Optimal Two-stage Auctions with Costly Information Acquisition

Transcription

1 Optmal Two-stage Auctons wth Costly Informaton Acquston Jngfeng Lu Lxn Ye December 2014 Abstract We consder an aucton envronment wth costly entry wheren the cost manly stems from nformaton acquston. Bdders are endowed wth orgnal estmates ( types ) about ther prvate values and can further learn ther true values of the object for sale by ncurrng an entry cost. We frst derve an ntegral form of the envelope formula as requred by ncentve compatble two-stage mechansms, based on whch we demonstrate that optmalty of the generalzed Myerson allocaton rule s robust to our settng wth costly nformaton acquston. Optmal entry s thus to admt the set of bdders that maxmzes expected vrtual surplus adjusted by both the second-stage sgnal and entry cost. We show that our optmal entry and allocaton rules are both IR and IC mplementable, and furthermore, n an mportant class of envronments, they can be mplemented va a two-stage aucton that s essentally a handcap aucton augmented wth an entry mechansm. Keywords: Two-stage auctons, entry, nformaton acquston, sequental screenng, handcap auctons, optmal mechansms. JEL Classfcaton: D44, D80, D82. 1 INTRODUCTION In hgh-valued asset sales, buyers often need to go through a due dlgence process before developng fnal bds. Due dlgence s usually a process to update or acqure nformaton about the value of the asset for sale or to prepare for the bddng process (e.g., to establsh qualfcatons to bd). Ths process s costly and s usually modeled as entry as t s closely montored by the auctoneer. For a sale of an asset worth bllons of dollars, the entry cost can run from tens of thousands to mllons of dollars. 1 We thank semnar partcpants at Unversty of Mchgan, the Conference n Honor of Paul Mlgrom s 65th Brthday, the Mdwest Economc Theory Conference, NSF/CEME Decentralzaton Conference, and n partcular, Drk Bergemann, Tlman Börgers, Yeon-Koo Che, Jeff Ely, L Hao, Preston McAfee, Davd Mller, Ilya Segal, Xanwen Sh and Juuso Tokka for very helpful comments and suggestons. All remanng errors are our own. Department of Economcs, Natonal Unversty of Sngapore, 10 Kent Rdge Crescent, Sngapore Tel: (65) , Emal: ecsljf@nus.edu.sg. Department of Economcs, The Oho State Unversty, 449A Arps Hall, 1945 North Hgh Street, Columbus, OH Tel.: (614) Emal: ye.45@osu.edu. 1 A more detaled descrpton of a typcal due dlgence process s provded n Secton 4. 1

2 Gven the substantal entry cost, t s unrealstc to assume that whoever s nterested would necessarly go through the costly entry process. The success of a sale thus very much reles on whether the most qualfed bdders would commt to the due dlgence process and partcpate n the fnal sale. Manly motvated by the need for entry screenng, varants of two-stage sellng mechansms have emerged n the real world. A leadng example of the two-stage aucton procedure s known as ndcatve bddng, whch s commonly used n sales of complcated busness assets wth very hgh values. It works as follows: the auctoneer actvely markets the assets to a large group of potentally nterested buyers. The potental buyers are then asked to submt non-bndng bds, based on whch a fnal set of bdders s shortlsted to advance to the second stage. The auctoneer then communcates only wth these fnal bdders, provdng them wth extensve access to nformaton about the assets, 2 and fnally runs the aucton (typcally usng bndng sealed bds). The use of ths two-stage aucton procedure s qute wdespread. For example, n response to the restructurng of the electrc power ndustry n the U.S. whch was desgned to separate power generaton from transmsson and dstrbuton bllons of dollars of electrcal generatng assets were dvested through ths two-stage aucton procedure over the last two decades. 3 Ths two-stage aucton procedure s also commonly used n prvatzaton, takeover, and merger and acquston contests. 4 Fnally, t s commonly used n the nsttutonal real estate market, whch has an annual sales volume n the order of $60 to $100 bllon. 5 Ye (2007) was the frst study of ndcatve bddng based on the assumpton of costly nformaton acquston. 6 Ye s analyss suggests that the current desgn of ndcatve bddng cannot relably select the most qualfed bdders for the fnal sale, as there does not exst a symmetrc, strctly ncreasng equlbrum bd functon n the ndcatve bddng stage. In a more recent paper, by restrctng ndcatve bds to a fnte dscrete doman, Qunt and Hendrcks (2013) show that a symmetrc equlbrum exsts n weakly-monotone strateges. But agan, the hghest-value bdders are not always selected, as bdder types pool over a fnte number of bds. Wthout safely selectng the most qualfed bdders for the fnal sale, the mechansm s hardly optmal n maxmzng expected revenue. What the optmal mechansm s n ths two-stage aucton envronment remans an open queston n the lterature, and ths paper seeks to provde an answer. We model the two-stage aucton envronment as follows. Before entry, each potental bdder s endowed wth a prvate sgnal, α, whch can be regarded as her pre-entry type. After entry (by ncurrng a common entry cost, c), each bdder fully observes her (prvate) value v, whch s postvely correlated to her pre-entry type. Gven costly entry, t s not feasble for all potental bdders to be ncluded n the fnal sale. As such, we consder the class of two-stage mechansms wth the frst stage allocatng entry 2 Data rooms, whch are descrbed n Secton 4, are typcally set up to facltate bdders due dlgence process. 3 A lst of ndustry examples usng ths two-stage aucton desgn can be found n Ye (2007). 4 Leadng examples nclude the prvatzaton of the Italan Ol and Energy Corporaton (ENI), the acquston of Ireland s largest cable televson provder Cablelnk Lmted, and the takeover contest for South Korea s second largest conglomerate Daewoo Motors. 5 See Foley (2003) for a detaled account. 6 Boone and Goeree (2009) provde an analyss of pre-qualfyng auctons, whch are smlar to ndcatve bddng. 2

