithammer: The Anderson School of Management, UCLA, Los Angeles, USA, Phone: (310) , robert.zeithammer@anderson.ucla.edu 3 Gerald Häubl: School of Business, University of Alberta, Edmonton, AB, Canada T6G R6, Phone: (780) , Fax: (780) , gerald.haeubl@ualberta.ca The authors thank Kursad Asdemir, Scott Fay, Yuanfang Lin, and Paul Messinger for their helful comments on an earlier version of this aer. 1

2 1 Introduction Reverse ricing (often also referred to as name-your-own-rice selling ) is an emerging market mechanism under which a consumer s bid for a secific roduct leads to a sale if this bid exceeds the seller s hidden accetance threshold, which in turn deends on the seller s marginal cost (Sann and Tellis 006). If a bid exceeds the seller s threshold, the consumer receives the roduct and ays the bid amount. Some current examles of reverse-ricing sellers are Priceline (which secializes in airline tickets, hotel accommodations, and car rentals), some ebay sellers (with ebay s Best Offer feature), and discount airlines such as Germanwings. Reverse ricing is oular as a sot market for handling unexected demand or suly fluctuations in industries with volatile cost and/or demand, such as the travel industry. Vendors such as Exedia ost retail rices based on demand redictions, but the residual uncertainty arising from consumers actual booking behavior frequently results in excess caacity, which airlines may sell through alternative channels (Koenigsberg, Muller, and Vilcassim 008). In this aer, we focus on selling excess caacity through a reverse-ricing intermediary retailer, such as Priceline, alongside osted-rice retailers. The marginal cost of the excess caacity varies with every seat sold and hotel room booked, and the strength of reverse ricing is its natural ability to accommodate these fluctuating costs, as we discuss next. The essence of reverse ricing is reversing the order of actions comared to a standard osted-rice market, in which the seller first sets the rice given his marginal cost and then waits for consumers. A reverse-ricing seller solicits consumer bids first and then evaluates them in light of current marginal costs. This rocess allows the seller to use a bid-accetance rule that only accets rofitable bids, that is, bids above the cost of getting the good from a roducer at the

3 moment. In this aer, we analyze the seller s roblem of setting such a bid-accetance rule to maximize rofits. The roblem is essentially static because our seller is an intermediary and neither the good nor the consumers ersist over time: the roducer s offer rice (the seller s marginal cost) is a temorary exloding offer, so the seller cannot kee the good and sell it later. The rosective customers are those in the market for that articular roduct or service at that moment, and they are likely to buy the good elsewhere if unsuccessful with reverse ricing. For examle, if Priceline does not sell a given seat on a flight to a customer interested in buying it right now, both the seat and the customer may disaear before another oortunity for a trade arises. We thus abstract away from dynamic yield-management issues that would arise if the seller were the roducer of the good. How should our seller design the reverse-ricing mechanism so as to maximize rofit? The secification of a seller strategy involves two key decisions. First, where in the interaction with consumers should the seller generate revenue: by relying entirely on the information rent from successful bids (i.e., the difference between the bid amount and cost), by setting a ositive minimum marku above marginal costs in secifying the bid-accetance threshold, by charging consumers a bidding fee (akin to a subscrition fee) for the right to submit a bid, or by some combination of these sources? Second, the seller must decide whether to hel or hinder the consumers in their learning about the bid-accetance threshold. Consumers may learn about the current threshold via reeated bidding (Fay 004) or through communication with other consumers (Hinz and Sann 008). Given that such learning is tyically considered destructive to rofits (Fay 004; Hinz and Sann 008; Segan 005), understanding whether the seller is in fact better off reventing consumers from such learning is imortant. The answer to this design question is intertwined with the answer to the revenue-source question: a seller relying on the 3

4 information rent would clearly refer to hinder learning, whereas a seller relying on an ufront bidding fee may benefit from such learning because resolving the consumers uncertainty may increase their willingness to ay the fee. We analyze both seller decisions using a model of a small intermediary seller who faces a oulation of consumers and a remium osted-rice cometitor. Regarding the first decision (revenue model), we find that under several different assumtions about consumers, bidding fees dominate minimum markus in terms of exected rofit. In other words, the seller should never try to commit to a ositive minimum marku. Instead, the seller is always better off lowering the minimum marku slightly and charging a corresondingly higher bidding fee. Bidding fees are a suerior source of revenue because they allow more trades to occur via an imlied lower bidaccetance threshold, thereby making the market more efficient. Although similar to the intuition for the sueriority of a two-art tariff to simle linear ricing (Tirole 1988,. 136), the argument for bidding fees is more subtle: unlike in the case of a simle two-art tariff, our reverse-ricing seller is not a monoolist and does not cature the entire consumer surlus because of strategic bid-shading (bidding less than valuation) by the consumers. In addition to their rofit advantage, bidding fees are also easier to imlement than minimum markus because they require less commitment from the seller: a minimum marku seller needs to credibly romise he will reject some rofitable trades in the future in order to increase bids today. Regarding the second decision, the seller who makes revenue through bidding fees sometimes refers to facilitate consumers learning about the bid-accetance threshold instead of trying to revent them from discovering it. We find that the seller refers to facilitate consumer learning about the threshold when the outside rice is relatively high, that is, when the outside cometition is relatively weak. The additional information assures that only consumers with 4

