Optimal selling strategies when buyers name their own prices

Transcription

1 Otimal selling strategies when buyers name their own rices Robert Zeithammer August 4, 04 Abstract: This aer models a name-your-own-rice (NYOP) retailer who allows buyers to initiate their retail interaction by describing a roduct and submitting a binding bid for it. The buyers have an outside otion to buy the same good for a commonly known osted rice that also acts as an informative uer bound on the cost the NYOP retailer faces. We concetualize a selling strategy of such a NYOP retailer to be the robability that a buyer s bid gets acceted that is a function of only the bid-level and the commonly known outside sot-market rice, but does not deend on the exact realization of the retailer s rocurement cost. Using mechanism design techniques, we characterize the otimal selling strategy and the equilibrium bidding function that best resonds to it. We show that the otimal strategy imlements the first-best ex-ost otimal mechanism: for every cost realization, the retailer can make as much rofit as he would if he could learn his cost first and use the otimal mechanism contingent on it. The comlexity involved in credibly communicating an entire bid-accetance function to buyers can make firstbest strategy imractical in some real-world markets, so we also analyze several simler NYOP strategies: setting a minimum bid, charging a articiation fee, and acceting all bids above cost. When both the distribution of buyer valuations and seller cost are uniform, the minimum-bid strategy dominates the other alternative strategies, and aroaches the first-best strategy in terms of rofit as the outside rice for the same good dros. Contact: Robert Zeithammer, UCLA Anderson School of Management. rzeitham@ucla.edu

2 . Introduction Name-your-own-rice (NYOP) selling allows buyers to initiate their interaction with a retailer by describing a roduct and submitting a binding bid for it. The retailer accets the bid whenever it exceeds his secret accetance threshold, and he kees the difference between the bid and his rocurement cost as rofit. NYOP selling was ioneered by Priceline.com in 998 in the travel industry, and since adoted globally by a range of online retailers in different roduct categories ranging from white goods (risminister.dk), restaurant meals (chiching.com), and designer fashion accessories (nyooly.com). This aer considers several strategies an NYOP retailer can use to increase his rofits. Most existing models of NYOP selling assume that the retailer accets all bids above his rocurement cost. Priceline does not officially disclose the details of its bid-accetance algorithm, but researchers knowledgeable about Priceline s suliers reort a olicy very similar to acceting all bids above cost (Anderson 009), ossibly with a minimum marku requirement whereby accetable bids need to exceed the cost by some minimum amount (Anderson and Wilson 0). While acceting all bids above cost is easy to exlain to bidders and guarantees ositive rofits for the NYOP retailer, the olicy may not be rofit maximizing for the retailer. For examle, the olicy is too simle to be otimal because it does not deend on the distribution of the buyer valuations (i.e. demand). Is there something else (such as the aforementioned minimum marku) an NYOP retailer can do in order to increase his rofits? How should we even concetualize the sace of ossible NYOP selling strategies? This aer defines a large set of such strategies, and uses mechanism design rinciles to characterize the rofit-maximizing one under general assumtions about the distributions of buyer valuations and retailer cost. We find that the first-best otimal NYOP selling strategy can be comlicated to exlain to buyers and requires relatively strong commitment by the retailer. In the second art of the aer, we restrict our attention to a articular distribution (uniform) and comare the first-best strategy to several second-best simler strategies that are easier to imlement. Amaldoss and Jain 008, Fay 009, Wang, Gal-Or, and Chatterjee 009, and others. Please see Anderson and Wilson (0) for a review of analytical aroaches to modeling NYOP selling.

3 Our model of the market in which the NYOP retailer oerates assumes the object sold by the NYOP retailer can also be obtained in an outside sot market for some commonly known rice. For examle, the object sold can be a articular designer handbag offered by Nyooly.com (the NYOP retailer), and the outside market rice is the lower of the rice osted in the designer s own store for the same handbag and that osted by third-arty retailers (if any). Alternatively, the object sold can be a seat on a articular flight sold by Priceline (the NYOP retailer), and the outside market rice is the lowest rice for that seat osted by other travel retailers (e.g. Exedia, Travelocity, Orbitz, etc.). We make the natural assumtion that the outside market rice is an informative uer bound on the rocurement cost the NYOP retailer can face at that moment. We define an NYOP selling strategy to be the robability the NYOP retailer accets a bid as a function of the bid-level and on the commonly known outside sot-market rice. For examle, the selling strategy imlied by acceting all bids over cost (the commonly assumed olicy described above) is just the cumulative distribution function of the rocurement cost. We roose that an NYOP retailer should actively manage his selling strategy and strive to communicate it credibly to the buyers. Our first contribution is a characterization of the otimal NYOP selling strategy and the buyers bidding function that best resonds to it. The NYOP retailer we consider needs to sets a single selling strategy for a range of ossible rocurement costs, but learns the exact cost realization before making his bid-accetance decision. There are two mutually related reasons for this modeling choice: First, NYOP retailers sometimes only query their suliers for current cost quotes after receiving a bid. For examle, according to its how it works age, chiching.com aroaches articiating suliers only after the bid is finalized. Several existing models make an analogous assumtion, e.g. Amaldoss and Jain (008) or Sann et al. (00). Second, even if the retailer can costlessly look u his current cost for any roduct offered and Priceline also makes its offering oaque (Fay 008) by hiding the airline name and exact time of dearture. Other retailers, e.g. byooly.com, risminister.dk or chiching.com, do not make the roducts oaque. We abstract away from oacity in this aer because it is an orthogonal issue. Please contact the authors for the otimal strategy when the retailer s offering is oaque.

