Resale in Second-Price Auctions with Costly Participation

Transcription

1 Resle in Second-Price Auctions with Costly Prticiption Gorkem Celik nd Okn Yilnky This Version: Jnury 22, 2015 Abstrct We study seled-bid second-price uctions with costly prticiption nd resle. Ech bidder chooses to prticipte in the uction if her vlution is higher thn her optimlly chosen prticiption cutoff. If resle is not llowed nd the bidder vlutions re drwn from strictly convex distribution function, the symmetric equilibrium (where ll bidders use the sme cutoff) is less effi cient thn clss of two-cutoff symmetric equilibri. Existence of these equilibri without resle is sufficient for existence of similrly constructed two-cutoff equilibri with resle. Moreover, these equilibri with resle re more symmetric nd (under suffi cient condition) more effi cient thn the corresponding equilibri without resle. JEL Clssifiction: C72, D44, D82 Keywords: Second-price uctions; resle; prticiption cost; endogenous entry; endogenous vlutions We thnk Bernrd Lebrun, Dn Levin, nd seminr prticipnts t Boğziçi University, Bilgi University, Bilkent University, ODTÜ, THEMA nd Istnbul Workshop in Economic Theory. Dong Ook Choi nd Nim Fzeli provided vluble reserch ssistnce. All remining errors re our own. Celik: Deprtment of Economics, ESSEC Business School nd THEMA Reserch Center, Cergy-Pontoise, Frnce; celik@essec.fr. Yilnky: Deprtment of Economics, Koç University, Istnbul, Turkey; oyilnky@ku.edu.tr.

2 1 Introduction We study resle in n independent privte vlues uction setting with costly prticiption, with prticulr focus on effi ciency. The seller uses seledbid second-price uction. Bidders re ex-nte symmetric: Their (use) vlues re drwn from the sme distribution function. After lerning their privte vlutions, bidders simultneously decide whether to prticipte in the uction or not. Bidders who choose to prticipte incur common rel resource cost. 1 2 In the bsence of resle opportunities, there is (unique) symmetric equilibrium of the second-price uction where ech bidder bids her vlution iff it is lrger thn prticiption cutoff tht is common to ll bidders. However, there my lso be symmetric equilibri with bidder-specific cutoffs. 3 We first show tht, when the vlutions re distributed ccording to strictly convex cumultive distribution function, there re symmetric equilibri which re exnte more effi cient thn the symmetric equilibrium. Existence of symmetric equilibri under strict convexity hs been estblished by Tn nd Yilnky (2006): For ny rbitrry prtition of the bidders into two groups, there exists n equilibrium where the bidders within group use the sme prticiption cutoff tht differs from the other group s cutoff. We complement this finding by showing tht these two-cutoff equilibri provide higher expected socil surplus (net of prticiption costs) thn does the symmetric equilibrium (Proposition 1). The relevnce of this result extends beyond second-price uctions, since Stegemn (1996) shows tht one of the equilibri of the second-price uction mximizes socil surplus within the clss of ll incentive-comptible lloction rules. 4 The second-price uction lloctes the object to the highest vlution bidder mong prticipnts in ll equilibri where prticipting bidders bid their vlues. Yet, when the equilibrium is symmetric, there is possibility tht 1 Purchsing bid documents, registering or pre-qulifying for the uction, being t the uction site, rrnging for finncing hed of time nd prepring bid (which is often detiled pln with documenttion, especilly in government procurement) re ll costly ctivities. 2 This set-up ws introduced by Smuelson (1985), nd studied by, mong others, Stegemn (1996), Cmpbell (1998), Tn nd Yilnky (2006, 2007) nd Celik nd Yilnky (2009). Also see Green nd Lffont (1984), where costs s well s vlutions re privte informtion. 3 See Stegemn (1996) for n exmple, nd Tn nd Yilnky (2006) for suffi cient nd necessry conditions for existence of symmetric equilibrium. 4 In Stegemn s (1996) exmple the symmetric equilibrium is more effi cient thn the symmetric one. Celik nd Yilnky (2009) provides chrcteriztion result for effi cient uctions. 1

3 non-prticipting bidder hs higher vlution thn the winner of the uction. This lloctive ineffi ciency implies tht there re potentil gins from further trde through resle. Hence we incorporte the possibility of resle (vi n optiml uction mximizing the reseller s revenue) nd study its impct on equilibrium behvior nd effi ciency. 5 Suppose tht there exists two-cutoff symmetric equilibrium of the secondprice uction without resle, where one group (ggressive prticipnts) hs low cutoff nd the other group hs higher one. We show tht there lso exists n equilibrium tht prtitions bidders the sme wy when resle is llowed. This resle equilibrium is more symmetric thn the corresponding no-resle equilibrium: The low cutoff decreses nd the high cutoff increses (Proposition 2). The prospect of reselling the good induces the low-cutoff bidders to enter even more ggressively nd the possibility of buying the object in the resle phse mkes the high-cutoff bidders even more hesitnt to enter. Moreover, there is overbidding by low-cutoff bidder types who hope to resell: They bid their djusted vlues (expected pyoffs inclusive of the resle phse), which re higher thn their use vlues. Fixing prticiption nd bidding behvior in the initil uction, resle enhnces effi ciency s the object is potentilly trnsferred to higher-vlue bidder. However, resle my lso ffect the equilibrium cutoffs nd bids. Nevertheless, provided tht suffi cient condition is stisfied, llowing resle improves ex-nte effi ciency: Whenever there is two-cutoff symmetric equilibrium without resle, the corresponding more symmetric equilibrium with resle yields higher socil surplus (Proposition 3). The effi ciency gins from resle re not solely the result of svings in prticiption costs. First, our sufficient condition is on the distribution of vlutions nd hence independent of the mgnitude of the prticiption cost. Second, llowing resle my ctully increse prticiption, nd hence totl prticiption cost incurred, s we show in our concluding remrks. Resle is commonly observed fter uctions in mny mrkets. There re few sources of gins from resle trde offered in the literture. New bidders or more informtion to existing bidders my rrive between the initil uction nd the resle stge (Bikchndni nd Hung, 1989; Hile, 1996 nd 2003; Bose nd Delts, 2006). Bidder symmetries my lso cuse ineffi ciencies in first-price uctions (Gupt nd Lebrun, 1999; Hflir nd Krishn, 2008; Cheng nd Tn, 2010; Lebrun, 2010; Virág, 2013). 6 5 We ssume tht there re no prticiption costs in the resle stge, nd discuss this ssumption in our concluding remrks. 6 Zheng (2002) studies the optiml uction with resle. Also see Lebrun (2012). 2

