Optimal Sequential Auctions with a Dynamic Bidder Population and Resale

Transcription

1 Optimal Sequential Auctions with a Dynamic Bidder Population and Resale Xiaoshu Xu y March, 2018 Abstract In practice, when bidders arrive sequentially and resale cannot be banned, the auctioneer s re-auction incentive naturally extends a standard one-shot auction to a dynamic game. If the initial auction is not successful, the auctioneer can re-auction the item to current and future bidders at a later date; otherwise, the winner can opt to resell it to future bidders. In a twostage private-value model where patient bidders arrive in two di erent batches, we characterize the optimal sequential ascending English auctions. There are three interesting ndings on the optimal reserve price path. First, the rst batch of bidders always have a second chance in the re-auction, although the auctioneer s updated belief on their value distribution becomes weaker after the initial auction fails. Second, the auctioneer would encounter the implementation issue because there exist multiple equilibria. We invent a novel selection rule to restore equilibrium uniqueness. Third, resale does not a ect the auctioneer s revenue in some situations because the optimal design does not give a possible winner any surplus from resale ex ante. Keywords: Optimal sequential auctions, Noncommitment, Sequential arrival, Patient bidders, Resale JEL Classi cations: D44, C7, D82 I thank Dan Levin, Lixin Ye, James Peck, Yaron Azrieli, Marco Ottaviani, Alessandro Pavan, Rakesh Vohra and Andy Skrzypacz for very helpful comments and suggestions. I also thank the support from National Natural Science Foundation of China (grant code: ). y College of Economics and Management, Shanghai Jiao Tong University, No Hua Shan Road, Shanghai , China. xuxiaoshu@sjtu.edu.cn 1

2 1 Introduction Extremely popular online auction platforms such as Ebay act as a focal point of demand where potential buyers can constantly search to determine whether the items in which they are interested are being auctioned. In such auctions, buyers thus arrive sequentially (or in di erent batches) 1, which could fundamentally change the incentives of the auctioneer and bidders. The auctioneer realizes that if there is no winner, a subsequent auction can be conducted at some future date when a new batch of potential bidders arrive. 2 Similarly, experienced current bidders also see future bidders as an opportunity: when the current auction admits a winner, the winner has the option of holding a resale. The re-auction incentive 3 and resale incentive are also pervasive in other auctions in the private sector, such as art pieces, precious wine, and antiques. The story is largely similar for auctions in the public sector: re-auction is not rare 4 and resale or resale-like actions (e.g., mergers) cannot be completely prohibited even with rm regulatory policies. Therefore, in real auctions, sequential arrival, reauction and resale are non-separable. Re-auction not only gives the auctioneer an opportunity to sell to new bidders, but it also serves as a second chance for the original bidders: when they anticipate that an unsuccessful initial auction will lead to a re-auction, they rationally wait if they are not forced to leave. 5 Our aim is to characterize the auctioneer s revenue maximization problem and its implications in this environment, which has not been studied in the previous literature. We propose an illustrative two-stage model with discounting: two batches of patient bidders (male, de ned as B1 bidders and B2 bidders) arrive sequentially, where the term patient refers to the assumption that B1 bidders can wait until the second stage without explicit costs. We focus on an ascending English auction (strategically equivalent to a second-price sealed-bid auction in this 1 We focus on the non-strategic arrival of bidders, meaning that they arrive for exogenous reasons (e.g., a new need for the item, being nancially incapable at the time of the previous auction). 2 Ebay even provides special guidance including tips to relist items, the fees charged, and how to make the auction more attractive. See for details. 3 McAfee and Vincent (1997) o er several good examples of re-auctions regularly being observed in auction houses. 4 For example, in timber auctions, there is a policy that unsold quantities need to be re-auctioned at later auctions after approximately six months. During the time elapsed, additional interested bidders could arrive. 5 There is no obvious reason for the auctioneer to exclude the original bidders or new bidders from the re-auction. In practice, there have been examples where re-auction is not open to new bidders because the winner of the initial auction failed to make the payment; however, this type of event is not the focus of this paper. 2

3 context) and assume that the auctioneer can only control reserve prices. 6 The second-stage game depends on the rst-stage outcome: when no B1 bidder drops out at or above the rst reserve price, the auctioneer (female) updates her belief about their values and posts a second reserve price in the re-auction to both batches of bidders; otherwise, the winner of the initial auction decides whether to consume the item immediately or attempt to resell it in the second stage. In case of an unsuccessful resale, he can still consume the item at the end of the second stage. 7 A key point of the model is that the reserve price in the re-auction is sequentially optimal. This is also the assumption of McAfee and Vincent (1997), although they restrict attention to a xed set of bidders. We consider perfect Bayesian equilibrium in which B1 bidders use symmetric and monotonic bidding strategies with a cuto in the initial auction: a B1 bidder remains active past the reserve price only when his value is higher than or equal to some cuto. On the equilibrium path, when the initial auction fails, the auctioneer consistently believes that all B1 bidders have values below that cuto. We explicitly address two challenges intrinsic to this dynamic game. The rst challenge comes from what we label endogenous asymmetry: even if two batches of bidders are symmetric in values ex ante, asymmetry is inevitably created ex post in the re-auction; thus, the auctioneer must decide whether to include B1 bidders as potential buyers when selecting the contingently optimal reserve price. Our analysis also reveals that this phenomenon causes a critical issue of equilibrium selection because some reserve prices in the initial auction can induce two equilibrium cuto s. The second challenge concerns resale. The resale opportunity will make B1 bidders bid more aggressively, which seems to bene t the auctioneer; however, any possible surplus from resale is extracted from B2 bidders that the auctioneer can serve herself by imposing a high reserve price in the initial auction. Whether resale can increase the auctioneer s revenue under her incentive for revenue maximization is an interesting yet intriguing question. We have several novel ndings. First, although the traditional one-to-one correspondence between the reserve price (in the initial auction) and the equilibrium cuto breaks down, we create a novel selection rule that focuses on the incentives of pivotal (de ned in Section 3) B1 bidders to restore 6 This assumption might seem restrictive, but it holds in many auctions in the private sector. 7 One implicit assumption about the item is that the winner cannot consume it in the rst stage and resell it in the second stage. An interpretation is that once it is "opened" (or consumed), its resale value drops to zero. Items such as wine, collectible toys and initial releases of certain books or DVDs have this property. 3