3 rghts and the second stage allocatng the asset. 7 Despte the potental complcaton due to both sequental screenng and costly nformaton acquston, we are able to completely characterze the optmal revenue-maxmzng two-stage mechansms. Our analyss benefts greatly from recent developments n the lterature of sequental screenng (e.g., Courty and L, 2000; Esö and Szentes, 2007; Bergemann and Wambach, 2014; and Pavan, Segal, and Tokka, 2014). 8 In partcular, our analyss follows Esö and Szentes closely, and our techncal contrbuton s to extend ther analyss to dynamc auctons wth costly nformaton acquston. We frst derve an ntegral form of the envelope formula as a necessary condton for ncentve compatblty for our twostage mechansms, whch extends the valdty of the envelope theorem to dynamc auctons wth costly nformaton acquston. Based on ths derved envelope formula, we are able to show that the optmal allocaton rule of the asset n our second stage s the same as that dentfed by Esö and Szentes, whch requres that, among the shortlsted bdders, the asset be allocated to the bdder wth the hghest vrtual value adjusted by the second-stage sgnal. Our analyss thus suggests that the optmalty of the generalzed Myerson optmal allocaton rule (adjusted by second-round sgnals) s robust to the dynamc aucton settng wth costly entry. The frst-stage entry rght allocaton mechansm s new to the orgnal Esö-Szentes framework, and we show that the optmal entry rule s to admt the set of bdders that gves rse to the maxmum expected vrtual surplus (adjusted by both the second-stage sgnal and entry cost). Alternatvely, gven the regularty assumpton and that buyers are ex ante symmetrc n our model, the optmal entry rule s to admt the bdders n descendng order of ther pre-entry types, the hghest type frst, the second hghest type second, etc., provded that ther margnal contrbuton to the expected vrtual surplus s postve. Therefore, the optmal number of shortlsted bdders typcally depends on the reported type profle from the potental bdders, whch s endogenously determned. We then show that specfc payment rules can be constructed n each stage to mplement both optmal entry and allocaton rules truthfully. For an mportant settng where one s value s lnear n her frst sgnal, Esö and Szentes show that ther optmal mechansm can be mplemented over two rounds va a so-called handcap aucton: n the frst round (before observng the second-stage sgnal), each buyer selects a prce premum by payng a fee accordng to a pre-announced schedule. In the second round (after observng the second-stage sgnals), buyers compete n a second-prce or Englsh aucton, where the wnner obtans the object at a prce equal to the second-hghest bd plus the prce premum selected from the frst round. In our settng wth entry, the mplementaton s presumably more complcated, as optmal entry needs to be mplemented pror to the fnal aucton. Indeed, now the mplementaton requres that an (optmal) entry rule be augmented to the handcap aucton. So n our case the optmal mechansm s mplemented va a two-stage aucton, wth the frst stage beng an aucton for entry rghts (as well as the prce prema) and the second stage 7 The focus on two-stage mechansms should be regarded as a constrant, whch s dscussed n Secton 4. 8 Early work on dynamc contractng wth a sngle agent are due to Baron and Besanko (1984) and Rordan and Sappngton (1987). 3