4 In second-price nd English uctions, even when resle is llowed, (use) vlue-bidding remins to be n equilibrium (see, for exmple, Hile, 1996). This equilibrium lloctes the object to the highest vlue bidder nd hence there is no resle on the pth of ply. However, bidding one s vlue is no longer wekly dominnt when resle is llowed. Grrtt nd Tröger (2006) identify lterntive equilibri where even specultor with no use vlue cn mke positive profits. Grrtt, Tröger, nd Zheng (2009) construct equilibri with designted bidder (potentil reseller) which cn then be used to support collusion (by rotting the designted bidder). 7 As we discussed bove, when there re prticiption costs, second-price uctions my hve symmetric equilibri in undominted strtegies, where ll prticipnts bid their vlues. We investigte the resle opportunities nturlly rising from these equilibri. In our equilibri, ll types of ll prticipnts bid their djusted vlues tht reflect potentil gins from the resle uction. The closest pper to ours is by Xu, Levin, nd Ye (2013), who study secondprice uctions with resle, where vlutions nd prticiption costs re both privte informtion. They show tht symmetric equilibrium exists nd is unique under some conditions. Prticipnts in the initil uction bid their djusted vlues described bove nd there is resle in equilibrium. Resle opportunities rise becuse of differences in prticiption costs: When low-cost bidder wins the object, she cn resell it to high-cost bidder with higher vlution (who did not prticipte in the initil uction). Further nlyticl results re diffi cult to obtin with two-dimensionl privte informtion. Their numericl nlysis suggests tht the effect of resle on effi ciency (nd on revenue) is mbiguous. In our model with commonly known prticiption costs, we show tht heterogeneity of costs is not necessry for equilibrium resle. Insted, resle opportunities re generted by symmetric equilibri. This setting lso llows us to obtin n nlyticl result on the impct of resle on effi ciency. In the next section, we describe the environment nd study the benchmrk cse, where resle is not llowed. We study resle in Section 3, nlyzing the optiml resle uction, nd the prticiption nd bidding behvior in the initil uction. We provide some concluding remrks in Section 4. All proofs re in the Appendix. 7 In Pgnozzi (2007), strong bidder my drop out of n scending-price uction before the price reches her vlue in order to improve her brgining position in the resle stge. 3

5 2 No Resle We consider symmetric independent privte vlues environment. There is risk-neutrl seller who owns n indivisible object nd is selling it vi seledbid second-price uction without reserve price. Her vlution is normlized to be 0. There re n 2 risk-neutrl (potentil) bidders. Let v i denote the (use) vlue of bidder i {1,..., n} for the object. Bidders vlutions re independently distributed ccording to the cumultive distribution function (cdf) F on [0, 1], with continuously differentible nd positive density function f. Bidders know their own vlutions. For simplicity, we ssume tht generlized version of the monotone hzrd rte condition is stisfied: For ny F (x) F (v) f(v) x (0, 1], is strictly decresing in v for ll v [0, x]. Note tht this condition utomticlly holds for convex distribution functions (for wekly incresing density functions f). There is prticiption cost, common to ll bidders, denoted by c (0, 1): Bidders must incur this rel resource cost in order to be ble to submit bid. 8 All bidders mke their prticiption nd bidding decisions simultneously. They know their vlutions when mking these decisions. 2.1 Equilibrium We first study the benchmrk cse where there is no resle possibility. If bidder decides to prticipte in the second-price uction, she cnnot do better thn bidding her own vlution. Accordingly, we only consider (Byesin- Nsh) equilibri in cutoff strtegies: Ech bidder bids her vlution if it is greter thn cutoff point nd does not prticipte otherwise. 9 However, there my be symmetric equilibri where bidders hve different prticiption cutoffs. In wht follows, we restrict ttention to equilibri with (t most) two cutoffs. Since our results re of the existence/possibility vriety, this restriction hs no bering on them, while simplifying the exposition considerbly. 10 Suppose tht l bidders use the low cutoff nd h = n l bidders use the high cutoff b in some equilibrium, with 0 b 1 nd 1 l n. These cutoffs re determined by indifference (to prticiption) conditions. To find 8 For our positive results bout equilibri in uctions with or without resle, c cn lso be interpreted s n entry fee (chrged by the seller). 9 We dopt the convention tht if bidder s cutoff is 0 (respectively, 1) ll (respectively, none of) her types re prticipting. The prticiption decisions of these zero mesure cutoff types re inconsequentil. 10 Tn nd Yilnky (2006) show tht, for ny c, log-concvity of F ( ) is suffi cient for the upper limit of distinct cutoffs to be two in ny (cutoff) equilibri. 4

6 them, first consider the prticiption decision of one of the l bidders who hs the lower cutoff nd whose vlution is lso. Suppose tht ll other bidders re following their equilibrium strtegies. She obtins the object iff she is the only bidder to prticipte, which hppens with probbility F () l 1 F (b) h, nd hence pys 0 if she wins. Her expected pyoff from prticiption is then F () l 1 F (b) h c. (1) Similrly, the expected pyoff of high-cutoff bidder with vlution b is given by F () l F (b) h 1 b + F (b) h 1 (b w) df (w) l c = F (b) h 1 [F () l + F (w) l dw] c. (2) Define the following functions: π L (, b) = F () l 1 F (b) h nd π H (, b) = F (b) h 1 [F () l + F (w)l dw]. For b to be equilibrium cutoffs, the following conditions must be stisfied: π L (, b ) c, with equlity if > 0, (3) π H (, b ) c, with equlity if b < 1. Any bidder with vlue lower thn c will hve negtive pyoff from prticiption. So, we know tht c > 0, nd the first condition will be stisfied with equlity. 11 We note the following observtion, which will be used lter: Remrk 1 π L (, b) nd π H (, b) re strictly incresing in nd b for > 0. ( π L (0, b) = 0 b nd π H (0, b) is strictly incresing in b.) Notice tht, since we llow for the possibility tht = b, the symmetric equilibrium is specil cse within the clss of (t most) two cutoff equilibri. There lwys exists symmetric equilibrium where ll bidders use the sme prticiption cutoff = b = v s (c, 1), where F (v s ) n 1 v s = c. (4) 11 We keep the conditions in their current forms to mke them directly comprble to the corresponding conditions in the resle cse, where bidder with vlution less thn c my prticipte in order to sell lter. 5