5 high-enough valuations enter and submit bids, greatly increasing market efficiency. A high outside rice gives the seller market ower over more consumers, allowing him to cature more of the efficiency gain. Bidding fees thus artially immunize the seller against the low-rofit consequences of consumers becoming informed about the threshold: a seller relying only on the information rents for revenue would always refer to hide threshold information from consumers. Most rior research on reverse ricing emirically analyzes bidding behavior and focuses on either behavioral asects (Chernev 003; Ding, Eliashberg, Huber, and Saini 005; Sann and Tellis 006), the inference of certain consumer characteristics from observed bidding behavior (Hann and Terwiesch 003; Sann, Skiera, and Schäfers 004), or the imact of information diffusion about the seller's bid-accetance threshold on bidding behavior (Hinz and Sann 008). Four recent articles are more closely relevant to the resent work in that they examine some asects of the seller strategy. Terwiesch, Savin, and Hann (005) discuss the influence of frictional costs on the otimal bid-accetance threshold. Wang, Gal-Or, and Chatterjee (009) and Fay (009) rovide two different rationales for the existence of the reverse-ricing channel in addition to a osted-rice channel, either used by the same retailer or used by two different retailers, resectively. Amaldoss and Jain (008) analyze joint bidding for multile items at a reverse-ricing retailer and find that such joint bids can increase retailer rofit. The most closely related iece of rior work is Fay (004), who examines whether the seller should limit consumers ability to learn his bid-accetance threshold. That work focuses on learning through rebidding as oosed to learning through communication among consumers. Contrary to the common belief that reeat bidding erodes rofits by revealing information about the seller s threshold, Fay s findings suggest that reeat bidding may leave rofits unaffected as long as the seller anticiates rebidding and adjusts his threshold accordingly. Moreover, they also 5

6 indicate that the seller may sometimes wish to encourage rather than discourage reeat bidding. The key assumtion of Fay s model is that only two ossible thresholds exist, arising from two ossible inventory levels. This assumtion both simlifies bidding and gives substantial ower to the seller through contingent ricing. In contrast, our model allows a continuum of thresholds in equilibrium, increasing realism. Our model is also comlementary to Fay (004) in that his seller is initially uncertain about inventory, whereas our seller is uncertain about marginal cost. In a nutshell, the contributions of this research relative to rior work on reverse ricing are as follows: We concetualize the reverse-ricing seller as an intermediary uncertain about his marginal costs. The resulting model allows analysis and otimization of two key arameters of a reverse-ricing mechanism that are under the seller s control: revenue model and information release. We show that the otimal reverse-ricing mechanism charges a ositive bidding fee and accets all bids above the marginal cost rather than charging a minimum marku above the marginal cost akin to a transaction fee. Moreover, we account for the ossibility of consumers discovering the bid-accetance threshold and show when it is in the seller s interest to facilitate such learning. The article is organized as follows: section.1 lays out our assumtions. In section., we develo a model of consumer bidding behavior in a reverse-ricing market with an outside otion. Section.3 then investigates the otimal strategy of a seller who cannot commit to a minimum marku. We then allow the seller to commit to a secific minimum marku in section.4, and ositive minimum markus are not otimal. Section.5 analyzes whether the reversericing seller should facilitate or hinder consumer learning of the bid-accetance threshold. We conclude in section 3. 6

7 Otimal Menu-Based Pricing for a Reverse-Pricing Seller.1 Assumtions We analyze the otimization roblem of a small erishable-roduct intermediary seller who cannot set osted rices contingent on his highly variable marginal costs. Caturing the idea of selling excess caacity beyond some outside market for the good, the seller sets his bidding fee and/or minimum marku after learning the revailing stable osted rice in an outside market but before learning current marginal cost w. Caturing the idea that the seller selling excess caacity is small relative to the outside market, the seller is a rice taker of both and w. This rice-taking assumtion ensures that our model does not degenerate into Bertrand rice cometition due to a lack of differentiation between the seller and the outside market. The seller intermediates between roducers and consumers of the roduct. The roduct (e.g., roducer s excess caacity) is erishable in that neither the consumers nor the roducer s offers ersist over time. Consequently, the seller can model his rofit-maximization roblem as a single-shot game between himself and the consumers who haen to be in the market at the moment. In the Priceline lane-ticket setting, the roducers are airlines, the roduct is a seat on a secific flight, and the roduct is erishable in that each airline makes a short-lived offer of a articular marginal cost for the seat to Priceline. The consumer is short-lived in that he moves on to buy the ticket elsewhere whenever Priceline does not accet his bid. Next, we discuss our assumtions about the outside market, consumers, and marginal costs in more detail, starting with the former. Outside market. The osted rice in the outside market is variable but set before the game begins. For examle, the osted rice for a lane ticket usually remains constant for several days and does not react immediately to activity on Priceline. This rice is informative to the consumers, who infer that the reverse-ricing seller s marginal cost is below. 7