4 uniqueness. Second, on the optimal reserve price path, B1 bidders always have a second chance in the re-auction after they fail the initial auction. Third, under some su cient conditions, the auctioneer s revenue-maximizing incentive would induce an equilibrium cuto high enough that no B1 bidder resells conditional on winning. Finally, even if resale is removed from this model, all analytical results continue to hold. This paper is closely related to the literature on optimal auctions with resale and revenue management with little (or no) commitment. Although resale has always been a widely discussed issue in the auction literature, most papers focus on bidders post-auction interactions and how they can in uence the equilibrium properties in the initial auction 8. There are only a few studies that address the issue of revenue maximization. Ausubel and Cramton (1999) prove that when inter-bidder resale of complete information cannot be banned, resale does not occur under the auctioneer s revenue-maximizing incentive. Zheng (2002) considers how the Myerson revenue can be achieved with inter-bidder resale. 9 Calzolari and Pavan (2006) study both the cases of inter-bidder resale and resale to a third party, but they focus on what disclosure rule the auctioneer should use to maximize revenue. Zhang and Wang (2013) consider which bidder the item should be assigned to in auctions with a regular bidder and a bidder whose value is publicly known. The new batch of bidders in our model are recognized as "outsiders" in previous studies; by naturally integrating them into the whole picture, we are able to examine in detail how the auctioneer s rst-mover advantage in selecting the reserve price a ects resale. Our result that resale does not occur in several circumstances on the optimal reserve price path shows that resale may have no e ect on the auctioneer s revenue when she can access the resale demand herself. The literature on revenue management with limited (or no) commitment and (a xed set of) strategic buyers includes Hörner and Samuelson (2011), Liu et al. (2013), McAfee and Vincent (1997) and Skreta (2006, 2015). Among these studies, McAfee and Vincent (1997) is the most related to ours. They consider an auction setting with nitely many periods during which the auctioneer cannot commit to future reserve prices. When sequential arrival and resale is removed, our model is their two-period case. Skreta s seminal work extends McAfee and Vincent (1997) to allow for general 8 Recent studies in this direction include Garratt et al. (2009), Hafalir and Krishna (2008), Haile (2000, 2003), Lebrun (1999), Pagnozzi (2007), and Xu et al. (2013). 9 Lebrun (2012) generalizes Zheng s result. 4

5 mechanisms, while Liu et al. (2013) extend McAfee and Vincent (1997) to an in nite-horizon auction model. Hörner and Samuelson (2011) consider a monopolist that has multiple objects to sell before a deadline, using posted prices. Our major contribution to this literature comes from the modeling of sequential arrival and the new ndings associated with it. Sequential arrival is a usual assumption in mechanism design with full commitment, but it has been ignored by the non-commitment literature, mainly for reasons of tractability. We show that endogenous asymmetry generates two critical new features. First, the earlier batch of bidders will always be included in later rounds of the auction. This result can be generalized since the intuition is not restricted to this model. Second, equilibrium selection is inevitable. We invent a novel rule and believe that its idea can be borrowed in other similar models. Finally, this paper is also related to the large and growing literature on mechanism design with a dynamic (but not necessarily patient) buyer population. Bergemann and Said (2011) summarize representative work in this eld. Board and Skrzypacz (2016) consider a monopolist s revenue maximization problem when selling a xed quantity of products to a dynamic and patient buyer population. The monopoly has full commitment power, and unsold quantities in any given period would be destroyed. They also prove that the optimal mechanism can be implemented by ascending English auctions and second-price sealed-bid auctions. 10 Without resale, our model is the two-period case of Board and Skrzypacz (2016) with a single item and limited commitment. Our research highlights that when limited (or no) commitment is combined with a dynamic and patient bidder population, the multi-equilibria phenomenon is inevitable, which instantly creates an obstacle to direct mechanism design: the existence of a maximum is not su cient since the maximum revenue in theory may not be achieved by an indirect mechanism when the designer and buyers have di erent preferences over the equilibria. Despite this concern, we believe that the results in this paper still serve as an important step toward further research on dynamic mechanism design without commitment. We proceed as follows: to illustrate di culties brought by this setting, Section 2 presents a simple example under the uniform distribution but without resale, including some results on comparative 10 Dilme and Li (2014) attempt to add noncommitment to a setting similar to that of Board and Skrzypacz (2016), but to maintain tractability, they restrict their attention to the posted price mechanism and further assume that buyers only have two possible types. Thus all high-valued buyers are served immediately upon arrival. The dual-type assumption subtly avoids endogenous asymmetry. 5