4 beng a second-prce or Englsh aucton for the asset. Other than the connecton wth sequental screenng and dynamc auctons mentoned above, our paper s related to the lterature on nformaton acquston n auctons (see, for example, Persco, 2000; Compte and Jehel, 2001; and Rezende, 2013). These papers focus on bdders ncentves to acqure nformaton n dfferent aucton formats. Our paper dffers from thers n that we follow the normatve approach to dentfy optmal mechansms wth nformaton acquston. To the extent that nformaton acquston s modeled as entry, our paper s closely related to the growng lterature on auctons wth costly entry. 9 Ths lterature can be summarzed nto three branches. In the frst branch, bdders are assumed to possess no prvate nformaton before entry and they learn ther prvate values or sgnals only after entry (see, for example, McAfee and McMllan, 1987; Engelbrecht-Wggans, 1993; Tan, 1992; Levn and Smth, 1994; and Ye, 2004). In the second branch, t s assumed that bdders are endowed wth prvate nformaton about ther values but have to ncur entry costs to partcpate n an aucton (see, for example, Samuelson, 1985; Stegeman, 1996; Campbell, 1998; Menezes and Montero, 2000; Tan and Ylankaya, 2006; Cao and Tan, 2009; and Lu, 2009). Fnally, n the thrd branch, bdders are endowed wth some prvate nformaton before entry, and are able to acqure addtonal prvate nformaton after entry (Ye, 2007; Qunt and Hendrcks, 2013). The framework n ths current paper nests all the models mentoned above as specal cases. Our paper thus characterzes optmal mechansms for a very general framework n the lterature on auctons wth costly entry. Our research s also related to a small lterature on auctons of entry rghts. In a poneerng work, Fullerton and McAfee (1999) ntroduce auctons for entry rghts to shortlst contestants for a fnal tournament. Ye (2007) extends ther approach to the settng of two-stage auctons descrbed above. Our current approach dffers from thers n the way the set of fnalsts s determned: whle n ther approach the number of fnalsts to be selected s fxed and pre-announced, n our entry rght allocaton mechansm the selecton of shortlsted bdders s contngent on the reported bd profle, makng the number of fnalsts endogenously determned. For ths reason the entry rght allocaton mechansm examned n ths research s more general. 10 In another relevant paper, Lu and Ye (2013) explore optmal two-stage mechansms n an envronment where bdders are characterzed by heterogenous and prvate nformaton acquston costs before entry. In that settng the pre-entry type s the entry cost, whch s nether correlated to nor part of the value of the asset for sale. As such, there s no beneft to make the second-stage mechansm contngent on the reports of the pre-entry types, resultng n a much smpler characterzaton of optmal mechansms. The settng n ths current paper s dfferent, as the pre-entry type s correlated to the value of the asset, hence there are potental gans to make the second-stage mechansm contngent on frst-stage reports. Indeed, n our current settng, the optmal allocaton and payment rules n the second stage do depend on the frst-stage reports. Therefore the characterzaton of optmal mechansms s more demandng n 9 See Bergemann and Välmäk (2006) for a thoughtful survey of ths lterature. 10 In fact, t resemble mult-unt auctons wth endogenously determned supply (see, e.g., McAdams, 2007). 4