8 Consumers. A unit mass of risk-neutral consumers exists. The consumers have rivate valuations of the roduct drawn from some continuous distribution H with suort on [0,1]. We rovide as many results as ossible for a general H but resort to H=Uniform[0,1] when a general result is not tractable. Given the normalization of utility to the [0,1] interval, we assume the outside osted rice is always below the highest valuation in the market (i.e. 1). Because the highest consumer valuation exceeds the outside rice, two distinct segments of consumers emerge: those with a valuation v less than ( low consumers ) can only buy the roduct from the reverse-ricing seller, whereas others with v ( high consumers ) can also obtain a ositive surlus by buying on the outside osted-rice market. Note that these segments are endogenous to, and so the same consumer can be a low consumer when is high and a high consumer when is low. Since some consumers cannot afford the outside rice, the outside market can be considered a remium otion. All consumers have unit demand in the sense that once they buy the roduct from the reverse-ricing seller, they do not buy it again on the outside market. Finally, consumers incur no frictional cost for bidding, an assumtion that can be relaxed while reserving the key qualitative results (lease contact the authors for details). Marginal Costs. The roducers offer the roduct to the reverse-ricing seller at marginal cost w, which is between 0 and the outside osted rice of. For analytical tractability, we assume marginal costs are distributed uniformly on [0, ]. The critical art of this assumtion is that and w are not erfectly correlated, so consumers remain uncertain about w even after observing. Timing of the Game. The difference between our model and a model of standard ostedrice retailing is that the marginal cost w is not known when the seller sets his strategy. The seller only queries the roducers when a bidder aears and makes a request for a secific roduct. 8

9 Thus the seller must maximize rofits by secifying his strategy (a bidding fee and a bidaccetance rule) before he knows w. 1 He has the following two sources of revenue at his disosal (see Figure 1): Figure 1: Seller Profit from Consumer s Bid seller rofit (Π )= margin + fee ( f ) margin marginal cost (w) threshold bid (b) min. marku (if m>0) information rent $ 1) The seller can charge each bidder a bidding fee f regardless of whether an individual s eventual bid is successful. This bidding fee enters the bidder s decision-making analogously to an entry cost in an auction (Samuelson 1985). Reverse ricing with a bidding fee is similar to charging consumers for a referral service that matches them with roducers but is not equivalent because the seller also kees the difference between the acceted bid and the marginal cost w. ) The seller can earn the margin between his marginal cost w and the consumer s bid. This margin consists of the seller s information rent as the difference between his bid-accetance threshold and the consumer s bid and a otential minimum marku between marginal costs and the bid-accetance threshold. The seller may or may not be able to commit to the minimum-marku strategy of only acceting bids that are at least a fixed marku m 0 above w. Commitment is necessary for the minimum marku strategy to work, because ex ost (i.e., once w is realized), this strategy foregoes otentially rofitable trades (when w < bid < w+m) 1 An equivalent assumtion to the seller announcing the bid-accetance rule is that the consumers infer it from recent exerience with the same seller and share this information through word-of-mouth (Hinz and Sann 008). 9

10 in order to induce higher bids ex ante. Therefore, the seller without commitment is restricted to m=0. In this aer, we first solve the game for a seller without this commitment, and then investigate how the seller s strategy would change with the commitment. Figure illustrates the timing of the game. First, at Stage 0, the osted rice is set. After seeing the rice, consumers udate their rior that the cost is uniformly distributed on [0, 1] to a belief that the cost is uniformly distributed on [0, ]. At Stage 1, the seller sets and announces his ( m, f ) strategy (the seller without commitment is restricted to m=0 as discussed above). At Stage, consumers make their entry decision. Some low consumers do not enter because their exected surlus is negative. The consumers who do enter then submit their bids. Once the seller receives a bid (Stage 3), he queries the roducers to determine the current marginal cost and then accets bids above w+m, using the m announced in Stage 1. The seller faces cometition from the osted-rice market because the bidders whose bids are unsuccessful, as well as those who chose not to submit a bid at all, have the oortunity to buy the roduct on the outside ostedrice market at Stage 4 for rice. Figure : Timing of the Game Stage 0 Stage 1 Stage Stage 3 Stage 4 time is set and revealed; consumers udate belief about w (w ) Seller announces entry fee f and minimum marku m One randomly drawn consumer decides whether to enter (for fee f ) & submits bid if enters Seller learns marginal cost w and accets bid if bid > w+m Consumer s otion to buy at outside osted rice 10

11 . Consumer strategy: entry decision and subsequent bidding strategy We solve the consumer s roblem in Stage using backward induction, starting with the bidding strategy given entry. We analyze the bidding strategy for the general case of m 0, with the consumers facing a non-commitment seller bidding as if m=0. Consumers are uncertain about w when they submit their bids in Stage. Therefore, they must otimize against a distribution of bid-accetance thresholds the seller s strategy in Stage 3 imlies, namely, Uniform[m, +m]. Note that this distribution is the correct equilibrium belief at this stage of the game, either because of the incentives of the seller without commitment to use m=0 or because of the seller s commitment to a articular m>0 in Stage 1. We extend the model of Sann and Tellis (006) to account for the outside otion and characterize consumers bidding behavior at the end of Stage as follows: Proosition 1: Consumers who enter bid b( v m) m+ v for m v< = m+ for v. The roofs of Proosition 1 and all subsequent roositions are in the aendix. This bidding roblem is analogous to the roblem of a bidder in a first-rice sealed-bid auction who faces one exogenous cometitor. The exogenous cometitor bids Uniform[m, +m]. To balance the surlus from winning against the robability of winning, the otimal bidding strategy in such a situation is to shade the bid down from one s valuation of the roduct. As in the first-rice sealed-bid auction, winning means aying one s bid, and shading is necessary to leave room for surlus. Since consumers with valuations below m cannot make a ositive surlus in this market, it is sufficient to characterize the bidding strategy for v>m, for which bid shading is aarent in See Krishna (00) for an exosition of bidding in a first-rice sealed-bid auction. 11