4 ost the otimal selling rice 3, the buyers he faces may refer to bid instead of urchasing at a osted rice. We roose that NYOP selling emerged as a natural resonse to such buyer activism, which was in turn enabled by the Internet. For modeling clarity, we adot the assumtion that the retailer needs to set his selling strategy before learning the exact cost realization, and base it only on the distribution of costs. However, all our results will also hold for a retailer who rivately knows his cost before a buyer bid arrives, but cannot feasibly ost a fixed selling rice contingent on it. To derive the otimal NYOP strategy, we adat mechanism-design techniques to accommodate the retailer s ex-ante lack of cost information. We first derive the otimal direct revelation mechanism and then characterize its NYOP imlementation. In the direct-revelation mechanism, the otimal allocation rule conditional on a cost realization is not surrising: the retailer should accet buyer valuations above the monooly rice imlied by the realized cost. However, NYOP selling is not a direct-revelation mechanism, because successful NYOP buyers ay their bids and hence bid strictly below their valuations. Interestingly, we find that under standard regularity conditions of Myerson (98), the otimal allocation rule from the directrevelation mechanism can be imlemented even when buyers ay their bids and bid strategically. The otimal NYOP selling strategy (i.e. the otimal bid-accetance robability function) involves a minimum bid above which higher bids result in higher robabilities of accetance. The otimal strategy is unique on an interval of low bids, but consists only of an uer bound on an interval of higher bids. The outside sot market influences both the buyer s bidding strategy (it involves a jum-discontinuity and ooling at a articular bid-level) and the retailer s bidaccetance robability (it accets high-enough bids with certainty, regardless of the cost realization), but we show that it does not hinder NYOP imlementation of the otimal allocation. The nature of our otimal solution suggests a theoretically interesting corollary: it imlies that NYOP can be an ex-ost otimal selling strategy. Riley and Zeckhauser (983) show that a monoolist who knows his rocurement cost and is selling one indivisible object to a single risk- 3 This would be the otimal cost-contingent mechanism for such a retailer (Riley and Zeckhauser 983). By assuming this ossibility away, we take the existence of NYOP bidders as exogenously given. 3

5 neutral buyer should set a osted rice and make a take-it-or-leave-it offer. Since NYOP selling results in the same allocation as such cost-contingent monooly ricing, the NYOP retailer makes as much rofit as he would if he could learn his cost first and use the otimal mechanism contingent on it (a strategy we exlicitly assume away as exlained above). The key intuition behind the ex-ost otimality of NYOP is that although the retailer does not know his cost at the time of setting his strategy, he learns it before making the bid-accetance decision, and hence eventually has all the ieces of information needed for imlementation of the otimal allocation. In other words, NYOP selling can accommodate buyer activism in the form of a desire to submit bids before the rocurement cost is realized without any loss of retailer rofit. The NYOP retailer needs to credibly communicate the strategy to rosective buyers. Credibility requires commitment to a articular accetance robability for every ossible bid level, but is does not require the retailer to credibly communicate his cost realization or commit to any action contingent on a cost realization. We devote a secial subsection (5.5) to a discussion of two mechanisms that can facilitate credibility in the real world: reutation and a third-arty auditor. The comlexity involved in credibly communicating an entire bid-accetance function to buyers can make first-best strategy imractical in some real-world markets. Our second main contribution is a thorough exloration of second-best NYOP selling strategies that are simler to credibly communicate. Secifically, we analyze two kinds of retailers who accet all considered bids above their cost, but can commit not to consider certain bids. First, we analyze a seller who can commit to only consider bids above some minimum level - a strict simlification of the firstbest strategy. Communicating the minimum bid is simler because the retailer only needs to ost a single number. Credibility is easier to achieve because the minimum bid is analogous to a ublic reserve in an auction a standard feature in real-world markets. Second, we consider a seller who can commit to only consider bids by buyers who aid a articiation fee a two-arttariff strategy roosed by Sann, Zeithammer, and Haübl (00) and shown by them to dominate the minimum marku described by Anderson and Wilson (0). Because a articiation fee dominates minimum marku (and any combination of a articiation fee and 4

6 minimum marku), our investigation of second-best strategies imlicitly covers the minimum marku strategy, as well. We also comare the two second-best NYOP strategies discussed in the revious aragrah to two benchmarks: ) the assive NYOP strategy of considering all bids and acceting all bids above cost, and ) the non-nyop strategy of osting a fixed selling rice before learning the cost realization. Not surrisingly, we document that all active strategies strictly outerform the assive strategy of acceting all bids above cost. More surrisingly, we find that the minimum-bid strategy dominates the articiation fee strategy for all levels of the outside osted rice. The minimum-bid strategy also weakly dominates the fixed osted rice benchmark. Finally, the relative advantage of the first-best strategy over the minimum bid strategy nearly vanishes when the outside rice is very low. We conclude that much of the heavy lifting of the first-best strategy is actually accomlished by the much simler minimum bid strategy, esecially when many otential buyers can afford the outside market rice.. Related literature While NYOP selling has generated a lot of academic interest as a novel environment for studying consumer decision-making (e.g. Chernev 003, Ding et al. 005, Sann and Tellis 006, Sann et al. 0, and others), relatively less is known about the selling strategy NYOP retailers should use to maximize their rofit (we review the relevant contributions in detail below). This aer contributes to the small but growing theoretical literature about otimal NYOP selling. As noted in the Introduction, most existing models of NYOP selling assume that the retailer accets all bids above his rocurement cost. All other existing aers restrict attention to secific variants of ossible strategies, e.g. the extent of randomization in the accetance rule (Shairo 0) or minimum markus and articiation fees (Sann,Zeithammer, and Haübl 00). In contrast, this is the first aer to aly mechanism design techniques to the roblem of NYOP selling, thereby considering a much larger set of ossible strategies. 5