7 Figure 1: F (v) = v 2, n = 2, c = Symmetric equilibrium with v s = Asymmetric equilibrium with = 0.11, b = Figure 2: F (v) = v 2, n = 2, c = 0.4. Symmetric equilibrium with v s = Asymmetric equilibrium with = 0.4, b = 1. 6

8 Tn nd Yilnky (2006) show tht strict convexity of F is suffi cient for existence of two-cutoff symmetric equilibri for ny l, the number of bidders using the lower cutoff. It my be helpful to go over their rgument with grphs, which we will lso utilize for our results. Consider the set = {(, b) : 0 b 1} in R 2 tht identifies fesible prticiption cutoff pirs, nd the curves given by π L (, b) = c nd π H (, b) = c in. When F is strictly convex, the second curve is steeper thn the first one t (v s, v s ), the symmetric equilibrium cutoffs. If these curves intersect in the interior of, their intersection yields the cutoffs for n symmetric equilibrium, stisfying (3) with equlities (Figure 1). Otherwise, we hve corner symmetric equilibrium with [c, v s ) nd b = 1, where π L (, 1) = c nd π H (, 1) c, s depicted in Figure Effi ciency We cn write down the socil surplus s function of the two cutoff points in the no-resle setting: S (, b) = vf (b) h df (v) l + 1 b vdf (v) h+l l (1 F ()) c h (1 F (b)) c. (5) The first integrl mesures the expected vlue of the object for the winner of the uction when she is low cutoff bidder with vlue on intervl [, b], nd the second one is the expected vlue for winner with vlution higher thn b. The lst two terms re expected prticiption costs incurred by ll bidders. Note tht the seller s vlution is normlized to be 0 nd the pyment mde by the winning bidder is just trnsfer to the seller. The derivtives of this socil surplus function with respect to its two rguments cn be written by referring to functions π L nd π H tht we just defined: S (, b) S (, b) b = lf () [ π L (, b) c], (6) = hf (b) [ π H (, b) c]. These first order conditions imply tht the socil surplus is decresing in nd incresing in b for the set of points where π H (, b) c π L (, b), i.e., the lens-shped res in Figures 1 nd 2. Accordingly, when F is convex, socil surplus will be higher on the symmetric equilibri identified by Tn nd Yilnky (2006) in comprison to the symmetric equilibrium. 7

9 Proposition 1 If F is strictly convex, then, for ny l {1, 2,..., n 1}, there exists n symmetric no-resle equilibrium where l bidders use one cutoff nd h = n l bidders use nother one tht genertes higher socil surplus thn the symmetric equilibrium. This result holds regrdless of the mgnitude of the prticiption cost c, the number of bidders n, nd the wy bidders re clssified into low nd highcutoff groups. For given levels of c nd n, strict convexity of F in Proposition 1 cn be replced with the weker locl condition tht F (v) is strictly incresing v t the symmetric cutoff v s, defined in (4). This locl condition is ll tht is needed to generte the lens-shped res such s those in Figures 1 nd 2. Strict convexity implies tht F (v) is strictly incresing for ll v. Finlly, v note tht the results identifies t lest n 1 distinct (ignoring permuttions of bidder identities) symmetric equilibri of the second-price uction, nd ech of them hs higher surplus thn the symmetric equilibrium Resle Asymmetric equilibri of the second-price uction with prticiption costs, such s those we discussed bove, hve the following feture: Even though the object is obtined by the bidder who hs the highest vlution mong prticipnts, nonprticipnt my hve higher vlution. Therefore, when the winner of the uction hs vlution which is lower thn the prticiption cutoff of nother bidder, there re potentil gins from further trde. To investigte this issue, we now llow for resle stge where the winner of the initil uction cn design her own resle uction for potentil bidders. Timing is s follows: Bidders mke prticiption nd bidding decisions simultneously in the initil uction. The winner designs resle uction if she chooses to do so. Others mke their prticiption nd bidding decisions in this resle uction. We ssume tht the highest bidder lerns tht she is the winner nd does not lern others bids. We mke this no-disclosure ssumption only for not- 12 Since the second-price uction hs n (ex-nte) effi cient equilibrium in this setting (Stegemn, 1996), Proposition 1 implies tht the effi cient uction is symmetric for strictly convex distribution functions. There is connection between optiml (revenue-mximizing) nd effi cient uctions. Following Myerson (1981), define J (v) = v 1 F (v) f(v) s the virtul vlue of bidder with vlue v. In Celik nd Yilnky (2009), we showed tht, if F (v) J(v) is strictly incresing, then the optiml uction is symmetric, implying (using the connection we mentioned) tht the effi cient uction is symmetric when F (v) v 8 is strictly incresing.