12 that b( v m) < v. Also note that the two consumer segments bid differently: because of the outside otion, all high consumers with v > mimic the behavior of a consumer with v=. Finally, the bidding strategy does not deend on the bidding fee f, because f is sunk uon entry. Given the bidding strategy, a comarison of the exected surlus from entering and bidding to the exected surlus from not entering governs the entry decision in the beginning of Stage (Samuelson 1985). Please see Figure 3 for the consumer s decision tree. The exected surlus is derived in the aendix, and it again deends on the consumer segment: (1) S( v f, m, ) ( v m) f for m v< 4 =. ( m ) v + f for v 4 The functional form of the surlus is instructive with resect to consumers entry decisions. First, low consumers face quadratic surlus because higher valuations and/or lower margins increase both the robability of bid accetance and the surlus uon accetance. These two comonents multily to roduce the convex function from which the bidding fee is simly subtracted as a direct transfer to the seller. Another insight concerns the high consumers: they receive the outside-otion surlus v - lus a gambling bonus equivalent to the surlus of the consumer they mimic in bidding, namely, v =. Since they can obtain the roduct for the rice without entering, the high consumers will enter only when the gambling bonus exceeds f. Since the gambling bonus is equivalent to the exected surlus of a consumer with v = and the exected surlus is increasing in v below, satisfying the consumer with v = is a rerequisite for the reverse-ricing seller to get any customers at all. When the v = consumer enters, all high consumers enter as well, and low consumers above a valuation threshold v = m+ f enter. 1

13 Figure 3: Payoff to Consumers ( v ) max,0 not enter enter & bid bid accet v bid f bid reject f Please note that even if consumers enter, their ayoff is uncertain. Thus, some consumers may ay an entry fee and bid but end u aying only the entry fee f when they are unsuccessful. This discussion concludes our analysis of the consumer side in Stage. The next section roceeds backwards to Stage 1 to solve the seller s rofit-maximization roblem..3 Otimal selling without commitment to a minimum marku (m=0) In this section, we derive the otimal strategy of a reverse-ricing seller for the case that he cannot commit to a minimum marku m > 0. At Stage 1, the reverse-ricing seller without commitment (m = 0) maximizes his exected rofit by selecting the otimal bidding fee f. The rofit-maximization is a screening roblem because the seller uses the fee to effectively set the entry threshold v = f. Increasing f slightly has the marginal benefit of all consumers above v aying a higher bidding fee. The marginal cost of increasing f slightly is the reduction in the number of consumers who enter. Secifically, the threshold consumer with valuation v does not enter anymore, which results in a 13

14 loss of both the marginal bidding fee the consumer would ay and the loss of the marginal rofit from that consumer not bidding. At the otimal fee level, the marginal benefit equals the marginal cost, and the following characterization results: Proosition : When ( ) ( ) x 1 H x is an increasing function of x, the otimal consumer h x 3 4 entry threshold of a non-commitment seller satisfies either whichever is lower. When H(x) = x on [0,], the imlied otimal fee is for < 13 3 f *( ) = min, =. ( 6) + for 13 3 ( + 6) 3 4 ( ) ( ) 1 H v v = or v =, h v Note that the secial-case assumtion H(x) = x on [0,] in Proosition laces no restriction on the distribution of high consumers. The assumtion is that there is a mass of low customers with v< distributed uniformly on [0,] and a mass of (1-) high customers with v > distributed arbitrarily above. There is no need for a secific assumtion about the high consumers valuations because they all mimic the v = tye. The screening nature of the rofit-maximization allows the seller to focus only on the marginal consumer v. The first condition of Proosition is a single-crossing condition analogous to a monotone hazard-rate condition on H. As argued in the revious section, the threshold must not exceed for the seller to make any money at all, hence the dichotomous solution: for low, the cometition from the outside market is so strong and the relative roortion of high consumers so high that the seller chooses to serve only the high consumers and sets v =. Figure 3 outlines the ossible ayoffs to consumers. 14

15 Since the roblem of setting the otimal fee is a screening roblem, we can easily comare it with other roblems of a similar structure, namely, g ( v ) InvHazardRate( v ) g( x) =. When = x, we get the equation for an otimal reserve of an auctioneer, who can obtain the roduct for zero marginal cost (Myerson 1981). When, on the other hand, g( x) x equation for an otimal fee of a zero-marginal-cost referral service that merely matches =, we get the consumers u with roducers without collecting any margin. We can therefore easily show that f *( ) is always less than the otimal referral fee but higher than the otimal reserve in a zeromarginal-cost auction..4 Otimal selling with commitment to a minimum marku (m 0) In this section, we show that at least in the uniform case of H( x) x 1 = on [0,], the otimal seller strategy constrained (by the lack of commitment) to m=0 and f f *( ) = of Proosition is in fact otimal even for the unconstrained seller, who can credibly commit to any minimum marku m 0. In other words, the seller would rather derive revenue from the bidding fee than from the additional margin arising from the minimum marku. Suose the seller can credibly establish commitment to any minimum marku m 0, for examle, through reutation or transarent billing. The consumers take m into account as we show in section., and the uside for the seller is that all consumers who enter raise their bids to comensate for the higher bid-accetance threshold. The downside is that fewer consumers now enter at all, as can be seen from the form of the entry threshold v = m+ f. The resulting rofit function of the seller is quite involved, but we can show that for any m>0, a corresonding otimal fee f *( ) ( ) m exists, and reducing the m slightly in favor of a small increase in f from f * m can hel the seller. In other words, the rofit of a seller who uses the strategy 15