6 statics; Section 3 proposes a general model; Section 4 o ers a brief discussion on possible extensions; Section 5 concludes. 2 A Uniform Distribution Example without Resale We begin with possibly the simplest example and delay all proofs until Section 3. To highlight the challenges created by endogenous asymmetry, we assume for the present that the auctioneer can explicitly ban resale. Notations in this section are consistent with their counterparts in Section 3. Consider a private-value model with two stages: an auctioneer (female) has an indivisible item, which she values at 0, to be sold to two potential buyers (male), denoted bidder 1 and bidder 2. Bidder i arrives at the beginning of the i th stage, with the valuation v i that is independently drawn from the uniform distribution on [0; 1]. The auctioneer holds at most two successive English (or sealed-bid second-price) auctions. She rst announces a reserve price r 1 to bidder 1: if bidder 1 does not drop out at 0, he wins at the price r 1, and the game ends; otherwise, the game proceeds to the second stage. The auctioneer then announces a reserve price r 2 to bidder 1 and the newly arrived bidder 2, where r 2 is sequentially optimal. Two bidders thus compete as in a normal English auction. We assume that all players are risk neutral with the common discount factor 2 (0; 1). In this indirect mechanism, the auctioneer s revenue crucially depends on bidders strategies and the reserve price path. In the second stage, we assume that each bidder adopts the weakly dominant strategy of dropping out at his value if it is higher than r 2. In the rst stage, we focus on a cuto strategy for bidder 1: he drops out at 0 when v 1 is smaller than some cuto v; otherwise, he immediately wins the item. Beliefs are constructed accordingly. Using backward induction, the auctioneer s task in the second stage is to determine the optimal r 2, with the belief that v 1 ~U[0; v] and v 2 ~U[0; 1]. Clearly, the auctioneer must compare her expected revenues generated by r 2 2 [0; v] and r 2 2 (v; 1] respectively. Intuitively, when v is relatively small, it is not wise to pick r 2 2 [0; v] at the cost of sacri cing some of the potential surplus that can be extracted from bidder 2. De ne the sequentially optimal r 2 to be r 2 (v). According to the proof of Proposition 1 in Section 3, there exists a threshold V T on the value of v such that the auctioneer s decision is cut into two disconnected pieces: 6

7 8 < r 2 (v) = : 1 2, when v 2 [0; V T ] v+1 4, when v 2 (V T ; 1], V T is approximately 0:415 (1) The intuition for the structure of r 2 (v) is straightforward. The revenue generated from just serving bidder 2 is xed while the revenue generated from serving both bidders increases as v increases. The latter equals the former at v = V T and it keeps increasing till v = 1. Therefore, r 2 (v) is piecewise. In the rst stage, bidder 1 is the last mover and decides whether to drop out at 0. Looking ahead, he anticipates that r 2 (v) is determined by v through equation (1). Denote his continuation payo from dropping out at 0 as l(v 1 ; v): l(v 1 ; v) is 0 when v 1 r 2 (v); otherwise, it is equal to R v1 0 (v 1 maxfv 2 ; r 2 (v)g)dv 2. Clearly, bidder 1 compares l(v 1 ; v) to v 1 r 1 (his payo from winning immediately) in making his decision. The construction of the equilibrium strategy requires that in equilibrium, v 1 r 1 > l(v 1 ; v) when v 1 > v; v 1 r 1 = l(v 1 ; v) when v 1 = v; and v 1 r 1 < l(v 1 ; v) otherwise. The correspondence between r 1 and v can thus be derived from the indi erence condition of a marginal bidder 1 with v 1 = v. De ne the corresponding r 1 of v to be r 1 (v), we have 8 < v, when v 2 [0; V T ] r 1 (v) = v l(v; v) = : v R v 0 (v maxfv 2; r 2 (v)g)dv 2, when v 2 (V T ; 1] In Proposition 3 of Section 3, we will prove that any pair of (r 1 ; v) satisfying equation (2) can be supported on the equilibrium path. For the present, we focus on the properties of r 1 (v). It can be immediately detected that r 1 (v) is not continuous at v = V T : it has a sudden drop in value because the term R v 0 (v maxfv 2; r 2 (v)g)dv 2 is strictly positive at v = V T. This property plus the fact that r 1 (v) increases in v over both [0; V T ] and (V T ; 1] together imply some r 1 has two corresponding equilibrium values of v. The left part of Pic_1 illustrates this property with r 1 on the X-axis, v on the Y-axis and = 0:9. The multi-equilibria phenomenon occurs for r 1 over (0:43; 0:45) approximately. (2) 7

8 Pic_1 This phenomenon immediately raises the equilibrium selection issue, which may bother the auctioneer. To see this point, denote by rev(r 1 (v); v) the auctioneer s expected revenue and by v a maximizer of rev(r 1 (v); v). If r 1 (v ) happens to fall into (0:43; 0:45), there exists some v 0 6= v such that r 1 (v 0 ) = r 1 (v ). In this case, the auctioneer cannot implement r 1 (v ) if bidder 1 prefers v 0 instead because bidder 1 moves after her in the rst stage, and her contingent belief on v 1 has to be consistent in equilibrium. A way to avoid the selection issue is to prove that r 1 (v ) lies outside the multi-cuto region. Fortunately, for any value of in this example, it can be proven that: rst, rev(r 1 (v); v) strictly increases in v for v 2 [0; V T ]; second, rev(r 1 (v); v) is strictly concave in v for v 2 (V T ; 1] with an interior maximizer which is also the global maximizer. This maximizer can be solved as v = 2 q ( ) In this example, it can also be proven for any value of that r 1 (v ) lies outside the multi-cuto interval. The right part of Pic_1 illustrates how rev(r 1 (v); v) (Y-axis) changes with v (X-axis) when = 0:9. Next, we present several results about comparative statics regarding v and r 1 (v ) (3) 8