7 The most closely related aer is by Sann, Zeithammer, and Haübl (00), who show that an NYOP retailer analogous to the one assumed herein rofits more from charging a articiation fee than from charging a minimum marku (or from some combination of a fee and a minimum marku). Whereas they assume both distributions that arameterize the model are uniform, we rove our results in full generality. We show that no articiation fee strategy can imlement the same allocation as the otimal direct-revelation mechanism, and so emloying articiation fees is strictly less rofitable than emloying our otimal mechanism. We take the existence of NYOP selling as given, but our results are relevant to the literature that tries to rationalize the existence of NYOP. One theoretical justification for NYOP selling is as a second-best solution to the otimal selling roblem that accommodates buyer activism enabled by the Internet. We show that under standard regularity assumtions about the distribution of buyer valuations, NYOP selling can accommodate buyer activism without comromising retailer rofits: Desite having to set his selling strategy (i.e., his schedule of bidaccetance robabilities) before learning his cost of roduction, the NYOP retailer can achieve first-best ex-ost rofits. Therefore, NYOP selling does not actually give any more market ower to the buyers. In other words, the first-best otimal strategy allows the retailer to recature his market ower desite buyers actively bidding. We roose that even if our strategy is not the most ractical, it serves as an imortant theoretical benchmark. In another closely related work, Shairo (0) shows that risk aversion is also a way to rationalize the existence of NYOP selling. Secifically, he shows that the NYOP monooly rofit is higher than the osted-rice monooly rofit when buyers are risk averse, because such buyers bid more than risk-neutral buyers to avoid the risk of not winning at all. Shairo s (0) model makes several of the same assumtions we do: his retailer can commit to a robabilistic bid accetance, and he sometimes faces a non-strategic remium osted-rice retailer akin to the one we assume. In contrast to Shairo (0), our buyers are risk neutral and our retailer is exante uncertain about his rocurement cost. Extending our model with ex-ante retailer uncertainty to the case of risk-averse buyers is not tractable to us, but Shairo s logic suggests an otimal 6

8 NYOP retailer would strictly outerform his osted-rice counterart instead of merely getting the same rofit as he does under risk neutrality. Several other aers consider NYOP retailers who consider all bids and accet all bids above their cost, and show that even such a assive NYOP strategy may dominate osted ricing in a cometitive setting (Fay 009) or as a rice-discrimination tool (Wang, Gal-Or, and Chatterjee 009; Shairo and Zillante 009). Such models rely on buyer heterogeneity in an inherent reference for buying via NYOP, erhas because of varying frictional costs (Hann and Terwiesch 003) or the varying imact of the risk arising from oacity of the NYOP offering (Fay 009). The buyers in this aer instead care only about their surlus, and do not heterogeneously favor osted-rice buying over NYOP, or vice versa. However, our results can be used as a building block in a cometitive model because we characterize the best resonse of an NYOP retailer to any regular demand function. Chernev (003) and Sann et al. (0) show that consumer behavior is different when the NYOP retailer allows any rice to be named (a true name your own rice in Chernev s nomenclature) comared to when the NYOP retailer resents a menu of rices (called select your rice by Chernev). The mechanism roosed here is more in line with select your rice. Moreover, we suggest the retailer should resent not only a menu of rices, but also the associated accetance robabilities. Such an institution facilitates credibility, simlifies bidding, and enables the retailer to better learn consumer references by fixing bidder beliefs. NYOP selling is also related to auction theory in that it corresonds to a first-rice sealed-bid auction with a single bidder (the buyer) and a stochastic secret reserve (the retailer s bid-accetance function). The imlementability of the otimal mechanism via NYOP selling can thus be interreted as an extension of the revenue equivalence between first-rice and secondrice auctions, itself a well-known result in auction theory (Vickrey 96, Myerson 98): the direct revelation mechanism we find is equivalent to a set of second-rice sealed-bid auctions, each with an otimal reserve rice (different for different costs) and only one bidder. Otimal NYOP selling can be thought of as a set of first-rice sealed-bid auctions, each of which is 7

9 constructed to accet a bid exactly often enough to maintain revenue equivalence with the corresonding second-rice auction. We analyze a static model in which each buyer is restricted to a single bid, but real-world bidders may manage to bid multile times. Fay (004) uses a stylized model to show that reeat bidding does not necessarily erode retailer rofits, as long as the retailer is aware of the behavior and uses the right dynamic thresholds. In a more recent aer on reeat bidding, Chen (0) suggests that Priceline s lockout eriod restriction, a design alleged to rotect sellers, can actually benefit customers. and links the issue to bargaining theory. Extending our setu to a dynamic environment is beyond the scoe of this aer, but we roose that something akin to Fay s (004) finding would relicate: as long as the retailer anticiates reeat bidding and conditions his accetance-robability strategy on the number of bids the bidder has submitted to date, the retailer should be able to extract first-best rofits from the reeated interactions. 3. Model Our notation follows Krishna (00) whenever ossible, and it is summarized for easy reference in Table. A risk-neutral buyer is interested in buying one articular indivisible object. The buyer s valuation x of the object is drawn from a continuous distribution f x and suort on, x x. Assume the virtual value x 8 F x with density x F x x is increasing, that is, f that the distribution F is regular in the sense of Myerson (98). The virtual value function is a central concet in the theory of mechanism design, and it reresents the marginal revenue a seller can extract from a buyer of tye x in a direct revelation mechanism (see Krishna 00 for a more detailed exosition). rice The object is readily available in an outside osted-rice market for a commonly known 0 x, where 0 is the rice a monoolist with zero marginal cost would charge for the object. When the buyer does not buy the object from either retailer, her ayoff is zero. Following Sann, Zeithammer, and Haübl (00) and Shairo (0), we assume the