10 tionl simplicity. Full-disclosure of ll bids (or ny other intermedite disclosure policy) would not ffect our equilibrium outcome. 13 We ssume tht there re no prticiption costs in the resle stge, nd discuss this ssumption in our concluding remrks. We look for (Perfect Byesin) equilibri of this gme where bidders re divided into two groups tht use two (potentilly distinct) prticiption cutoffs in the initil uction, just like before. Similrly, we restrict ttention to nturl equilibri in which prticipnts in the initil uction bid their djusted vlues (gross expected pyoff inclusive of the resle stge). 14 We nlyze the optiml resle uction first (given the restrictions bove), followed by bids nd equilibrium prticiption cutoffs in the initil uction. 3.1 Optiml resle uction Suppose tht l bidders use cutoff nd h = n l bidders use cutoff b in the initil uction, with 0 b 1 nd 1 l n. Suppose further tht bids re monotone incresing in vlutions nd tht bidders with identicl vlutions bid the sme mount (if they prticipte). In this equilibrium we re constructing, there re opportunities for resle only if bidder wins the initil uction with vlue between nd b. The bidders who re using the higher cutoff b in the initil uction re the potentil buyers in the resle stge. The winner of the initil uction (one of the l bidders who use s the cutoff nd who hs vlution in [, b]) hs lerned tht none of these high-cutoff bidders hve vlue higher thn b, otherwise they would hve prticipted in the initil uction nd cquired the good. Therefore, the problem she is fcing is finding the optiml uction for h 1 bidders whose vlutions re independently distributed on [0, b] ccording to the cdf F ( ) F (b) See Lebrun (2010b) for discussion of importnce of bid disclosure policies, especilly in first-price uctions with resle. 14 Note tht bidding this djusted vlue is no longer the dominnt strtegy even conditionl on prticipting, since this vlue is clculted using the equilibrium expected pyoff from the resle uction. However, given n equilibrium of the resle uction (which itself will be in dominnt strtegies for the resle stge), bidding the djusted vlue will indeed be dominnt conditionl on prticiption, s usul, explining our use of the word nturl. 15 The generlized monotone hzrd rte condition we ssumed ensures tht we re in the regulr cse with incresing virtul vlutions. Notice lso tht when h = 0 (or = b), we hve symmetric prticiption in the initil uction, nd hence there is no room for resle. 16 We re describing the resle stge only on the equilibrium pth (or rther when the initil uction behvior is described by two prticiption cutoffs nd monotone bid functions). We do not formlly define the full strtegies nd the beliefs t the resle stge to keep the exposition simple. There re mny resle-stge beliefs tht would support the equilibri we 9

11 If the reseller s vlution were commonly known, this would be the stndrd optiml uction problem à l Myerson (1981). However, it is not known, nd so we hve n informed principl problem. Fortuntely, it is possible to show tht this does not mtter in this independent privte vlues setting. It is optiml for ech type of the reseller in [, b] to choose stndrd optiml uction for tht type. 17 There re mny uctions which re expected-pyoff equivlent for the reseller nd the bidders (the revenue equivlence theorem), but we will focus on second-price uction with n optiml reserve price r(w) for the reseller with vlution w [, b], stisfying r = w + F (b) F (r). (7) f (r) The monotone hzrd rte condition implies tht, for ny b, there is unique vlue for r (w) (w, b]. Note tht we hve r (w) (0, 1) nd r (b) = b. A bidder with vlue v prticiptes in the resle uction if v r, nd bids v. It is strightforwrd to clculte the expected pyment she mkes to the reseller if she wins the resle uction: 20 α (v, r) = v v r F (x) h 1 dx. (8) h 1 F (v) 3.2 Equilibrium bids in the initil uction Now tht we discussed the optiml resle uction on the equilibrium pth, we re redy to study bidding in the initil uction. As we mentioned bove, we re describing, including the pssive one where the reseller does not updte when she sees n off-the-equilibrium-pth bid. 17 Yilnky (1999) shows this in the brgining context, i.e., when h = 1. The sme rgument pplies for h > 1 (Yilnky, 2004, vilble from the uthors upon request): The Myerson uction is optiml when the seller s vlution is common knowledge. It is lso the seller s ex-nte optiml mechnism. Myerson s principle of inscrutbility (1983) implies tht it will be the informed principl s optiml mechnism. Also see the discussion in Mylovnov nd Tröger (2014) nd Mskin nd Tirole (1990). 18 On the equilibrium pth, of course, it does not mtter which of the optiml uctions is used in the resle stge. However, it my mtter when considering potentil devitions in the initil uction. 19 We suppress the dependence of r on b for nottionl simplicity. 20 This expression my be fmilir s the equilibrium bid function in first price uction with h bidders whose vlutions re distributed ccording to F ( ) F (b) on [0, b]. Revenue equivlence theorem implies tht this is the expected pyment of the winner in the second-price uction. 10

12 look for n equilibrium in which bids re given by gross expected pyoffs, tking the resle stge into ccount. When the winner is bidder with vlution higher thn the high cutoff b, there is no room for resle nd the winner s pyoff is equl to her (use) vlue. Therefore such bidder would bid her vlue in the initil uction. On the other hnd, when the winner is low-cutoff bidder with vlution v [, b], her gross pyoff is equl to the expected continution pyoff in the resle stge. In the Appendix (Lemm 1), we show F (r(x)) tht this continution pyoff is b h dx, where r (x) is the optiml F (b) h v reserve price for bidder with vlution x, by using the revenue equivlence theorem. Accordingly, the equilibrium bids of the low-cutoff bidders in the initil uction re β (v) = b v No if v < F (r(x)) h F (b) h dx if v < b v if b v, (9) where No denotes not prticipting. Since high-cutoff bidders prticiption precludes resle stge, their bids will reflect only their use vlues: ˆβ (v) = { No if v < b v if b v. (10) The equilibrium bid of low-cutoff bidder is higher thn her use vlue when it is in [, b). Bidders with such vlutions re wre of the possibility tht they cn resell the good to high-cutoff bidder who hs higher use vlue. Since this possibility is decresing in the vlution of the low-cutoff bidder, the extent of overbidding is decresing in v (nd it is eliminted for v b). Bidding functions β ( ) nd ˆβ ( ) imply tht the initil uction is ex-post effi cient in constrined sense: It lloctes the good to the bidder with the highest vlution mong prticipnts. Any ineffi ciency in the initil uction (therefore, ny resle opportunity) is due to possible symmetric prticiption behvior, which we discuss next. 3.3 Equilibrium prticiption in the initil uction A bidder s prticiption decision in the initil uction will depend on the comprison of the prticiption cost with the pyoff differentil generted by her prticiption, tking into ccount the resle stge. We show in the Appendix (Lemm 2) tht, for ech bidder, this pyoff differentil is (wekly) incresing 11