9 1. For general n and m with the uniform distribution, v decreases in n when n is relatively large. It increases in m and. This result can be proved analytically. We attach the proof at the end of the appendix. 2. With n = m = 1, r 1 (v ) increases in. Intuitively, as increases, there are con icting e ects because both the auctioneer and bidder 1 are more patient. On the one hand, the auctioneer has incentive to raise v and thus r 1 (v ) because bidder 2 appears more attractive now. On the other hand, since bidder 1 also intends to wait, r 1 has to decrease to keep a marginal bidder 1 indi erent. With n = m = 1, it can be proven that r 1 (v ) strictly increases in, as illustrated by Pic_2. Pic_2 A useful remark is that when there is no new arrival, McAfee and Vincent (1997) prove that with two periods, the optimal reserve price in the initial auction decreases with. The intuition is straightforward: without new arrival, the auctioneer has no incentive to delay the sale so she has to give bidders extra incentive to bid now. However, in our model, the existence of bidder 2 completely alters this result. 9

10 3. r 1 (v ) decreases in n and increases in m for relatively small, but it increases in both n and m for relatively large. This result is achieved numerically with n 5 and m 5, while ranges from 0:1 to 0:9. It is clear that r 1 (v ) should increase with m: the re-auction becomes more attractive to the auctioneer so she has incentive to delay the sale; meanwhile, B1 bidders are more reluctant to wait which means the auctioneer can extract more surplus from them in the initial auction. However, the e ect of n on r 1 (v ) is very tricky. We know that v decreases in n when n is relatively large. This result holds in this numerical example with only one exception. Nonetheless, as n increases, there are two con icting e ects on a B1 bidder s option value of waiting: the sequentially optimal reserve price r 2 (v) drops, but the degree of competition in the re-auction from other B1 bidders increases. The second e ect dominates the rst: as Bulow and Klemperer (1994) show, an auctioneer may bene t from attracting one more bidder rather than attempt to impose an optimal reserve price; thus a B1 bidder is hurt more in the re-auction by a larger n. Therefore, B1 bidders su er from the increased n. From equation (2), we can conclude that the change in v dominates the change of B1 bidders incentive when is relatively small, while the reverse conclusion holds for a relatively large. Finally, we brie y discuss the challenges that endogenous asymmetry raises for revenue maximization. The rst challenge is whether it is optimal to set an a ordable reserve price for bidder 1 in the re-auction. This concern is inevitable with limited commitment and a dynamic and patient bidder population, and it would become more severe when the game has more than two stages. Board and Skrzypacz (2016) study this setting under full commitment and characterize a stable condition across periods. With limited commitment, stableness would be impossible to achieve since backward induction keeps adding more batches of earlier bidders into consideration. Another related study is Liu et al. (2013), who use the traditional approach with a constraint representing sequential rationality to characterize the optimal auction with non-commitment and an in nite horizon. However, to keep the analysis tractable, they do not consider sequential arrival. The second challenge arises from the equilibrium selection issue. Consider a case in which the auctioneer selects some r 1 with two equilibrium cuto s. If the auctioneer and bidder 1 s interests diverge, she can implement the cuto that she prefers in a direct mechanism but not in the indirect mechanism because bidder 1 moves after her. This concern implies that the traditional mechanism design approach cannot be applied in this setting. 10

11 In Section 3, the generalization of the number of bidders and allowing for resale make the analysis less tractable. We are nevertheless able to prove that B1 bidders still have a chance in the re-auction on the optimal reserve price path. However, the equilibrium selection issue cannot be avoided. Instead, we invent a novel selection rule that focuses on pivotal B1 bidders and prove that their interests coincide. Results on comparative statics in the general model are not available although basic intuition in this example can still go through. McAfee and Vincent (1997) have clearer intuition on comparative statics since their model does not have new bidders, but they still point out that analytical proof is available only with two periods when their general results allow for any nite many periods. 3 The General Model Based on the model in Section 2, we make the following generalizations. First, the number of B1 bidders is n, and the number of B2 bidders is m. Second, bidders values independently and identically follow some arbitrary F () on [0; 1], where the conditional distribution F (xjx a) satis es the monotonic likelihood ratio property for any a Third, at the end of the rst stage, the winner (if any) can choose to consume the item (in which case his gross payo is his value) or to resell it in the second stage (in which case his gross payo is the discounted expected resale payo, including the event that he consumes the item when no bid meets the reserve price) to B2 bidders and B1 losers. Resale takes the form of an optimal ascending English auction. For notational convenience, in the following analysis, we do not include the parameters n; m; in mathematical terms unless speci ed. We also list major de nitions at the beginning of each subsection for quick reference. The solution concept we adopt is perfect Bayesian equilibrium (PBE). In the rst stage, we still focus on a symmetric and monotonic bidding strategy with a cuto : given r 1, there exists a cuto v such that a B1 bidder with value v drops out at 0 when v < v and drops out at some (v) otherwise, where (v) strictly increases with v and (v) r In the second stage, regardless of the auction- 11 This implies that x F (a) F (x) f(x) is a strictly increasing function. It ensures that there is a unique optimal reserve price in resale. Hafalir and Krishna (2008) also impose this assumption. 12 Note that although (v) can be strictly higher than r 1, dropping out at (v) or r 1 does not change the contingent payo. We assume that these bidders drop out at (v) (if not at zero) for continuity. 11