10 NYOP retailer is small in that he takes the osted rice as fixed, and the outside sot market does not adjust its osted rice in resonse to the NYOP retailer s strategy. 4 An NYOP retailer can rocure the object for a rocurement cost c~ H c, where the distribution H has full suort on 0,. In other words, the outside osted rice is a ublic uer bound on the NYOP retailer s rocurement cost. For examle, the NYOP retailer can be Priceline selling excess caacity on a articular flight, whereas is the rice of a seat on the same flight osted by Exedia. Alternatively, the NYOP retailer Nyooly could be reaching rice-conscious consumers of designer handbags, whereas is the rice of the same handbag at the designer store. 5 The entire model is thus arameterized by the two distributions F and H, where the suort of the latter deends on. A lower is both bad news (tougher cometition) and good news (lower exected cost) for the NYOP retailer. Note that in our baseline model, we abstract from the fact that NYOP roducts are often oaque a feature ioneered by Priceline, and a otential source of differentiation between the two retailers (Fay 008, Shairo and Shi 008). In our seat-on-a-flight examle, the buyer knows the same amount of information about the seat before urchasing it from either retailer. Oacity is not intertwined with NYOP selling in the real world: none of the other three retailers we mention in the Introduction (chiching.com, risminister.dk, nyooly.com) are oaque. It can be shown that oacity does not change our main result qualitatively, lease contact the authors for details. Timing of the game is as follows (lease see the bottom timeline in Figure ): In the beginning of the game, the NYOP retailer announces his bid-accetance strategy 4 Since the two cometitors are selling the same object, Bertrand cometition would result if the outside cometitor resonded. To revent a comlete collase of rofits, we could introduce horizontal differentiation arising from heterogeneity in buyer inherent reference for NYOP over osted ricing similar to Hann and Terwiesch (003) or Fay (009). Within such a larger model, our aer characterizes what the NYOP best resonse would look like. 5 The underlying assumtion is that the NYOP retailer does not have a secial technology for roducing the object, but rather obtains the object from the same sulier as his osted-rice cometitors. Even after learning the osted rice, uncertainty about c remains because is a relatively stable rice, set to reflect long-run revenue-management considerations and quite ossibly a larger set of customers (as in Sann, Zeithammer, and Haübl 00). 9

11 Pr accet b A b for all ossible levels of bid b submitted by the buyer for the object. The buyer then submits a binding bid. After receiving a bid, the retailer queries his suliers for a cost quote to learn his actual cost c and decides whether to accet the bid. At any time during the game, the buyers can choose to buy from the osted-rice outside market and ay the rice. Figure highlights the key contrast with osted ricing in the timing of the cost information. Figure : Timing of the NYOP game, as comared to standard osted ricing Standard osted-rice selling Seller learns his cost c Seller commits to a take-it-or-leave-it rice ρ(c) Buyer accets or rejects ρ(c) time Seller hands the object over iff ρ(c) acceted Buyer can buy same object for in outside market Name-your-own-rice (NYOP) selling time Seller commits to bid-accetance olicy Ab Pr bacceted Buyer submits bid b(x) Seller learns his cost c Seller follows rule Pr b acceted c such that Ab E c Pr b acceted c To derive the otimal mechanism, we follow Myerson (98) and the rest of the mechanism-design literature, and assume the NYOP retailer can commit to any bid-accetance strategy Ab Pr accet b. In the second half of the aer, we relax this assumtion in several ways and exlore the imact on retailer rofits. When the retailer has no re-commitment ability, he obviously accets all bids above c, so everyone knows Ab Pr b c, and the retailer s announcement in the beginning of the game contains no new information for the buyers. 0

12 Before solving for the otimal bid-accetance strategy, we summarize the imact of the outside sot market on the demand the NYOP retailer faces. For all bid-accetance strategies, buyers with x> all mimic the tye x=, because they have a real otion to buy in the outside market when the rice they name is rejected. In other words, the NYOP retailer faces buyers with a distribution of net valuations F on 0, and F exected surlus U of an x> buyer who bids b is,, mass at. To see this fact, note the U b x A b x b A b x x U b () where Ub, is the utility of the buyer with x=. It is immediate that the same b that maximizes Ub, also maximizes, U b x. Intuitively, the buyer thus receives all of his valuation in excess of as surlus, and his articiation with the NYOP retailer is akin to free gambling in hoes of randomly getting a rice below. 4. Otimal direct-revelation mechanism We use the revelation rincile (Myerson 98) to restrict attention to direct-revelation mechanisms whereby the buyer reorts her valuation truthfully. Much of the material in this section is standard, so the details are relegated to the Aendix. The only deviation from a textbook treatment (e.g., Krishna 00) is the fact that the retailer does not know his cost c when he sets his strategy, but he does learn c before making his bid-accetance decision. Knowing the cost before the accetance decision needs to be made allows us to first otimize the contingent bid-accetance rule x, c Pr allocate object to x when cost is c. Not knowing the cost at the outset restricts the retailer to robabilistic assignments that are only a function of x. Let q x be the robability that a buyer with x receives the object. A retailer who is lanning to use a given x, c can only commit to qx x, cdhc 0. The aforementioned atom at also rovides an interesting wrinkle relative to the textbook in that the otimal q is discontinuous at. The otimal contingent bid-accetance rule is as follows (lease see the Aendix for all roofs):