13 in her vlution when other bidders re using cutoff strtegies. Therefore, it is suffi cient to consider prticiption incentives for bidders whose vlutions re equl their respective cutoffs. Consider one of the l low-cutoff bidders who hs vlution equl to her cutoff. This bidder cnnot buy the object in resle uction since ll the equilibrium reserve prices will be higher thn. When she enters in the initil uction, she would be the winner if she is the only prticipnt. Given the other bidders prticiption decisions, the probbility of this hppening is F () l 1 F (b) h. This sole prticipnt does not mke ny pyment, nd her expected pyoff in the continution gme is β (). Therefore, her pyoff differentil for prticiption in the initil uction is π L (, b) = F () l 1 F (b) h β (). (11) Now consider one of the h high-cutoff bidders who hs vlution equl to her cutoff b. If she prticiptes (nd bids b), then her expected pyoff will be F (b) h 1 [F () l b + [b β (w)] df (w) l ]. (12) To see this, notice tht she wins only if none of the h 1 high-cutoff bidders prticiptes nd ll the l low-cutoff bidders hve vlutions less thn b. She pys 0 if none of the low-cutoff bidders prticiptes (when ll of them hve vlutions less thn ). Otherwise she pys the bid of the highest-vlution low-cutoff bidder, β (w). 21 On the other hnd, if she stys out, then her expected pyoff will be F (b) h 1 [b α (b, r (w))] df (w) l, (13) since resle uction occurs if none of the h 1 other high-cutoff bidders prticiptes in the initil uction nd the highest vlution mong low-cutoff bidders is in [, b]. The bidder who hs this highest vlution w sets the reserve price r(w), so the expected pyment of type-b bidder is α (b, r (w)) (see (8)). 21 There will not be resle stge if she prticiptes, since resle hppens only if lowcutoff bidder wins the object with vlution in [, b], nd these bidder types bid less thn b, i.e., β (w) < b for w < b. 12

14 is Therefore, the pyoff differentil for high-cutoff bidder with vlution b π H (, b) = F (b) h 1 [F () l b + We prove the following in the Appendix. [α (b, r (w)) β (w)] df (w) l ]. (14) Remrk 2 π L (, b) is incresing in nd b for > 0, nd π H (, b) is incresing in. We re finlly redy to identify the conditions tht equilibrium cutoffs nd b must stisfy, nlogous to conditions (3) in the no-resle setting: π L (, b ) c, with equlity if > 0, (15) π H (, b ) c, with equlity if b < 1. These conditions dmit symmetric solution, with = b = v s, the symmetric equilibrium cutoff of the benchmrk no-resle cse. There is no resle in this equilibrium, since the bidder with the highest vlution receives the object. Our next result is bout the existence nd properties of equilibri with symmetric cutoffs, where resle is n equilibrium phenomenon. Whenever there is n symmetric equilibrium (with two cutoffs) in the benchmrk noresle cse, there will lso be n symmetric equilibrium with resle. Moreover, the equilibrium with resle will be more symmetric. Proposition 2 Suppose tht there exists n symmetric equilibrium in the benchmrk cse of no-resle with l {1, 2,..., n 1} bidders using the cutoff nd h = n l bidders using the cutoff b >. Then there exists n symmetric equilibrium with resle, where l bidders use cutoff nd h bidders use cutoff b >. Moreover, < nd b b (with strict inequlity if b < 1). A key step in our proof is showing tht π L (, b) > π L (, b) nd π H (, b) < π H (, b) for b > > 0. These inequlities imply tht the symmetric noresle equilibrium cutoffs (, b ) lie below the curve given by π H (, b) = c nd bove the curve π L (, b) = c. The reminder of the rgument is very similr to our discussion of symmetric equilibri in the no-resle benchmrk. If these two curves intersect in the interior of = {(, b) : 0 b 1}, their intersection yields the cutoffs for n symmetric equilibrium with resle, 13

15 Figure 3: F (v) = v 2, n = 2, c = Symmetric equilibrium with v s = Asymmetric equilibrium with = 0, b = Figure 4: F (v) = v 2, n = 2, c = 0.4. Symmetric equilibrium with v s = Asymmetric equilibrium with = 0.044, b = 1. 14

16 stisfying (15) with equlities. Otherwise, we hve corner equilibrium either with = 0 (Figure 3) or with b = 1 (Figure 4). The resulting equilibrium cutoffs (, b ) re more symmetric thn the corresponding no-resle equilibrium cutoffs (, b ) in the sense tht they re further wy from the symmetric equilibrium cutoff v s. In n equilibrium such s the one described bove, low-cutoff bidder prticiptes nd bids more ggressively in the initil uction due to the opportunity to resell to high-cutoff bidder. This opportunity in turn is supported by some types of the high-cutoff bidder remining out of the initil uction to buy lter in the resle stge. This symmetry in behvior rises s n equilibrium phenomenon even though bidders re ex-nte symmetric, s it is the cse in the no-resle benchmrk. A similr specultive motive lso ppers in the symmetric equilibrium of Xu, Levin nd Ye s (2013) model with two possible (privtely-known) prticiption costs: Bidders with high cost tend to sty out of the initil uction nd the low-cost bidders over-enter nd over-bid with the hope of reselling the object. With Proposition 2 we show tht this specultive motive nd resle cn rise in equilibrium even when ll bidders hve the sme prticiption cost. 3.4 Effi ciency with resle To exmine the welfre effects of resle, we consider the socil surplus s function of two prticiption cutoffs. This surplus function is constructed under the ssumption tht, once the bidders enter in or sty out of the initil uction ccording to these prticiption cutoffs, they follow the equilibrium bidding nd resle strtegies described bove. [ ] S (, b) = F (r (w)) h w + vdf (v) h df (w) l (16) + 1 b r(w) wdf (w) h+l l (1 F ()) c h (1 F (b)) c. The first integrl term refers to the expected surplus if the initil uction lloctes the good to low-cutoff bidder with vlution in [, b]. This expected surplus is clculted by tking the possibility of resle into ccount. The second integrl term is the expected surplus when the highest vlution mong ll bidders is higher thn cutoff b. The lst two terms mesure the expected cost of prticiption. 15