12 holder s identity, we assume that all bidders follow the weakly dominant strategy of sincere bidding. All beliefs are consistent. Therefore, on the equilibrium path, when n > 1, the winner of the initial auction believes that all losers values are below his; thus, in resale, he only takes B2 bidders into account. We discuss the possibility of other possible equilibria in Section Equilibrium Analysis: Second Period We make the following de nitions for the present. Definition 1 F (j) k (j) and f k : the distribution and density function of the j V T order statistic out of k bidders with the value distribution F () r 2 (v) : the sequentially optimal reserve price in the re-auction r : the unique solution to x = 1 F (x), which is the optimal reserve price in f(x) a standard IPV auction with the distribution F (): V T : the value of v at which the auctioneer is indi erent between serving both batches of bidders and just serving B2 bidders in the re-auction r(v) : the optimal reserve price in the resale when the winner s value is v w(v) : the expected payo from resale when the winner s value is v The seller s identity in the second auction depends on the outcome of the initial auction. We rst consider the case of re-auction. The following proposition characterizes r 2 (v): Proposition 1 (The Contingently Optimal Re-auction) m[1 F (x)]+n[f (v) F (x)] De ne (v) as the unique solution to x = (m+n)f(x). r 2 (v) is piece-wise 13 : 8 < r, when v 2 [0; V T ] r 2 (v) = : (v), when v 2 (V T ; 1] (4) 13 Although r 2(v) is expressed as a function, r, and (v) generates the same second-stage revenue when v = V T 12

13 Proof. The sketch of the proof is as follows: When the auctioneer is forced to include B1 bidders (namely, r 2 < v), (v) is optimal; when she is forced to exclude them, r is optimal. As v increases, the expected revenue from including them increases, and it equals the expected revenue from excluding them when v hits V T. (v) strictly increases in v with (1) = r. Details are relegated to the Appendix. From the analysis in Section 2, Proposition 1 is the starting point at which our results depart from previous related studies. Inevitably, the piece-wise structure projects back to the auctioneer s optimization problem in the initial auction and creates the multiple equilibria phenomenon through backward induction. Note that according to Myerson (1981), the optimal direct mechanism would assign the item to a bidder with the highest positive virtual value, which may not be ex post e cient. Even if we allow for more stages for possible resale among the mixed bidder population (and exclude more batches of bidders), Zheng (2002) shows that the Myerson revenue can typically only be achieved with two bidders. We next consider the case when the initial auction admits a winner. If he opts to hold a resale, on the equilibrium path, he consistently believes that only B2 bidders are potential buyers; thus, when his own value is v, his expected payo from resale can be derived as w(v) = max r2[v;1] F m (r)v + mf m 1 (r)[1 F (r)]r + The optimal reserve price is clearly the unique solution to x Z 1 r F (a) xf (2) m (x)dx (5) F (x) f(x) = v. 3.2 First Period The analysis proceeds as follows: using backward induction, B1 bidders contingent payo s in the second stage can be derived, which can be used to construct the indi erence condition for a marginal B1 bidder (whose value equals the cuto v); thus, the correspondence between r 1 and v can be derived. We then prove that any pair of r 1 and v linked through the indi erence condition can be supported on the equilibrium path. As illustrated in Section 2, the traditional one-to-one correspondence breaks down because of endogenous asymmetry. We will impose a natural equilibrium selection rule. In 13

14 short, this rule assumes that all pivotal (de ned below) B1 bidders are the key players in selecting the cuto when there are multiple cuto s. We further prove that they prefer the same cuto Equilibrium Existence We track a B1 bidder with value v. The following de nitions are made: Definition 2 H(v) : H(v) = maxfv; w(v)g; which can be interpreted as the actual worth of the item v() : the point at which a winner is indi erent between holding a resale and consuming the item w (v; v) : the expected payo from dropping out at maxfh(v); H(v)g l (v; v) : the expected payo from dropping out at 0 v H and v L : the larger(smaller) cuto when a single r 1 induces two The next proposition addresses what his bid should be when he does not drop out at 0. We postpone the discussion of whether he is better o by dropping out at 0 to the proof of Proposition 3. Proposition 2 implies that in equilibrium, the symmetric bidding function () should be H(). Proposition 2 When other bidders drop out at H() if their values are no less than v, a B1 bidder should drop out at maxfh(v); H(v)g if not at 0. Proof. In the proof, we assume that this bidder mimics type ~v. Clearly, if he does not drop at 0, ~v has to be no less than v. We then prove that if v v, the best ~v is v, and thus, this bidder should drop out at H(v). For v > v, we prove that the best ~v is v by ruling out the overbidding and underbidding case, and thus, this bidder should drop out at H(v). The underbidding case is more di cult because even if he loses, he still has a chance of buying back the item through a resale if the winner holds one. Details are relegated to the Appendix. Note that when v > v, in equilibrium, H(v) is the gross payo conditional on winning. However, if such a bidder loses in equilibrium, he has no hope of winning the resale, and thus, his payo conditional on losing is 0. Therefore, the intuition of Proposition 2 is straightforward: a B1 bidder with v > v bids the actual worth of the item in equilibrium. Clearly, this bidding strategy has the 14