13 Proosition : The otimal contingent bid-accetance rule in a direct-revelation mechanism is when x and c x or when x xc, 0 otherwise For every cost c, this rule achieves the same rofit as a osted-rice monoolist who knows c before setting his rice, namely, F * * c where * min c,. The allocation rule for x< is familiar from the mechanism-design literature: Myerson s (98) otimal reserve rice alies for low buyers: for every x<, the retailer should sell (i.e., set xc, ) iff x c x monooly rice c. In addition to sometimes serving some of the low buyers, the retailer should also always sell to all high-value buyers (x ) because c holds for all c by construction. The intuition for x, c cost c<, who faces demand goes back to a simle osted-rice monoolist with a marginal Fz for all rices z<, and a oint mass of F customers willing to ay exactly. Such a monoolist charges recisely maximize his rofit. Because the retailer can condition his x, c c min, to on c, he can effectively get the ex-ost monooly rofits. In other words, for every cost c, he can get the same rofit as a osted-rice monoolist who knows c before setting his rice. Note that the otimal allocation alies regardless of the distribution of retailer cost c, but the imlied bid-accetance strategy will deend on H. Examle: F and H Uniform Throughout the aer, we will use the examle of F=Uniform[0,] and H=Uniform[0,] to illustrate the findings in closed form. When F is uniform on [0,], x x, so the comlete otimal allocation is c sell x or x. Note that only buyers with x

14 have any chance of winning, so the retailer effectively sets a minimum bid of ½. The retailer Ec x c dx. When H is also uniform, the retailer s c min, makes a rofit of * exected rofit becomes 3 * xc xc 6 dx dc dx dc 0 c Note that regularity of F in the sense of Myerson (98) is not required for osted ricing to be the otimal strategy contingent on c: Riley and Zeckhauser (983) show that the otimal x, c is a ste function with a single ste even when x is not increasing in x. Had we not assumed regularity in this section, Proosition would be modified to xc x x * for some x* that satisfies c x*, when either, or when x= (see Proosition of Riley and Zeckhauser 983). Exosition is easier with a regular F, and the next section will show that regularity is actually a necessary condition for an imlementation of the otimal allocation rule through NYOP selling. The form of the otimal allocation rule in Proosition is not surrising. The key question of this aer is how to imlement it within the NYOP institution. The retailer could simly romise to charge a rice of c min, to all bidders with bids that exceed it, reserving the incentive to bid truthfully. In other words, the retailer could run a Becker, DeGroot, and Marschak (964) rocedure with a carefully selected distribution of rices. However, an NYOP retailer romises to charge buyers their bids whenever a sale occurs. In resonse to aying their bids, buyers shade their bids below their rivate valuations. In the next section, we derive the buyer s bidding function and the retailer s bid-accetance rule that imlements the otimal mechanism. 3

15 5. Imlementation of the otimal mechanism through NYOP An NYOP retailer romises to charge buyers their bids whenever a sale occurs. One advantage of acceted buyers aying their bids (vs. the otimal monooly rice conditional on the realized c) is that the retailer does not need to credibly communicate his c to the buyers. However, how buyers will resond is not a riori clear. Will they bid according to an increasing (and hence invertible) bidding function that allows the retailer to imlement his otimal allocation, or will they somehow obfuscate their tye to avoid being exloited? In this section, we show constructively that the buyers best-resonse bidding function is invertible and the NYOP retailer can thus imlement the otimal allocation rule described in Proosition if and only if x the virtual value is increasing in x, irresective of H. In other words, regularity of F in the sense of Myerson (98) is necessary and sufficient for an NYOP imlementation of the otimal mechanism. The roof roceeds in three stes, each catured in a searate lemma: ) Derivation of the bidding strategy for low-valuation bidders (x<) ) Derivation of the bidding strategy for high-valuation bidders (x=) 3) Secification of the otimal bid-accetance robability for intermediate bid levels 5. Bidding strategy for low-valuation bidders (x<) First, consider a buyer with 0 x who follows a bidding strategy x shows that his robability of winning in the otimal mechanism is, q x x c dh c H x 0 x x x U x q t dt H t dt x t dh t x x x. Proosition. The roof of Proosition shows that his exected utility is, where the last equality follows from integration by arts. For NYOP selling to be revenue equivalent with the otimal directrevelation mechanism, the bidding strategy x must satisfy: U x Uxqxxxx x () q x 4