17 For fixed cutoffs, the possibility of resle increses totl welfre: S (, b) > S (, b) for < b, since S (, b) S (, b) = r(w) (v w) df (v) h df (w) l. (17) This difference is simply the surplus gin of trnsferring the object from low-cutoff bidder with vlue w to high-cutoff bidder with higher vlue v in the resle phse. Consider n equilibrium in the benchmrk cse of no-resle with symmetric cutoffs nd b. As we just observed, if bidders were to use the sme cutoffs when resle is llowed, then the surplus would be higher, i.e., S (, b ) > S (, b ). However, the possibility of resle my lso chnge equilibrium prticiption behvior of the bidders. Therefore, we need to know how the vlue of function S (, b) chnges s we move from the no-resle equilibrium cutoffs (, b ) to resle equilibrium cutoffs (, b ). With our next result, we provide suffi cient condition for the socil surplus to increse when resle is llowed. Proposition 3 Suppose tht there exists two-cutoff symmetric equilibrium in the benchmrk cse of no-resle nd tht vf(v) is wekly incresing. Then F (v) there exists n symmetric equilibrium with resle which genertes higher socil surplus thn does this symmetric no-resle equilibrium. To prove the proposition, we first show tht the socil surplus function S (, b) is decresing in nd incresing in b whenever π H (, b) < c < π L (, b), if vf(v) is wekly incresing. The result then follows from inequlities F (v) < nd b b. Proposition 3 provides suffi cient condition on the distribution of vlutions only. Hence, it is independent of the mgnitude of the prticiption cost c, the number of bidders n, nd how these bidders re divided into two groups. The condition is stisfied when the cdf for the vlutions is power function, i.e., F (v) = v α for α > 0 (since vf(v) = α). F (v) 4 Concluding Remrks We study resle nd show tht it cn be n equilibrium phenomenon in symmetric second-price uction with costly prticiption. The equilibrium with resle is more symmetric thn the corresponding one without resle due to specultive motives. When resle is not llowed, we identify symmetric 16

18 equilibri tht re more effi cient thn the symmetric one if the cdf of bidders vlutions is strictly convex. We provide suffi cient condition for resle to improve (ex-nte) effi ciency. Therefore, when F is strictly convex nd this suffi cient condition is stisfied, we hve rnking: Symmetric equilibrium (resle llowed or not) is less effi cient thn the symmetric no-resle equilibri we identified, which in turn re less effi cient thn the corresponding symmetric resle equilibri. We ssume tht there re no prticiption costs t the resle stge. One possible justifiction is tht the reseller my follow bidder qulifiction procedure which is less stringent thn tht of the originl seller, e.g., due to the originl seller being public entity (see Xu, Levin, nd Ye, 2013, who lso ssume costless resle). However, the min reson for our ssumption is to keep the nlysis simple. When there re prticiption costs, the optiml uction t the resle stge would be more complicted thn stndrd uction; in prticulr, it could be symmetric (Celik nd Yilnky, 2009). These complictions side, we expect the resle equilibri nd their min structure to remin intct in the presence of costly resle. There is mbiguity for the effect of resle on the expected number of prticipnts, nd hence on the expected prticiption cost incurred. Consider the exmples depicted in Figures 3 nd 4, where n = 2 nd F (v) = v 2. When c = 0.01, no-resle symmetric equilibrium cutoffs re (, b ) = (0.11, 0, 301), while with resle equilibrium cutoffs re (, b ) = (0, 0.454), with decrese in prticiption. However, resle increses prticiption when c = 0.4. Equilibrium cutoffs chnge from (0.4, 1) to (0.044, 1): The high-cutoff bidder does not prticipte in either cse while the low-cutoff bidder is more likely to prticipte when there is resle. This exmple lso demonstrtes tht the effi ciency gins from resle re not only due to svings in prticiption costs. The effect of resle on expected revenue is lso uncler. Opportunity to resell the object induces higher prticiption nd higher bids by some of the bidders, leding to positive impct on revenue. Other bidders, however, would prticipte less since they might hve the option to buy the object lter, identifying counterviling fctor. Therefore, the net effect of resle on revenue is mbiguous, s cn be seen in the following clsses of exmples: In the symmetric no-resle equilibrium we construct, if some bidders never prticipte in the uction regrdless of their vlues nd t lest two other bidders prticipte with positive probbility (i.e., if b = 1 nd l > 1), revenue would be higher in the corresponding equilibrium with resle (since the prticipting bidders will enter with higher probbility nd bid more). On the other hnd, llowing for resle would eliminte ll revenue if the symmetric equilibrium with resle hs only one bidder prticipting with positive probbility (i.e., 17

19 if b = 1 nd l = 1). Finlly, in our model, bidders mke their prticiption nd bidding decisions simultneously. Another possibility is bidders mking their prticiption decisions first nd then bidding fter hving observed the number of prticipnts (e.g., Xu, Levin, nd Ye, 2013). This lterntive scenrio (or even observing the identities of prticipnts) would not chnge our results, since in equilibrium ll bidder types bid their djusted vlues inclusive of the pyoff from resle stge. 5 Appendix Proof of Proposition 1 (no-resle) Following the proof of Proposition 3i in Tn nd Yilnky (2006), for ll b [v s, 1], define Φ (b) implicitly s the vlue of which solves π L (, b) = c. Notice tht function Φ (b) is continuously differentible, strictly decresing, nd tht Φ (v s ) = v s. For ll b [v s, 1], lso define g (b) = π H (Φ (b), b) c. This function is lso continuously differentible with g (v s ) = 0. According to the no-resle equilibrium conditions (3), the two equilibrium cutoffs re identified s b [v s, 1] nd = Φ (b ) (0, v s ] such tht g (b ) 0, with equlity if b < 1. This confirms the existence of the symmetric equilibrium t = b = v s. Tn nd Yilnky (2006) show tht when F is strictly convex, function g ( ) is decresing round v s. This implies tht the equilibrium conditions re stisfied for t lest one pir of symmetric cutoffs: Either there exists b (v s, 1) such tht g(b ) = 0 (thus there exists n symmetric equilibrium with b s the high cutoff nd Φ (b ) s the low cutoff), or g (1) 0 (thus there exists n symmetric equilibrium with 1 s the high cutoffnd Φ (1) s the low cutoff). Now consider the smllest vlue of b tht stisfies these equilibrium conditions: b = min {b > v s : g (b) 0, with equlity if b < 1}. To see tht surplus is higher under cutoffs b nd Φ ( b ), write S ( Φ ( b ), b ) S (v s, v s ) s 18