5 ths work, and the mplementaton of the optmal mechansm s also more sophstcated. The rest of the paper s organzed as follows. Secton 2 presents the model. Secton 3 characterzes the optmal mechansm and ts aucton mplementaton. Secton 4 dscusses some restrctons n our analyss. Secton 5 concludes. 2 THE MODEL The nformaton structure n our model s closest to that n Esö and Szentes (2007). The man dfferences are that n Esö and Szentes, the addtonal nformaton s controlled by the seller, and they focus on the seller s ncentve to dsclose (wthout observng) addtonal sgnals to the buyers. In our settng, however, t s costly for the bdders to acqure addtonal nformaton, and we focus on the bdders ncentve for nformaton acquston (entry). In addton, all buyers are ncluded n the fnal sale n Esö and Szentes, but due to costly entry n our settng, not all buyers wll be wllng to partcpate n the fnal aucton. As such, we wll addtonally consder entry mechansms whch s the major dfference from the analyss n Esö and Szentes. Formally, a sngle ndvsble asset s offered for sale to N potentally nterested buyers. The seller and bdders are assumed to be rsk neutral. The seller s own valuaton for the asset s normalzed to 0. Buyer s true valuaton for the asset s v. However, ntally she only observes a nosy sgnal of t, α, whch s her prvate nformaton and can be nterpreted as her orgnal type. After ncurrng a common nformaton acquston cost (or entry cost) of c, bdder fully observes her ex post value, v. The pars, v ) are assumed to be ndependent across. 11 Ex ante, α follows dstrbuton F( ) wth ts assocated densty f ( ) on support α,α]. 12 We assume that f s postve on the nterval α,α] and satsfes the monotone hazard rate condton; that s, f /(1 F) s weakly ncreasng. Gven α, the ex post value v follows dstrbuton H α H( α ) wth ts densty h α h( α ) over support v,v] R. 13 The values N and c and dstrbutons F and H α are all common knowledge. Followng the sgnal orthogonalzaton technque ntroduced by Esö and Szentes (2007), 14 there exst functons u and s, such that u, s ) v, where u s strctly ncreasng n both arguments, and s s ndependent of α. In partcular, s can be constructed as follows: s = H(v α ), 11 As n Esö and Szentes (2007) and Pavan, Segal, and Tokka (2014), ths assumpton rules out the possblty of full rent extracton (Crémer and McLean, 1988). 12 Esö and Szentes allow α s to be drawn from dfferent dstrbutons. Our procedure can be extended to accommodate asymmetrc dstrbutons for α s. For ease of characterzng our optmal entry rght allocaton rule, we assume that α s are drawn from a common dstrbuton, so that bdders are ex ante symmetrc. Note that wth dfferent realzatons of α s, bdder heterogenety before entry s stll captured n our model. 13 Followng the dynamc mechansm desgn lterature, we assume that the support of v s ndependent of the frst-stage sgnal α. 14 The use of ths technque becomes standard n the lterature (see, e.g., Pavan, Segal, and Tokka, 2014, and Bergemann and Wambach, 2014). 5

6 whch s the percentle of the value realzaton to bdder. 15 valuaton can be computed as v = H 1 α (s ) u, s ). Thus gven type α and sgnal s, the We wll denote the c.d.f. of s by G. 16 We mantan the followng assumptons that are adopted n Esö and Szentes (2007): Assumpton 1. ( H α (v)/ α)/h α (v) s ncreasng n v. Assumpton 2. ( H α (v)/ α)/h α (v) s ncreasng n α. Esö and Szentes show that Assumpton 1 s equvalent to u 12 0 and Assumpton 2 s equvalent to u 11 /u 1 u 12 /u 2. Assumpton 1 thus states that the margnal mpact of the new nformaton on buyer s value s decreasng n her type α. Assumpton 2 mples that an ncrease n α, holdng u, s ) constant, weakly decreases the margnal value of α. Assumptons 1 and 2 can thus be nterpreted as a knd of substtutablty n buyer s posteror valuaton between α and s. Example 1. (Ye, 2007): Each potental bdder s endowed wth a prvate value component α before entry; after entry, each buyer learns another prvate value component s, where s s ndependent of α. The ex post value u, s )=α + s. By the lnearty of u, s ), Assumptons 1 and 2 hold. Example 2. (Adapted from Esö and Szentes, 2007): v s drawn from a normal dstrbuton wth mean µ and precson (nverse varance) τ 0. The pre-entry type, α, s normally dstrbuted wth mean v and precson τ v. After entry, the buyer can observe her true value, v. It s clear that v and α are strctly afflated. The dstrbuton of α, whch s normal, satsfes the hazard rate condton. The cdf of v condtonal on α, H α, s normal wth mean (τ 0 µ+τ v α )/(τ 0 + τ v ) and precson τ 0 + τ v. Defne s = H α (v ) and let u, s )= H 1 α (s ) v. Obvously u s strctly ncreasng n s. It can be verfed that u 1, s )=τ v /(τ 0 + τ v ), whch s a constant. Therefore, u s lnear and strctly ncreasng n α. Hence Assumptons 1 and 2 hold. Snce nformaton acquston s modeled as entry n our settng, we consder a mechansm desgn framework n whch the seller exercses entry control. In addton, we restrct our analyss to two-stage mechansms: the frst stage s the entry rght allocaton mechansm, and the second stage s the prvate good provson mechansm. Note that n ths mechansm desgn framework, the second-stage mechansm can be made contngent on the frst-stage reports. Followng Myerson (1986) and Pavan, Segal, and Tokka (2014), we restrct to drect mechansms where agents report ther types truthfully at each stage on the equlbrum path. We assume that the frst-stage reported profle α s revealed to all admtted bdders so that the frst-stage entry allocaton 15 It s easly seen that s s unformly dstrbuted over 0,1], and s hence statstcally ndependent of the ntal nformaton α. 16 G could be assumed to be unform on 0,1]. More generally, all s s satsfyng u,s ) v are postve monotonc transformaton of each other (Lemma 1 n Esö and Szentes). 6