15 same avor as "sincere bidding" in a regular English auction, although it is not a weakly dominant strategy, and the proof is much more di cult. We next derive the indi erence condition for a marginal B1 bidder. The calculation of w (v; v) is simple, while the calculation of l (v; v) is very tedious: when this bidder loses, the contingent game could be a re-auction, a resale, or no resale. The detailed derivations of w (v; v) and l (v; v) are presented in the proof of Lemma 1. In equilibrium, the indi erence condition is w (v; v) = l (v; v) (6) The correspondence between v and r 1 can thus be derived, as characterized by Lemma 1. Lemma 1 De ne l(v; v) as this bidder s expected payo (undiscounted) from the re-auction when the initial auction fails. De ne l(v; v) + as the strictly positive part of l(v; v). As r 1 increases, it rst induces a single cuto, then two cuto s, and nally, a single cuto again, as described by the following 8 system: H(v) = r >< 1, when r 1 2 [w(0); H(V T ) l(v T ; V T ) + ) H(v L ) = r 1, w(v H ) l(v H ; v H ) = r 1, when r 1 2 [H(V T ) l(v T ; V T ) + ; H(V T )] >: w(v) l(v; v) = r 1, when r 1 2 (H(V T ); 1 l(1; 1) + ] Proof. See Appendix. We call [H(V T ) l(v T ; V T ) + ; H(V T )] the multi-cuto region. Before resolving the equilibrium selection issue, it is necessary to prove that any pair of (r 1 ; v) satisfying Lemma 1 can be supported on the equilibrium path. Proposition 3 formally presents the PBE. Proposition 3 Given r 1, the following strategies and beliefs constitute a PBE: all B1 bidders follow the proposed cuto strategy, where (v) = H(v) and the cuto v satis es Lemma 1. When there is no winner, the auctioneer believes that all B1 bidders have values following the truncated distribution of F () over [0; v] and selects r 2 (v) as the reserve price; otherwise, the winner decides whether to hold a resale with the reserve price r(v). All losers in the initial auction do not proceed to resale. In the second-stage game, all bidders apply the sincere bidding strategy. All beliefs are consistently constructed. 15

16 Proof. The proof is divided into two cases: v V T and v > V T. For v V T, the di cult part is to prove that a B1 bidder with v > v has no incentive to drop out at 0: even with this deviation, he may still have a chance in the re-auction or in the resale (if there is one). For v > V T, the di cult part is to prove that a B1 bidder with v v has no incentive to drop out at H(v): with this deviation, he may be able to become the winner in the initial auction and manages to earn a positive pro t from resale. Details are relegated to the Appendix Equilibrium Selection It is clear that the multi-equilibria phenomenon is irrelevant to the auctioneer s revenue concerns only when the optimal r 1 never falls in the multi-cuto region: namely, regardless of whether v L or v H is coordinated on, the optimal v corresponds to some r 1 outside that region. However, as illustrated in the right part of Pic_1 of Section 2, con rmation of the above pattern requires a full characterization of the expected revenue as a function of v. To avoid this generally complicated issue, an equilibrium selection rule is needed. The usual Pareto ranking rule cannot be easily applied because it still requires detailed examination of the payo functions of the auctioneer and B1 bidders. Nonetheless, we are able to invent a novel rule that is natural. As we have pointed out in Section 2, since B1 bidders move after the auctioneer announces r 1, she cannot control which cuto they follow. Among B1 bidders, those with v 2 [0; v L ) [ (v H ; 1] do not actually have the freedom to select the cuto : their strategies in the initial auction do not vary with the choice of v L or v H. Therefore, only B1 bidders with v 2 [v L ; v H ] are key players in determining the cuto. In the following, we call them pivotal to highlight their importance. We assume that all players follow the cuto that pivotal bidders prefer. The next concern is whether all pivotal bidders interests coincide for any r 1 in the multi-cuto region. Proposition 4 shows that they do share the same interest. Fortunately, Proposition 4 Given r 1 2 [H(V T ) l(v T ; V T ) + ; H(V T )], v H is always preferred to v L by any pivotal bidder. Proof. We track a bidder with value v 2 [v L ; v H ]. When the other n 1 bidders and the auctioneer 16

17 follow v H but he deviates from dropping out at zero in the initial auction, according to Proposition 2, his best deviation is to drop out at H(v) and obtain a deviation payo of w (v; v H ). Clearly, this deviation payo should be less than his equilibrium payo l (v; v H ). Next, consider the case in which the other n 1 B1 bidders and the auctioneer follow v L. If this bidder also follows v L, in all of his winning events, he also wins when the other B1 bidders and the auctioneer follow v H but he deviates to v L. In fact, he has more winning events with this deviation because B1 bidders with values in between v and v H previously dropped out at zero but do not now. Moreover, in every winning event, his expected payment in the deviation case is weakly lower while the expected gross payo remains the same. The reason is that, with the deviation, his expected payment conditional on winning is always r 1. However, when v L is believed to be the cuto, there is a positive probability that there exists at least one B1 bidder with a value in between v L and v, resulting in a higher payment. Therefore, w (v; v H ) > w (v; v L ), where w (v; v L ) is the equilibrium payo when the other B1 bidders and the auctioneer follow v L. However, because l (v; v H ) is this bidder s equilibrium payo when the cuto is v H, l (v; v H ) w (v; v H ). By transitivity, l (v; v H ) > w (v; v L ). Therefore, any pivotal bidder prefers v H to be the equilibrium cuto. This completes the proof. We provide a rough intuition for Proposition 4 while xing non-pivotal B1 bidders values. For pivotal bidders, the choice of v H and v L leads to di erent layouts of the game. If v H is selected, a pivotal bidder cannot win in the initial auction. However, he still has a chance of winning in the re-auction, with a lower reserve price 14 but extra competition from B2 bidders. By contrast, if v L is selected, a pivotal bidder could win immediately with a payment of no less than r 1 and the resale option. Therefore, v H 0 s advantage is the lower reserve price, while v0 Ls advantage is that B2 bidders serve as a source for extracting surplus instead of rivals in competing for the item. However, the contingently optimal r 2 can be accurately predicted while B2 bidders exact values are not. certainty serves as the primary factor in determining the relative advantage of v H. As a result of Proposition 4, we impose the following selection rule to restore equilibrium uniqueness: 14 It is obvious that when v H is selected as the equilibrium cuto, r 2 is strictly smaller than r 1; otherwise, a marginal bidder faces a larger reserve price, competition from B2 bidders and the discount in the re-auction, in which case he cannot be indi erent. This 17