20 = = ( S (Φ (b), b) v s b + S (Φ (b), b) Φ (b))db ( hf (b) [ π H (Φ (b), b) c] lf () [ π v s }{{} L (Φ (b), b) c] Φ (b))db }{{} g(b) 0 v s hf (b) g (b) db. By definition of b, the integrnd is positive for ll b ( v s, b ), proving tht S ( Φ ( b ), b ) > S (v s, v s ). Lemm 1 Consider low-cutoff bidder with vlue w [, b] who won the initil uction. Her expected pyoff from the resle phse is β (w) = b w F (r (x)) h F (b) h dx. (18) Proof Consider the stndrd optiml uction problem where the seller s vlue is w [, b], there re h 1 bidders whose vlutions re independently distributed on [0, b] ccording to cdf F ( ), where 0 b 1. Note tht F (b) (18) is just the expected pyoff of the seller, obtined from the stndrd formultion (see Myerson (1981)) by using chnge of vribles to incorporte the reserve price. Here we give heuristic rgument s well: The revenue equivlence theorem implies tht the continution pyoff for the winner of the initil uction is continuous function of her vlution nd its derivtive t ech vlution is equl to the probbility tht the bidder will keep the good t the end of the resle phse. Bidders with v b will not resell the object if they win the initil uction. Accordingly, the continution pyoff of bidder with vlution b is equl to b. A reseller with vlue w [, b], who sets her reserve price optimlly to r(w), does not sell the object if ll h bidders hve vlutions less thn r(w), which hppens with probbility F (r(w))h. Accordingly, F (b) h her continution pyoff is b w F (r(x)) h dx. F (b) h Lemm 2 Suppose tht ll bidders except one re following their equilibrium strtegies. For the remining bidder, the pyoff differentil between prticiption in the initil uction nd stying out of it is wekly incresing in her vlution. 19

21 Proof It follows from the revenue equivlence theorem tht bidder s non-prticiption pyoff is continuous in her vlution nd its derivtive is equl to the probbility tht this bidder will cquire the uctioned object during the resle phse. Her prticiption pyoff is continuous in her vlution s well nd its derivtive is equl to the probbility tht she receives the good nd keeps it fter the end of the initil uction nd the resle phse. To conclude tht the pyoff differentil is wekly incresing in vlution, it will be suffi cient to show tht probbility of cquiring the good is t lest s lrge for ll bidder vlutions if the bidder were to prticipte in the initil uction. Suppose tht the bidder s vlution is v. If this bidder stys out of the initil uction, she will cquire the object only when the vlutions of ll the high-cutoff bidders (excluding the bidder in question, in cse tht she is high-cutoff bidder) re lower thn v nd the vlution of the highest low-cutoff bidder (excluding the bidder in question) is between nd r 1 (v). Now notice tht, when the vlutions of the other bidders stisfy this condition, the sme bidder would hve cquired the object by prticipting in the initil uction (either by overbidding the highest low-cutoff bidder or t the resle phse) s well. This proves tht entering in the initil uction does not decrese the probbility of cquiring the object for ny vlution. Proof of Remrk 2 i) π L (, b) is incresing in nd b for > 0. Using (9) nd (11), we hve π L (, b) = F () l 1 F (b) h β() = F () l 1 F (b) h [b F (r (x)) h dx]. F (b) h Let > 0. π L (, b) is incresing in, since β() is incresing in. π L (, b) b = hf () l 1 [F (b) h 1 f (b) b = hf () l 1 f (b) [F (b) h 1 b F (r (x)) h 1 f (r (x)) F (r (x)) h 1 r (x) dx] r (x) dx] b hf () l 1 f (b) [F (b) h 1 b F (b) h 1 dx] = hf () l 1 f (b) F (b) h 1 > 0, 20

22 where the second equlity follows from f (r (x)) r(x) = f (b) r (x) (using the b implicit function theorem for (7)), nd the inequlity follows from F (b) F (r (x)) nd r (x) (0, 1). ii) π H (, b) is incresing in. From (14), π H (, b) = lf () l 1 f () [F (b) h 1 b α(b, r()) + β()]. Using (8), (9), nd chnge of vribles (x r(x)), π H (, b) = lf () l 1 f () [F (b) h 1 b 1 F (r (x)) h 1 (F (r (x)) F (b) r (x))dx] F (b) lf () l 1 f () [F (b) h 1 b F (b) h 1 dx] = lf () l 1 f () F (b) h 1 > 0, where the inequlity follows from F (b) F (r (x)) nd r (x) (0, 1). Proof of Proposition 2 (Existence of Asymmetric Equilibri with Resle) The proof will use the following lemm. Lemm 3 If 0 < < b then π L (, b) > π L (, b) nd π H (, b) < π H (, b). Proof Recll tht ((1),(2),(11),(14)) π L (, b) = F () l 1 F (b) h π L (, b) = F () l 1 F (b) h β () π H (, b) = F (b) h 1 [F () l b + π H (, b) = F (b) h 1 [F () l b + (b w) df (w) l ] [α (b, r (w)) β (w)] df (w) l ]. Let 0 < < b. The first inequlity follows from β () >. The second inequlity follows from b α (b, r (w)) nd β (w) > w for w [, b). 21