16 ( *( ), ) f m m is decreasing in m for all non-negative m. The details of this result form the roof of Proosition 3 the central result of this aer: Proosition 3: Suose H(x) = x on [0,]. Even if the seller can credibly commit to a ositive minimum marku, the otimal selling strategy uses zero minimum marku and the ositive fee f *( ) derived in Proosition. As both the fee and the minimum marku screen out some low consumers, they might aear equivalent at first glance. However, the minimum marku also changes the entrants bidding strategy. Because consumers shade their bids to halfway between their valuation and m (Proosition 1), the minimum marku increases the bids of the entrants less than it increases the accetance threshold w + m. Therefore, increasing the minimum marku reduces the amount of trading the entrants achieve on average, thus diminishing market efficiency (i.e., the size of the ie). This reduction in efficiency is large enough to hurt seller rofit as well. Consequently, generating revenue through a bidding fee, which does not distort bidding, is always better than generating revenue through a minimum marku, which does. Note that we analyze the extreme case of erfect and effortless commitment to a margin, and find that it is not useful. Therefore, intermediate levels of commitment arising from reutation in an alternative dynamic model would also not be useful to the seller. As discussed above, the roof of Proosition 3 relies on showing that seller rofit is decreasing in m along the locus of ( *( ), ) f m m strategies. Since the rofit is decreasing even at m=0, a seller who could credibly subsidize the consumer bids would do so. We emhasize this interesting result in the following corollary: 16

17 Corollary to Proosition 3: Suose H(x) = x on [0,]. A seller who charges a bidding fee would find it rofitable to credibly subsidize bids and accet at least some bids below cost. Illustrative Examle. For concreteness, suose the seller is Priceline and the roduct is a lane ticket from New York to London, for which = $1000 the lowest outside osted rice cometitors such as Exedia and Travelocity offer. Assume the highest valuation that corresonds to v = 1 is $1500, indicating that two thirds of the rosective Priceline customers are low consumers in that they value the ticket below and one third are high consumers who value it above. Table 1 dislays the results for (1) the otimal bidding fee given the minimum marku set to zero, () the case of an otimal minimum marku given no bidding fee, and (3) for the case of the otimal fee to use given the minimum marku suggested in (). Table 1: Illustrative examle of a lane ticket from New York to London that costs $1000 on Exedia and that consumers value uniformly between $0 an $1500. otimal bidding fee, no minimum marku otimal minimum marku, otimal minimum marku and its associated otimal bidding fee no bidding fee Bidding fee ( f ) $0 $0 $10 Minimum marku (m) $0 $60 $60 Screening level (v) $900 $60 $900 Social welfare (gains from trade) $16 $15 $99 Seller rofit $109 $60 $84 Please see the aendix for details of the calculations that roduced Table 1. The results in Table 1 show that the social welfare with a minimum marku and no fee is almost as high as with the otimal fee and no minimum marku, but the seller only makes about half of this welfare in rofit, much less than with the otimal fee. Moreover, charging f=0 is not otimal for the seller who uses the minimum marku of $60. Instead, using a ositive fee of $10 combined with this 17

18 minimum marku of $60 is otimal, which results in lower social welfare, because more consumers are screened out, but higher rofit (comare third and fourth column of Table 1). However, rofit is still considerably lower than in the case of the otimal bidding fee with no minimum marku (comare the second and fourth columns of Table 1). This comarison illustrates the key oint of Proosition 3: the lack of market efficiency when a rofit-maximizing seller uses a ositive minimum marku as well as a bidding fee hurts rofitability. The uniform assumtion does not drive the key qualitative result that the seller refers fees over minimum markus. We have roven an analogy of Proosition 3 for the exonential distribution (not reorted), and we roose that the economic intuition of increased market efficiency is quite general. This intuition is analogous to the intuition behind two-art tariffs outerforming uniform ricing in osted-rice markets..5 Should the seller who charges a bidding fee facilitate or hinder consumer learning about the current bid-accetance threshold? As discussed in the introduction, consumers may learn the bid-accetance threshold through communication with recent buyers (Hinz and Sann 008). Consumers who know the accetance threshold never bid more than it, erasing the seller s information rent comletely. Therefore, a seller who derives rofit only from the information rent (m=f=0) obviously has a strong incentive to hide his marginal cost w from consumers. 3 This incentive to hide threshold information does not immediately extend to a seller who imlements a bidding fee following Proosition 3: facilitating consumer learning may increase articiation by high-value bidders, in turn allowing the seller to charge a higher bidding fee and make a higher rofit overall. In this subsection, we 3 We believe this to be the basis of Priceline s no-rebidding olicy, as well as the rationale for its randomization of the accetance rule reorted in Anderson (009) and Segan (005). 18