7 and payments are mmedately verfable. 17 Ths revelaton polcy turns out to be optmal, n the sense that no other revelaton polcy (e.g., not revealng or partally revealng α) can generate hgher expected revenue to the seller. For ths reason, our restrcton to fully reveal α s wthout loss of generalty n our search for optmal mechansms. A detaled dscusson s relegated to Secton 4. As n Pavan, Segal, and Tokka (2014), the revelaton polcy concerned n ths paper s about the frst-stage nformaton and outcome, ncludng the agents frst-stage reports, ther payments, and the agents beng shortlsted. In our paper, the prncpal has no control over the ways n whch the second-stage new nformaton s revealed to bdders. A shortlsted bdder wll be fully nformed about her true value v after ncurrng the entry cost. As such, we are not concerned about the dscrmnatory nformaton dsclosure ssue studed n L and Sh (2013). As n Esö and Szentes, we can focus attenton on equvalent drect mechansms that requre bdders to report s s, rather than v s. Note that reportng,v ) s equvalent to reportng, s = H α (v )). Let N=1,2,..., N} denote the set of all the potental buyers and 2 N denote the collecton of all the subsets (subgroups) of N, ncludng the empty set, φ. The frst-stage mechansm s characterzed by the shortlstng rule A g ) and payment rule x ), = 1,2,..., N. Gven the reported profle α, the shortlstng rule, A g : α,α ] N 0,1], assgns a probablty to each subgroup g 2 N, where g 2 N A g ) = 1. The payment rule x : α,α ] N R, specfes bdder s frst-stage payment gven the reported profle α. 18 Gven the frst-stage reported profle α, the second-stage mechansm s characterzed by p g,sg ), the probablty that the asset s allocated to buyer g, and t g,sg ), the payment to the seller made by buyer g, g 2 N. 3 THE ANALYSIS We start wth the second stage. Suppose group g s shortlsted, and the profle α reported n the frst stage s revealed as publc nformaton to the shortlsted bdders. Frst, suppose α s truthfully reported at the frst stage and group g s shortlsted. Assume that they follow the recommendaton and ncur the nformaton acquston cost c to dscover s g. 19 Gven the announced α and s, defne the nterm wnnng probablty and expected payment rule as P g, s )= E g s p g,sg ) and T g, s )=E g s t g,sg ), where s g = sg \s }, g and g 2 N. Then bdder s second-stage nterm expected payoff when she observes s but reports ŝ s as follows: π g ;s, ŝ )=E g s u, s )p g, ŝ,s g ) tg, ŝ,s g )]= u, s )P g, ŝ ) T g, ŝ ). 17 In Esö and Szentes, there s no such need for nterm verfcaton, as ther allocaton and payment rules are executed at the end of the mechansm. 18 Note that our shortlstng rule A g ) s more general than specfyng each bdder s probablty of beng shortlsted gven ther reported type profle α. 19 As wll be shown, the equlbrum expected proft from gong forward s postve for a buyer upon entry, so n equlbrum, a bdder does have an ncentve to follow the recommendaton to acqure (costly) nformaton and partcpate n the fnal aucton once admtted (as droppng out only results n zero proft). 7