16 The structure of the candidate bidding function in equation () yields the following result: Lemma : For bidders with x<, the NYOP bidding strategy imlied by the otimal directrevelation mechanism of Proosition is increasing iff ψ is increasing, and it can be characterized by x Ec c c x. Together, Proosition and Lemma show that when F is regular, the otimal mechanism can be imlemented via NYOP. No additional assumtions about F are needed, and the H distribution of costs merely needs to have full suort on 0,. The exact x function in Lemma follows from lugging the q and U from Proosition into equation (). In words, Lemma shows that the otimal bid by a buyer with valuation x< is the average (over c) of monooly rices the buyer would be willing to ay, namely, rices below x. Given the regularity of F we assume throughout, the bidding function is increasing, and hence invertible, so the retailer can infer each bidder s x and aly the allocation rule of Proosition. Given the bidding function, we can also solve for the cost-contingent bid-accetance rule imlied by the x, c in Proosition : cost c b b c b E cost cost c (3) That is, the retailer accets bids over the average monooly rice he would charge for all costs below his actual cost realization. 5. Bidding strategy for high-valuation bidders (x=) Now consider a buyer with x=, and recall that the otimal allocation rule is to sell to this buyer for all levels of c. Because xx for all x, the limit of the above accetance rule as x aroaches from below involves an accetance robability below one for all x other than x=x secial because x x. In other words, A Pr c, where lim x To imlement the needed c,, a bid level. x must exist such that buyers with 5

17 x= refer to bid, but buyers with x< do not deviate from bidding according to the above x. Our next lemma shows that a unique such does exist: Lemma : For bidders with a net valuation of the NYOP offering equal to the outside rice, the NYOP bidding strategy imlied by the otimal direct-revelation mechanism of Proosition. 0 H z dz is 5.3 Otimal bid-accetance robability for intermediate bid levels The otimal mechanism determines the accetance robability of all bids in 0,,. No bidders should submit bids in the intermediate region of, and, but the NYOP retailer still needs to secify a bid-accetance strategy function for such bids, ensuring they indeed remain off equilibrium. One simle rule is to simly reject such bids. Another simle rule that kees Abnon-decreasing is to simly let Pr accet b H for all b,. Our next Lemma derives the uer bound on robability accetance of every intermediate b: A b H z dz b Lemma 3: When b,. Plugging b for b, 0, no buyer submits a bid into the bound in Lemma 3 shows that the uer bound aroaches as b aroaches. Therefore, the full Ab can actually be continuous and increasing on 0,. Lemmas -3 comlete the roof of the first main result of this aer: 6

18 Proosition : For every regular continuous F on x, x and every outside osted rice 0, the following bid-accetance robability function imlements the ex-ost otimal b lim x : H b x H z mechanism: Ab lim xb : anything dz x b 0 b : where the bidding function x is the unique best resonse to bid b whenever b x dh c 0, : x : H zdz 0 x c 0 H x Ab. After learning his roduction cost c, the retailer accets a c for b lim x, and he accets with certainty. x Note that the NYOP imlementation of the otimal mechanism consists of a air of a bidaccetance robability minimum bid imlied by examle: Ab and a bidding function x and A b is 0 x that best resond to each other. The. It is useful to consider a closed-form 5.4 Examle: F and H Uniform It is easy to show that an F=Uniform[0,] imlies 0, and hence for x, 0 x x h z c h c c c x z dz dc E x H x H x The bid of the x= bidders is (4) c c c c Pr Pr Ec (5) 7

19 When H is also uniform on [0,], x 4 x for all x<, and. Please see 4 Figure a for this bidding function. Figure a: Bidding functions that imlement the otimal mechanism, for five levels of the outside market rice F Uniform0,, H Uniform0, bid ( β(x) ) x x x : 4 x : Note to Figure: The subscrit of the bidding function indicates. The emty circles indicate the jum discontinuities from z z for all z<q. to. For q<, valuation (x) q Given this bidding strategy, we can also solve for the bid-accetance rule in closed form: 8

20 c b : accet when c b b 4 4 bc, b : accet with Pr b b : accet always 4 (6) Therefore, the retailer only accets bids over /, accets bids over with certainty, and is 4 willing to sell below cost whenever c cc. The imlied bid-accetance strategy is 4 3 Ab 0 for b 4b for b 4 below for b 4 b 4 4 for b 4 (7) Interestingly, the bid level at which the accetance robability jums to is 4 exactly the bid level at which lim x; x; x. This finding may contain deeer insight. Desite the aarent comlexity of the imlementation, the exected rofit given c is just c 4. Figure b illustrates the otimal A for different levels of. 5.5 Two real-world mechanisms for making the first-best strategy credible The otimal strategy outlined above requires the retailer to communicate a articular bidaccetance function and commit to it. One way to facilitate this would be for the NYOP retailer to ost a table with the robabilities of different bids being acceted, and allow buyers to easily reort osted robabilities and accetance outcomes to a third-arty auditor. The auditor could 9

21 then ost the summary statistics, ensuring the emirical robabilities match the romised ones. For examle, the auditor could simly reort a single grah, relating the stated robabilities to recency-weighted emirical estimates of the actual robabilities. As long as that grah is a 45- degree line, the retailer is not under- or over-stating the buyers chances. Note that the auditor only needs to know the stated robability of success and the outcome (accet or reject), not the retailer s eventual cost realization or anything secific about the roduct at hand or the bid level. We roose the NYOP retailers interested in using our otimal selling strategy should facilitate the information flow to such a third-arty auditor and erhas even suort its oerations with a flat subsidy. Figure b: Otimal bid-accetance strategy, for five different levels of outside rice A(b)=Pr(accet bid) bid Note to Figure: An illustration of equation (7). The numbers next to the lines indicate the five levels of. Each line reaches at the resective. For bids between and, the lines indicate the maximum accetance robability such that no bidder bids at those levels. 0