19 sharen this intuition and qualify its validity with a bound on the intensity of outside market cometition catured by in our model. To analyze the seller s incentive to hinder consumer learning under bidding fees (f >0), we consider the limiting case of consumers knowing the ucoming bid-accetance threshold with certainty rior to making their entry decision. That is, we analyze consumers who learn w at the beginning of Stage (see Figure ). We hoe this simle limiting case qualitatively catures what would haen with more realistic learning that would still leave the consumers somewhat uncertain. When consumers know the current accetance threshold w, they enter and bid min ( w, ) whenever min (, ) v w> f. The seller accounts for this bidding strategy and charges a different bidding fee than one that would be otimal with uncertain consumers. We denote the bidding fee otimal under the informed-consumer scenario f I. 4 It is relatively simle to show (lease see Aendix for details) that f I satisfies ( ) fi = E min, v v> fi, that is, the seller should charge half of the exected net valuation (minimum of v and ) of consumers who have a ositive robability of trading (valuations above f I ). In the uniform low-consumer case of * H(x)=x on [0,], the otimal fee is fi ( ) ( ) = , which can be bounded by * * < I ( ) <. In the lane-ticket examle of Table 1, I ( ) 4 f f = $4. The lane-ticket examle suggests the bidding fee is much higher when the otential consumers learn the marginal cost w in advance. This inequality extends to all as long as the low consumers are uniformly distributed: when H( x) * I ( ) 4 *( ) * = x on [0,], f ( ) > f *( ) for every because f > > f. The higher bidding fee makes sense because informed consumers do not I 4 While we focus on bidding fees (f>0), note that the argument for facilitating consumer learning is equally valid for a seller with commitment to a minimum margin (m>0): with consumers informed about w, bidding fees and minimum margins are interchangeable in that any combination that results in the same f+m yields the same consumer behavior and rofit. 19

20 have to gamble at entry time and shade their bids afterward. They are thus willing to ay more for entry. Our final roosition characterizes when facilitating consumer learning about marginal costs is more rofitable for the seller than hindering it: Proosition 4: Suose H(x) = x on [0,]. A unique outside rice 13 3 < < 1 exists such that facilitating consumer learning about the seller s current bid-accetance threshold is rofitable for the seller when >, and vice versa. Numerical aroximation reveals that Recall that when < , the reverse-ricing seller only serves high consumers (Proosition ). Therefore, when the outside rice is so low that the reverse-ricing seller serves only high consumers, Proosition 4 shows that hindering consumer learning is always otimal. Figure 4: Seller rofit under two consumer-information scenarios 0.08 seller rofit ( Π ) Consumer knowledge about seller cost: uncertain informed outside osted rice ( ) 13 3 Note to Figure 4: This figure illustrates Proosition 4. The two curves reresent seller rofits as a function of the rice on the outside osted-rice market. The solid curve involves consumers informed about seller cost before making their entry decision. The dashed curve involves consumers uncertain about seller cost at the time of their entry decision. 0

21 Not surrisingly, facilitating consumer learning increases the efficiency of the market for all (roof omitted for arsimony). The intuition for the rofitability of facilitating consumer learning only with high is as follows: a high means both relatively fewer high consumers and a larger efficiency gain from the added information. The seller can cature enough of the efficiency gain to comensate for the lost information rent rovided by the uninformed consumers. As dros and the roortion of high consumers increases, the efficiency advantage gradually disaears, but the information rent from uninformed consumers remains. Figure 4 illustrates Proosition 4. 3 Discussion This aer analyzes the design of an otimal reverse-ricing mechanism from the ersective of a rofit-maximizing seller. Secification of such a mechanism involves two key decisions: (1) the revenue model, namely, whether to set a minimum marku above cost (and thus set the bidaccetance threshold) and/or charge a fee for the right to bid, and () whether to facilitate or hinder consumer learning about the bid-accetance threshold. Consumers may learn about the accetance threshold through reeated bidding or communication with other consumers. Two key imlications for reverse-ricing sellers result from our analyses. First, the otimal revenue model is to charge a bidding fee and then accet all bids above the marginal cost instead of charging a minimum marku above the marginal cost. A bidding fee is referable because it allows more trades to occur and because it does not require commitment to reject otentially rofitable trades. In contrast, a minimum marku increases the bid-accetance threshold (to cost + minimum marku) without equally increasing bids, thus reducing the amount of trading achieved (i.e., reducing efficiency). 1