18 Assumption 1 (Equilibrium Selection) When r 1 induces two equilibrium cuto s, we assume that the auctioneer and B1 bidders adopt v H, which generates a higher continuation payo for the 8rest of the game for pivotal bidders. This restriction implies that v is determined by r 1 according to < H(v) = r 1, when r 1 2 [w(0); H(V T ) l(v T ; V T ) + ) : H(v) l(v; v) + = r 1, when r 1 2 [H(V T ) l(v T ; V T ) + ; 1 l(1; 1) + ] According to this rule, some values of v can never be implemented. E ective cuto values lie in disjoint intervals: v 2 [0; H 1 (H(V T ) l(v T ; V T ) + )] [ [V T ; 1]. Thus, any r 1 generates a unique equilibrium cuto, which further induces a unique contingently optimal r 2. Therefore, the auctioneer s maximum revenue is determined by r 1. Although we establish this intuitive rule to avoid complications involved in a detailed characterization of the revenue function, it is possible that the rule has no e ect on the maximum revenue: rst, any r 1 such that the auctioneer s and B1 bidders interests diverge will never be chosen because the auctioneer can anticipate that B1 bidders will follow their preferred cuto ; second, v H may also be preferred by the auctioneer; and third, it is possible that the optimal r 1 always lies outside the multi-cuto region, regardless of which cuto we select over that interval, as in Section Optimal Sequential Auction Because of equilibrium selection, nding the optimal r 1 is equivalent to nding the optimal v. Since values of v lie in two disconnected intervals after the selection, it su ces to nd the optimal cuto in each interval and compare the revenues that they generate. We seek to answer two questions by characterizing the optimal reserve price path. The rst question has been presented in Section 2: should the auctioneer give B1 bidders another chance in the re-auction? The second question focuses on how resale a ects the auctioneer s revenue Second Chance for B1 Bidders? Unlike noncommitment studies with a xed set of bidders in which it is sequentially optimal to constantly give bidders more chances (as in McAfee and Vincent, 1997), there are pros and cons of doing so in this model. On the one hand, other things being equal, including more bidders improves revenue in the re-auction; on the other hand, including B1 bidders means a lower contingently optimal r 2, which reduces the expected surplus extracted from B2 bidders. It seems that the relative 18

19 strength of these countervailing forces largely depends on n and m: if m is large relative to n, it may be optimal to exclude B1 bidders ex ante and vice versa. Definition 3 r 1 (v) : the corresponding r 1 of v Rev(r 1 (v); v) : the expected revenue as a function of v v : the globally optimal cuto value The next proposition shows the aforementioned intuition does not hold: the auctioneer s decision does not depend on n and m. Proposition 5 v V T, or equivalently, r 2 (v ) < v. Therefore, on the optimal reserve price path, B1 bidders are always included as potential buyers after they fail in the initial auction.. Proof. The sketch of the proof is as follows. We rst prove that Rev(r 1 (v); v) strictly increases in v for v V T. We then prove that there exists some v (2) > V T such that Rev(r 1 (v (2) ); v (2) ) Rev(r 1 (v); v) for any v > V T ; thus, v (2) is the maximizer when v > V T. Finally, we show that Rev(r 1 (v (2) ); v (2) ) Rev(r 1(V T ); V T ) strictly increases in and is positive at = 0. Therefore, v (2) is also the global maximizer v, implying that the auctioneer optimally gives B1 bidders a second chance in the re-auction. Details are relegated to the Appendix. The key in understanding Proposition 5 is that the auctioneer can only control r 1. When B2 bidders are not attractive compared to B1 bidders, clearly she should still give B1 bidders a chance in the re-auction. However, even if B2 bidders are very attractive (e.g., m is su ciently larger than n or is su ciently close to 1), the initial auction has to fail for the auctioneer to access them. Ironically, the only way to increase the chance of a re-auction is to raise r 1, while in equilibrium a high r 1 induces a contingently optimal r 2 that is a ordable to B1 bidders. Therefore, it is never wise to preclude B1 bidders in the re-auction on the optimal reserve price path. Since the intuition does not depend on n, m or, we believe that this result can be extended to any nitely many periods (with a batch of patient bidders arriving in every period), which may be a good starting point for future research regarding non-commitment and sequential arrival. 19