22 Without an official auditor, consumer word of mouth on websites such as biddingfortravel.com can serve as an unofficial auditor, and facilitate credibility via reutation. This is already haening in the case of Priceline, at least to some degree: when a buyer selects a hotel region to bid on, the website flashes a message saying that some erson recently won a hotel in the selected area for a certain rice. When the buyer then tyes in a very low bid, Priceline automatically flashes a message saying Based on recent data, your rice has almost no chance of being acceted. Thus, the website communicates at least the suort of rices that have a ositive but still uncertain chance of being acceted. User forums such as betterbidding.com and biddingfortravel.com contain many reorts of how accurate this information is, making sure Priceline does not exaggerate the range of suggested rices it reorts. One of the main managerial messages of this aer is that instead of obfuscating the bidders chances of winning, being clear and truthful about ones selling strategy may actually serve an NYOP retailer well as long as he uses a good strategy and does not merely assively accet all bids above his rocurement cost. 6. Simler second-best selling strategies: four alternatives The comlexity of the first-best NYOP selling strategy characterized in the revious section may be imractical for some real-world markets. In this section of the aer, we consider three simler NYOP selling strategies and one obvious non-nyop benchmark: first, retailers may not be able to credibly communicate a full bid-accetance strategy, but commit to only considering bids above some minimum level akin to a ublic reserve in an auction. Second, the retailers may be able to charge a articiation fee, that is, ask to be aid for considering bids, as roosed by Sann, Zeithammer, and Haübl (00). Third, we analyze a assive retailer who can only credibly reject unrofitable bids (bids below cost). The rofits available to such a retailer are the relevant baseline from which all of the above retailers imrove. Considering the rofits of the assive retailer also isolates the dual role of in the model. Finally, an imortant non-nyop benchmark is offered by a seller who sets a fixed osted rice desite his ex-ante uncertainty about cost.

23 6. Particiation fee: a two-art tariff strategy Sann, Zeithammer, and Haübl (00, 04) assume F=Uniform[0,] and H=Uniform[0,], and show that the retailer rofits more from charging a articiation fee than from charging a minimum marku. They also show that the articiation fee alone is more rofitable than any combination of a articiation fee and a minimum marku. We now reeat their main result in our notation. The otimal fee e is (Proosition of Sann, Zeithammer, and Haübl 04): 4 for 0.56 e* min, for (8) and the resulting retailer rofit is: * e 3 4 for for (9) The articiation fee screens low-valuation buyers out of the market, and the imlied entry 4 threshold is e* min,. The retailer then accets all bids above cost, so his NYOP 7 selling strategy is Ab Pr b c, and the uniform assumtion on H imlies buyers who enter bid x x. Therefore, the comlete allocation rule of the underlying mechanism is: q e x 0 for x 0,4 7 x for x4 7, (0) To rank retailer rofits relative to the first-best otimal strategy, it is enough to comare the allocation rule qe x to that in the otimal mechanism imlied by Proosition : x x x x 0 for, 0 H and F 0 for x 0, uniform qxh x for x 0, x for x, for, for x, ()

24 Because q x qx e, no articiation fee can imlement the otimal mechanism. Figure 3 shows the two contingent (on c) allocation rules for an outside rice fixed at = 3. Figure 3: Allocation rules: the otimal mechanism vs. simler strategies c c x (buyer s rivate valuation) otimal mechanism articiation fee minimum bid 0 c (NYOP seller s marginal cost) 3 Note to Figure: The osted rice is set to throughout the figure. The suort of (x,c) is in 3 the x rectangle, and F(x) is uniform. The shaded area indicates the allocation rule of the otimal mechanism. The area filled by vertical lines indicates the allocation rule of the otimal articiation fee. Note that all consumers with x 4 enter and ay the articiation fee. 7 3

25 It is clear from Figure 3 that the otimal mechanism is more efficient than the articiation fee when the retailer cost is high. However, there is also a small region of the (c,x) sace where the fee strategy does result in a trade while the otimal mechanism does not. Therefore, Figure 3 is not sufficient for relative efficiency ranking. However, by the revelation rincile, the fact that the two allocations are not the same is enough to rank the rofits. We will return to exlicit rofit comarisons in Section Minimum bid and fixed rice strategies While the revious section mostly reviewed an existing result, this section resents a new contribution to the literature on NYOP selling. Two closely related strategies are analyzed in turn: setting a minimum bid and charging a fixed osted rice. Suose that the NYOP retailer cannot commit to a selling strategy in the form of a robability schedule, but can credibly refuse to consider bids below a certain minimum level m. Once he considers a bid, he accets all bids above his cost. Credibly communicating a minimum bid is easier than credibly communicating an arbitrary Ab, because minimum bid is a ure strategy, and hence can be verified on a case-by-case basis. Moreover, a minimum bid can also be made credible via common knowledge that getting cost quotes from suliers is costly (a low bid is then not worth considering because it does not cover the cost of a sulier quote in exectation). Finally, minimum bid is clearly a realistic strategy because it is equivalent to a ublic reserve in auctions a commonly observed feature in most auction markets. Another reason to analyze the minimum-bid strategy is the fact that minimum bids are a salient feature of the fully otimal olicy. It is interesting to investigate how much of the rofitincreasing otential of the fully otimal olicy is achieved via the minimum bid alone, and how the added rofit from the fully otimal olicy deends on the outside market rice. We now solve for the otimal minimum-bid strategy. Buyers with x<m do not enter the NYOP market, because they cannot earn a non-negative surlus. Buyers with x m solve x arg max Hb xb Hb max 0, x bm 4