22 The second major imlication of our analyses is that a seller relying on an ufront bidding fee can benefit from consumer learning (e.g., via word of mouth) about the bid-accetance threshold. This result is in shar contrast with a seller who relies only on information rents as source of revenue such a seller always wants to hinder consumer learning. A seller relying on an ufront bidding fee can benefit from such learning due to an increase in market efficiency. The seller can charge a higher bidding fee because informed consumers do not have to gamble at entry time and shade their bids afterward. Consequently, consumers are willing to ay more for entry. Thus, sellers may facilitate consumers learning of these thresholds, for examle, by suorting online communities such as or However, the result of the seller benefiting from such an early revelation of the threshold deends on the distribution of valuations across consumers and the rice in the outside market. As the outside rice dros and the roortion of high consumers thus increases, the seller may be better off hindering consumer learning about the bid-accetance threshold, because the information rent from these consumers dominates the efficiency gains. The results of our analyses also rovide interesting insights into the henomenon of ( live ) shoing communities (for examle, in which retailers may charge consumers a fee for the oortunity to enter and buy short-term romotional offers. Furthermore, retailers may even charge consumers a fee for the discovery of the selling rice. For instance, on the online shoing site Dubli.com, consumers have to ay.80 to learn the current rice of a roduct (and simultaneously, they lower the rice by.5 ). Swooo.com combines an ascending oen auction with bidding fees where consumers have to ay a fee in order to submit a bid. Bidding fees can be viewed as analogous to subscritions, which are quite common in discount retailing (BJ s, Sam s Club, etc.). When interreting the bidding fee as a subscrition,

23 the key simlifying assumtion is that the consumer s surlus from bidding at Stage is an accurate characterization of the consumer surlus that arises from her aggregate urchases over the lifetime of the subscrition, with these urchases reflecting successful reverse-ricing bids rather than osted-rice transactions. By showing that bidding fees are otimal in such a situation, we have rovided a model of a subscrition-based reverse-ricing discounter. We derive our results under the simlifying assumtion of no frictional costs for ease of exosition. This assumtion is not a limitation because, as we can show (not reorted), our basic qualitative results would not change if we relaxed it. In contrast, the assumtion that the outside market is not a strategic layer is necessary for our results, and what would haen if more active cometition were allowed is unclear. The analysis of the dynamics of cometition between reverse-ricing and osted-rice sellers (see Fay 009) is an imortant area for future research. Our analyses are based on the (common) assumtion of rational, utility-maximizing consumers. Follow-u work is needed to emirically validate our analytical results. Psychological factors might cause actual bidding behavior in a reverse-ricing context to deart from the normative model we use to characterize it. For instance, examining consumer reactions to bidding fees would be an interesting line of research. We susect consumers might (irrationally) be more willing to ay a minimum marku than to ay a bidding fee for entry. For examle, consumers may use an overriding rule to never ay for anything other than the roduct itself (e.g., information or service), a situation akin to the one Amir and Ariely (007) describe. A dee understanding of the behavioral asects of bidding in reverse-ricing markets will require much emirical work, but the analytical findings we reort here rovide a normative benchmark against which to develo and test behavioral deartures and extensions. 3

24 The analyses we have resented in this article have imortant imlications for sellers who wish to aly a reverse-ricing mechanism. Our findings rovide guidance as to how such firms should determine the otimal reverse-ricing mechanism in terms of how to set a bidding fee or minimum marku, whether to facilitate or hinder consumer learning about the bid-accetance threshold, and how best to adjust the revenue model when such consumer learning is anticiated. Aendix: Proofs of Proositions Proof of Proosition 1: It is sufficient to characterize bidding for valuations that exceed the minimum marku m, because bidders with valuations below m cannot obtain a ositive surlus in the market. A bidder with valuation v who has decided to enter and bid an amount x has the exected surlus (,, ) Pr ( acceted )( ) 1 ( ) Pr ( denied )( ) S v f m = x m v x + v> x m v f = x m + m x = ( v x) + 1 ( v> ) ( v ) f. Maximizing this exected surlus yields the bidding strategy b( v m) m+ v for m v< = m+ for v QED. Derivation of consumer surlus: Plugging the otimal bidding strategy b( v m ) into the surlus function S( v f, m, ) yields S( v, f, m, ) ( v m) f for m< v 4 =. ( m ) v + f for v > 4 4

25 Proof of Proosition : The seller maximizes his exected rofit Π ( f ) by selecting the otimal bidding fee f. As long as f, that is, as long as at least consumer v = enters, the seller obtains the following total exected rofit: (A1) ( f ) Pr ( v v) f E 1( b w)( b w) Π = < + wv, > v > = v v = 1 H( v) f + 1 > w w dwdh( v) + 1 H( ) w dw=, v 0 0 ( ) π ( ) ( ) ( ) π ( ) = 1 H f f + v dh v + 1 H v 0 f v v where π ( v) w dw= is the rofit from a bidder of valuation v. 8 The rofit Π ( f ) consists of three intuitive arts. The first term is the revenue from bidding fees, the second is the exected revenue from low entrants, and the third is the exected revenue from high entrants. The second term of the rofit function in equation (A1) requires further dissection and exlanation. It considers all ossible valuations v of consumers who will enter, and for each such v, it comutes the seller s ex-ante exected rofit from a articular entrant with that valuation. This rofit, denoted π ( v), arises from the seller integrating over all marginal costs that lead to acceted bids ( ws.t. w< b( v) ) while collecting the contribution b( v) w for each such w. The otimal bidding fee satisfies the following first-order condition: (A) ( + ) f FOC f : h f 1 H f ( ) = ( ) The FOC catures the following intuition: increasing f slightly has the marginal benefit of all consumers above v aying a higher bidding fee (RHS of equation (A)). The marginal cost of increasing f slightly (LHS of equation (A)) is the threshold consumer with valuation v not entering anymore, which results in a loss of both the marginal bidding fee consumer would ay f ( dv df ) = f and the loss of the marginal rofit from that consumer not bidding π ( f v) ( dv df ) = ( f ). Both of these comonents of the marginal cost are weighted by the density h( v ) of the marginal consumer occurring in the first lace.. 5