23 Proof of Proposition 2. For ll b [v s, 1], define Φ (b) = { : πl (, b) = c if π L (0, b) < c 0 otherwise. Notice tht Φ (b) is continuously differentible, strictly decresing whenever it tkes positive vlues, nd tht Φ (v s ) = v s. For ll b [v s, 1], lso define g (b) = π H (Φ (b), b) c. This lst function is lso continuously differentible with g (v s ) = 0. The resle equilibrium conditions re stisfied (with b s the high cutoff nd Φ (b) s the low cutoff) if nd only if g (b) 0, with equlity if b < 1. Recll tht π L (, b ) = c. Since function π L (, b) is incresing in nd is lrger thn π L (, b) for 0 < < b, it must be tht Φ (b ) <. Consider g (b ) = π H (Φ (b ), b ) c < π H (, b ) c < π H (, b ) c 0. The first inequlity follows from monotonicity of π H in its first rgument, nd the second one from π H (, b) < π H (, b) for 0 < < b. Finlly, g (b ) < 0 implies the existence of b b (with strict inequlity if b < 1) such tht g (b ) 0 (with equlity if b < 1). Accordingly, there exists n equilibrium where h bidders use cutoff b nd l bidders use cutoff = Φ (b ). To complete the proof, notice tht Φ (b ) <. Proof of Proposition 3 (Welfre under Resle) The proof of the proposition will follow from these lemms: Lemm 4 S(,b) Proof S (, b) lf () [π L (, b) c]. = [F (r ()) h + r() = lf () [F (r ()) h F () l 1 + vdf (v) h ]lf () l 1 f () + lf () c r() F () l 1 vdf (v) h c]. Reclling tht π L (, b) = F () l 1 F (b) h β () = F () l 1 F (b) h [b w we need to show F () l 1 [F (b) h b F (r ()) h F (r (x)) h dx] F () l 1 22 r() F (r(x)) h dx], F (b) h vdf (v) h, or

24 F () l 1 xdf (r (x)) h F () l 1 r (x) df (r (x)) h, where the left hnd side is obtined by using integrtion by prts nd the right hnd side chnge of vribles. The inequlity holds since x < r (x) for ll x < b. Lemm 5 If vf(v) F (v) Proof is wekly incresing in v, then S(,b) b hf (b) [π H (, b) c]. Step 1: If vf(v) F (v) is wekly incresing in v, then F (r (w)) F (b) r (w) 0 for ll w < b, nd the inequlity is strict if w > 0. Totl differentition of (7) revels tht r (w) = 1, 2 + [F (b) F (r)] f (r) f(r) 2 where r equls the optiml reserve price for vlution w. So we need to show F (b) F (r (w)) r (w) = 2F (r) + [F (b) F (r)] f (r) f (r) 2 F (r) F (b) F (r) F (r) + [F (b) F (r)] f (r) f (r) 2 F (r), which is identicl to [1 f (r) F (r) f (r) 2 ] [F (b) F (r)] F (r) [ f (r)2 f (r) F (r) ] [r w] F (r). f (r) The lst line follows from (7). Recll tht r w 0. If f (r) 2 f (r) F (r), then the left hnd side is t most zero nd the inequlity is stisfied. Otherwise, showing [ f (r)2 f (r) F (r) ]r F (r) f (r) is suffi cient for the proof. This lst inequlity is identicl to F (r) f (r) + f (r) F (r) r f (r) 2 r 0. The left hnd side is equl to the numertor of the derivtive of rf(r) with respect to r. The denomintor of the derivtive is F (r) F (r) 2 > 0. Therefore if vf(v) is wekly incresing in v for ll vlues lower thn F (v) b, then F (r (w)) F (b) r (w) 0, concluding the proof of this first step. 23

25 Step 2: If F (r (w)) F (b) r (w) 0 for ll w [0, b], then S(,b) b hf (b) [π H (, b) c]. S (, b) b = hbf (b) h+l 1 f (b) + hf (b) c [ ] b hbf (b) h 1 f (b) + hwf (r (w)) h 1 f (r (w)) dr(w) + db hr (w) F (r (w)) h 1 f (r (w)) dr(w) df (w) l = hf (b) [F (b) h 1 F () l b c db (r (w) w) F (r (w)) h 1 r (w) df (w) l ]. where the lst equlity follows from f (r (x)) r(x) b implicit function theorem for (7)). Reclling tht = f (b) r (x) (using the π H (, b) = F () l F (b) h 1 b + F (b) h 1 we need to show [α (b, r (w)) β (w)] df (w) l = F () l F (b) h 1 b + 1 F (r (x)) h 1 [F (r (x)) F (b) r (x)] dxdf (w) l, F (b) w F () l F (b) h 1 b + 1 F (b) F (b) h 1 F () l b + w F (r (x)) h 1 [F (r (x)) F (b) r (x)] dxdf (w) l (r (w) w) F (r (w)) h 1 r (w) df (w) l. The first terms on either side of the inequlity cncel out. Since F (r (x)) F (b) r (x) is non-negtive for x b nd F (r (x)) h 1 F (r (w)) h 1 for x w, it is suffi cient to show 1 b F (b) F (r (w)) h 1 w [F (r (x)) F (b) r (x)] dxdf (w) l (r (w) w) F (r (w)) h 1 r (w) df (w) l. 24

26 F (r (x)) F (b) r (x) cn be written s [F (r (x)) F (b)] r (x) + F (x) [1 r (x)] [F (r (x)) F (b)] = f (r (x)) r (x) + F (r (x)) [1 r (x)] f (r (x)) = [x r (x)] f (r (x)) r (x) + F (r (x)) [1 r (x)] d [x r (x)] F (r (x)) =, dx where the second equlity follows from (7). Therefore w [F (r (x)) F (b) r (x)] = [r (w) w] F (r (w)). So showing the below inequlity would be suffi cient for the result: 1 b F (r (w)) h (r (w) w) df (w) l F (b) which cn be rewritten s (r (w) w) F (r (w)) h 1 r (w) df (w) l, [F (r (w)) F (b) r (w)] F (r (w)) h 1 (r (w) w) df (w) l 0. This inequlity holds since ll terms in the integrnd re positive for ll w (, b) under the hypothesis of the lemm. Proof of Proposition 3. We know from the proof of the Proposition 2 tht g (b ) < 0 nd there exists b which stisfies the resle equilibrium conditions with Φ (b). Now consider the smllest such vlue of b: b = min {b b : g (b) 0, with equlity if b < 1}. To see tht surplus is higher under cutoffs b nd Φ (b ), write S (Φ (b ), b ) S (, b ) s Φ(b ) S (, b ) b ( ) S (Φ (b), b) S (Φ (b), b) d + + Φ (b) db b b + Φ(b ) b lf () [π L (, b ) c] d + lf () [π L (Φ (b), b) c] Φ (b) db. b hf (b) [π H (Φ (b), b) c] db The inequlity bove follows from the previous two lemms nd tht Φ (b) 0. Moreover, π L (, b ) > π L (Φ (b ), b ) = c for ll (Φ (b ), ], estblishing 25

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