26 where the second term reresents the otion value of buying in the outside osted-rice market. Let b* solve the first-order condition of the unconstrained roblem, namely, H b* b* min x,, and assume H satisfies the usual conditions for b* to be the argmax of h b* H b x b is x max b*, m. Adding the constraint that b m imlies that the buyers otimal bidding strategy as long as m. In other words, a mass of buyers with valuations just above m all ool on bidding m as long as this is a better deal than the outside market. Assume for tractability (and for a direct comarison with the articiation-fee result) that H is uniform on [0,]. Then x with x m,m min x, max, m. In other words, when m</, buyers all bid m. But such low values of m often do not maximize the retailer rofit. In d dm, which holds, for examle, the Aendix, we show that 0 Fm Fm mf m for the Uniform F distribution. When m>/, every buyer who enters the market bids m, simlifying the rofit to c m Fm m cmcd Fm () 0 Maximizing the rofit in equation without the m constraint is achieved by the following first-order condition: m F m (3) f m It is immediate that the otimal minimum bid exceeds the the monooly rice the retailer would charge if his marginal cost were zero, namely condition in (3) is solved by m* min,. The resulting retailer rofit is: When F is uniform, the first-order m, so the otimal minimum bid the retailer should use is 3

27 : 3 7 * m : 3 (4) We collect the overall conclusion of this subsection in Proosition 3: Proosition 3: When F is uniform on 0, and H is uniform on 0,, the otimal minimum bid is min, 3. All buyers with valuation above the minimum bid enter the NYOP store and submit the minimum bid. At least in the Uniform-uniform case, the equilibrium otimal minimum bid is so high that nobody bids above it: the seller effectively charges m to all buyers who can afford it. However, the minimum-bid strategy is not equivalent to charging a osted-rice because the minimum-bid retailer is not obliged to sell to every buyer willing to ay m. Instead, the sale is made only when m haens to exceed the actual rocurement cost c. This is guaranteed (by definition of H) only when the minimum bid is equal to. To illustrate the difference between the minimum bid and the osted rice, note that a retailer forced to charge a fixed osted rice irresective of cost would maximize the exected rofit Fr rec subject to r is uniform on 0, and H is uniform on, and so charge r* min E c,. When F 0,, the otimal osted rice would thus be min,. Noting that, we have the following result: Proosition 4: When F is uniform on 0, and H is uniform on 0,, the otimal fixed osted rice strategy is to charge for 4 3 and otherwise. 3 6

28 The otimal fixed osted rice strategy thus coincides exactly with the otimal minimum bid strategy for low, and involves charging a higher rice than the otimal minimum bid for high in order to comensate for rising costs. It is also obvious that the fixed rice strategy is less rofitable than the minimum bid strategy. 6.3 Passive retailer We conclude our investigation of simler strategies with a assive retailer who cannot commit to anything and cannot charge a articiation fee, and hence accets all bids above cost. F=Uniform[0,] and H=Uniform[0,], such a retailer makes the rofit: x x c c 3 0 c cd dx c c d (5) Note that this rofit is hill shaed, and maximized at =0.75. This non-monotonicity demonstrates how a lower outside-market rice is both bad news (tougher cometition) and good news (lower exected cost) for the NYOP retailer. The cometitive effect is stronger for low, the cost-reduction effect for high. 7

29 Figure 4: Exected rofit of NYOP retailers: four alternative strategies 0. exected rofit of retailer otimal mechanism + x minimum bid only fixed rice articiation fee assive baseline rice of the outside otion () Note to Figure: Illustration of Proosition 5. F is uniform on 0, and H is uniform on 0,. 6.4 Profit comarison of the four alternative strategies Figure 4 lots rofits of all four alternative selling strategies and the fully otimal strategy as a function of the rice of the outside otion (under the F=Uniform[0,] and H=Uniform[0,] assumtion), and illustrates our last roosition: 8

30 Proosition 5: When F is uniform on 0 0, and H is uniform on 0,, as aroaches from above, the relative rofit advantage of the first-best strategy over the minimum bid strategy vanishes. The relative ranking of the second-best strategies is as follows: a) the minimum-bid strategy strictly dominates the articiation-fee strategy for all, and the rofit advantage is decreasing in. b) the minimum-bid strategy is equivalent to the fixed-rice strategy for and strictly 3 dominates the fixed rice strategy otherwise. c) there is a * such that the fixed-rice strategy strictly dominates the articiation-fee strategy for *, and vice versa. d) the assive strategy is strictly dominated by all other strategies. To gain intuition for the minimum bid aroaching full otimality as aroaches half, note that for = ½, both the fully otimal contingent bid-accetance rule and the otimal minimum bid strategy reduce to osted ricing because the retailer only serves the high-valuation consumers and charges them. Both the fully otimal NYOP retailer and his minimum-bid counterart abandon low-valuation buyers comletely, focusing on stealing the high-valuation buyers from the outside market. The full otimality of the minimum bid strategy at =½ also immediately imlies its dominance over articiation fees for low. Intuitively, the minimum-bid strategy is good at raising everyone s bid, which is useful when the outside otion is attractive to many buyers. The difference between the two second-best NYOP strategies declines as rises because the retailer who uses a articiation fee is better able to extract rofit as increases. Both retailers face rising costs as increases, but the minimum-bid retailer does not benefit from a comensating increase in bids beyond =/3. To see why, note that for >/3, the outside rice does not affect entry and bidding when a retailer commits to a minimum bid: buyers with xm* enter and 3 9