20 3.3.2 How Resale A ects the Auctioneer s Revenue? In auctions with resale, a typical question is how the resale incentive would a ect bidders bidding behavior and thus the auctioneer s revenue. In the symmetric equilibrium we focus on, resale makes B1 bidders to increase the bid from v to maxfv; w(v)g. Therefore, for a xed r 1, resale bene ts the auctioneer when r 1 is not su ciently large. However, under the auctioneer s incentive for revenue maximization, the impact of resale is not apparent. We next analyze this problem with two steps. First, if the total surplus that can be extracted from B2 bidders is thought to be a pie, does the auctioneer have incentive to share it with B1 bidders? Second, if not, how can the auctioneer mitigate B1 bidders possible gain from resale? The answer to the rst question is "No", because of two reasons. Obviously, B1 bidders have no information advantage about B2 bidders values over the auctioneer; thus the existence of the resale opportunity does not make any contribution to the ex ante size of the pie. Moreover, although it seems that the auctioneer can bene t from B1 bidders in ated bids, the in ated part is actually a mark-up where the winner serves as an intermediary. However, there is no necessity of that when the auctioneer can wait till the second period to serve B2 bidders herself. The second question is tricky. Although the auctioneer has no incentive to share the pie with B1 bidders, there is no extra measure she can take beyond the reserve price. Moreover, the reserve price, and thus the induced cuto, only serves as a lower bound for selecting the winner s value. To see whether resale increases the auctioneer s maximum revenue, we can check whether a marginal B1 bidder s resale incentive (with v = v ) is mitigated. Using the envelop theorem, it can be proven that the term x w(x) strictly increases in x. Recall that v() is the unique point where v = w(v), if v satis es v v(), resale does not a ect the auctioneer s revenue because it does not occur. However, even though v v() is only a su cient condition, it is generally impossible to be veri ed. Instead, we present three independent conditions for this inequality to hold. Proposition 6 Any of the following conditions alone is su cient for implicitly banning resale: i) v() 2 [0; V T ] ii) v() 2 (V T ; r ]; R r(v T ) V T (x 1 F (x) f(x) )df m (x) 0 iii) There exists some ~ su ciently close to 1 such that when 2 ( ~ ; 1], resale does not occur. Proof. Condition i): Proposition 5 shows that the optimal cuto is v = v (2) > V T. By transitivity, 20

21 v > v(). A winner s value has to be higher than v in equilibrium; thus, resale is implicitly banned. Condition ii): By transitivity, it su ces to show that v = v (2) > r. Since Rev(r 1 (v); v) reaches its maximum at v (2), it su ces to show that Rev(r 1(v); v) increases in v for v 2 [V T ; r ]. expression for Rev(r 1 (v); v) can be found at the beginning of the proof of Proposition 5 in the Appendix. We next discuss two cases: a) When v 2 [v(); r ], drev(r 1(v);v) dv > nf n 1 (v)f(v)f 1 F (v) f(v) vg(1 ) 0. b) When v 2 [V T ; v()), a su cient condition for drev(r 1(v);v) R r(vt ) V T (x 1 F (x) f(x) )df m (x) 0. Therefore, when condition ii) holds, drev(r 1(v);v) dv Condition iii) dv The > 0 when v minfv(); r g is > 0, implying that v (2) > r v(). When = 1, clearly the auctioneer should wait to gather all n + m bidders in the second stage; thus, the optimal cuto should be 1 in the initial auction. By continuity, when is su ciently large, resale does not occur on the equilibrium path either. When is small, the in ated part of B1 bidder s bids cannot make up for the auctioneer s loss from imposing a low r 1, which explains condition i). Condition ii) characterizes a more struggling situation. When is relatively large, it may not be rational to impose v v() because the marginal cost to forgo B1 bidders bid increments is also large. Therefore, the marginal bene t to wait for the second period has to increase to give the auctioneer more incentive. The second part of condition ii) adds to B2 bidders attractiveness because it partly represents the marginal revenue that the auctioneer can extract from the most competitive B2 bidder. Condition iii) is straightforward: when both parties are su ciently patient, the auctioneer has an incentive to set an extremely high r 1 to delay the sale. This r 1 induces a cuto su ciently close to 1 that leaves no room for resale when there is a winner. It remains an open question whether resale does not a ect the auctioneer s maximum revenue for any parameter values. The closest conclusion in the spirit of this conjecture was achieved by Ausubel and Cramton (1999) but with a xed set of bidders and complete information in resale. Garratt et al. (2009) also mention that controlling the reserve price can help dampen a speculator s resale incentive. An educated guess is that since the information about B2 bidders is symmetric among the auctioneer and B1 bidders and because the auctioneer has the rst-mover advantage in selecting the reserve price, resale does not increase her revenue in this private-value model. 21

22 In reality, there may be some exogenous shocks after the initial auction, the exact in uence of which cannot be fully predicted ex ante. For example, if the item auctioned is a house and the government releases a policy reducing mortgage rates after the initial auction, additional demand could expand the potential market for resale. This type of event may incentivize a winner who does not bene t from resale ex ante. Whether this kind of shock can bene t the auctioneer depends on her belief on it. 4 Discussions We make several assumptions, some of which can be relaxed. 1. Private-value To some extent, this is a usual assumption in the resale literature made for the sake of tractability 15. Real items normally have a common value part and a private value part. When values are a liated, a usual assumption is that the auctioneer has more accurate information about the veiled "common-value-part" of the item than do bidders. The reserve price and decision to delay sale (which would require a secret reserve price) would thus have a signaling e ect 16. Moreover, when considering optimal auctions in this environment, it is almost necessary for the auctioneer to know the true value of the common value part, as in Horstmann and LaCasse (1997), otherwise her option value of keeping the item also depends on her posterior beliefs. Distributional assumptions also have to be made to keep the analysis tractable. Even so, the signaling e ect alone makes it very di cult to determine the optimal public or secret reserve prices in general. 2. Deterministic n and m In reality, n and m could be random: the auctioneer may only have a rough estimate of the actual numbers of arrivals. If n cannot be observed throughout, as long as B1 bidders and the auctioneer have the same information about m s distribution, the analysis would still hold. If n can be observed after the initial auction, the auctioneer needs to determine the optimal contingent r 2 for any possible 15 For example, see Hafalir and Krishna (2008), Garratt et al. (2009), Pagnozzi (2009) and Xu et al. (2013). 16 For example, see Horstmann and LaCasse (1997) and Cai et al. (2007). 22