26 Rewriting the FOC in terms of the entry threshold v = f comletes the roof of the = xon [0,] yields f *( ) = ( ( + 6) ). The imlied general case. Plugging in H( x) threshold is less than when 4 ( 6) +, above which v = is otimal. QED Proof of Proosition 3: At Stage 1, the reverse-ricing seller maximizes his exected rofit ( m, f ) Π by selecting the otimal combination of bidding fee f and minimum marku m. As long as m+ f (i.e., as long as consumer v= enters), the seller obtains the following total exected rofit: (A3) ( ) ( ) 1 ( ) ( ) ( )( ) Π m, f = Pr v> v f + Ewv, > v b v m > w+ m b v m b w = ( ) π ( ) ( ) ( ) π ( ) = 1 H m+ f f + m v dh v + 1 H m m+ f ( m v) where π v m v+ m v m v+ 3m w dw= 4. To maximize Π ( m, f ) 0, the seller first finds the otimal bidding fee for every ossible level of minimum marku m. This contingent otimal fee satisfies the following first-order condition that equates the marginal cost of raising f slightly with its marginal benefit: (A4) Let H( x) ( + ) f FOCf : m+ h m+ f = 1 H m+ f ( ) ( ) = xon [0,]. In other words, assume low consumers are distributed uniformly on [0,] and an additional mass ( 1 ) of arbitrarily distributed high consumers are above. For this uniform distribution of low consumers, the otimal fee contingent on m imlied by becomes (A5) f *( m) ( ) ( 6) m 1+ = +,. FOC f 6

27 which obviously holds for m low-enough, secifically m 1 ( 1 ) +. When m is larger, a ositive fee is not otimal because m alone already screens many consumers sub-otimally. The imlies an entry threshold of FOC f 4+ m 3m v =. Since the entry threshold is linear in m and the 6 + otimal fee is quadratic in m, the rofit function ( m) ( m, f *( m) ) of oints that satisfy the FOC f is only cubic in m : Π Π restricted to the locus (A6) 1 1 Π ( m) = ( 1 v) f + ( v m)( v 3m) dv ( m)( 3m) = 8 1 ( ) v ( ) ( ) ( ) ( ) m = > 0 ( ) ( )( ) m ( + ) + ( 6+ ) ( ) m To rove Proosition 3, it is sufficient to show that for all 0 m 1 ( 1+ ), ( m) in m, that is, ( m) 0 ( m) Π <. Π (m) can be evaluated as Π is decreasing ( ) ( 3) ( 8+ 3) m + ( 3 )( ) m+ 3( 1+ ) + ( 6+ ) ( 3 ) Π = > 0 We start by showing that ( m) 0 ( ) ( + ) 86 Π < for the two extreme values of m: At the uer limit of m, Π = < + ( ) + > Second, the rofit is also decreasing at the lower limit of m: (A7) ( ) ( + ) + ( + ) ( ) 86 ( + ), which obviously holds for all Π = < > Inequality (A7) clearly holds for =0 and =1. To see that it also holds for all 0< < 1, let λ ( ) = Evaluating ( ) λ 0 = 3> 0 and λ () 1 = 31< 0 is simle. Since λ is a quadratic with a ositive coefficient on, it aroaches ositive infinity as grows very large or very small. Therefore, λ has at least one root above 1 because it is negative.. 7

28 at =1 and continuous. Now suose a 0 * 1 < < exists such that ( ) λ * < 0. Then λ must actually have all three remaining roots between 0 and 1. In other words, λ must start out ositive around =0, turn negative (λ must decrease below zero toward *), turn ositive again (to let λ increase again toward its ositive value at 1), and then turn negative near =1. λ starting out ositive, ending negative, and having three roots between 0 and 1 in turn imlies that λ has exactly two roots between 0 and 1. Since λ ( ) λ ( ) 0 = 144 < 0 and 1 = 68 > 0, λ having exactly two roots between 0 and 1 is imossible: a cubic that is negative at 0 and ositive at 1 must have either one or three roots between 0 and 1. Therefore, the existence of a * such that 0 * 1 < < and ( ) every 0 1. λ * < 0would lead to a contradiction, and we have shown that Π (0)<0 for It remains to be shown that ( m) 0 < m< ( + ). Since ( m) Π < for all non-extreme m values as well, that is, for Π is cubic in m, Π (m) is at most quadratic in m. When = 3, Π (m) is actually constant at ( m) 41 0 Π = <. When 360 3, the coefficient on m in Π (m) is unequivocally negative, so Π (m) is concave and has a global maximum at ( ) Π m* = 0. Simle algebra shows this maximum is m* = When < 3, 1 m* >, so 1+ Π <Π < 0 for 1+ Π (m) is monotonically increasing on the 0,1 ( 1 + ) interval and ( ) 1 m ( ) 0< m< When, on the other hand, > 3, then m * < 0, so Π (m) is monotonically decreasing on the 0,1 ( 1+ ) interval and Π ( m ) <Π ( 0 ) < 0 for 0 m 1 ( 1 ) < < +. Since rofit Π ( m) is thus decreasing in m along the locus of ossible solutions, the globally otimal solution is catured by the FOC f given m = 0. QED 8