8 The second-stage ncentve compatblty (IC) condtons requre that π g ;s, ŝ ) π g ;s, s ), g,α,s, ŝ. (1) Frst, the followng lemma s standard n the tradtonal screenng lterature: Lemma 1. Suppose α s truthfully revealed from the frst stage and P g, s ), g, s contnuous and weakly ncreasng n s where g denotes the group beng shortlsted, then the second-stage ncentve compatblty condton (1) holds f and only f s π g ;s, s )= π g ;ŝ, ŝ )+ ŝ u 2,τ)P g,τ)dτ, s >ŝ, g. (2) (2) s an ntegral form of the envelope formula. Next, we consder the case when ˆα nstead of α s reported by bdder whle others report ther types truthfully. As demonstrated n Esö and Szentes (2007), whenever a bdder had msrepresented her type n the frst stage, she would correct her le n the second stage. Formally n our settng, suppose α s truthfully revealed from the frst stage and the second-stage mechansm s ncentve-compatble gven a truthfully revealed α. Then buyer of type α who reported ˆα n the frst round wll report ŝ = σ, ˆα, s ) f she observes s n the second stage such that 20 u, s )= u( ˆα,σ, ˆα, s )). (3) Reportng ŝ after a le ˆα s equvalent to revealng v truthfully regardless of the frst-stage report. The optmalty of ths strategy has been establshed n general for the Markov envronments by Pavan, Segal and Tokka (2014). Our two-stage settng resembles the Markov envronment defned n Pavan, Segal, and Tokka snce the agents payoffs only depend on ther second-stage true types (v s) and the allocaton outcome, but not on ther frst-stage true types. For ths reason, an agent s reportng ncentve n the second stage depends only on her current type and her frst-stage report, but not on her frst-stage true type. Snce t s optmal for the agent to report her value truthfully when the past report has been truthful, t s also optmal for her to report her value truthfully even f she has led n the frst stage. Note that ŝ does not depend on α, g, or s g. Defne π g, ˆα ;s, ŝ ) = E g s u, s )p g, ˆα, ŝ,s g ) tg, ˆα, ŝ,s g )] = u, s )P g, ˆα, ŝ ) T g, ˆα, ŝ ); π g, ˆα ;α ) = E s π g, ˆα ;s, ŝ = σ, ˆα, s )). π g, ˆα ;α ) s the expected second-stage payoff for the type-α bdder f she reported ˆα n the frst 20 The exstence of σ (,, ) reles on the assumpton that the support of v does not depend on the frst-stage sgnal α. 8

9 stage (and everyone else reported truthfully). Parallel to Lemma 5 n Esö and Szentes, we can show the followng lemma: Lemma 2. Suppose α s truthfully revealed from the frst stage and the second-stage mechansm s ncentve-compatble gven a truthfully revealed α. If buyer of type α who reported ˆα n the frst stage s shortlsted n group g, her expected payoff from the second stage s gven by α π g, ˆα ;α )= π g ( ˆα, ˆα ;α )+ ˆα u 1 (y, s )P g ( ˆα,α,σ (y, ˆα, s ))d ydg (s ). (4) Throughout, g wll be used to denote the group ncludng bdder. (4) should agan be regarded as an ntegral form of the envelope formula: the wnnng probablty (P g ) s now obtaned when evaluatng at ŝ = σ (y, ˆα, s ) (whch s optmal gven the frst-round le ˆα ). We are now ready to consder the frst-stage IC mechansm. Let π, ˆα ) be the expected payoff (net of the entry cost) for a type-α bdder who reports ˆα n the frst stage. By (3), we have π, ˆα ) = E α g A g ( ˆα,α ) π g, ˆα ;α ) c] x ( ˆα,α ) } (5) = E α g A g ( ˆα,α ) E s ( u, s )P g, ˆα, ŝ ) T g, ˆα, ŝ ) ) c ]} x ( ˆα ), where ŝ = σ, ˆα, s ) and x ( ˆα )= E α x ( ˆα,α ). The followng lemma characterzes the bdder s expected payoff n an IC two-stage mechansm wth costly entry. Lemma 3. If the two-stage mechansm s ncentve compatble and E α A g,α )P g,α, s ) s contnuous n α then buyer s expected payoff (as a functon of her pre-entry type) can be expressed as π,α )=π,α)+ α α u 1 (y, s ) Eα A g (y,α )P g (y,α, s ) ] dg (s )d y. (6) g Proof. See Appendx. Note that g Eα A g (y,α )P g (y,α, s ) ] s buyer s equlbrum probablty of eventually wnnng the asset wth sgnals (y, s ) n our settng. Thus (6) s also an ntegral form of the envelope formula. Under a set of regular condtons, Pavan, Segal, and Tokka (2014) show that the envelope formula contnues to hold n the dynamc mechansm desgn settng. Lemma 3 can be regarded as an extenson of ther result to a dynamc mechansm desgn settng wth costly nformaton acquston. 9

10 3.1 The Optmal Two-stage Mechansms We are now ready to derve the seller s expected payoff from an IC two-stage mechansm. By Lemma 3, we have Eπ,α ) = π,α)+ α α α α u 1 (y, s ) Eα A g (y,α )P g (y,α, s ) ] dg (s )d ydf ) g u 1, s ) α 1 F ) = π,α)+ Eα A g,α )P g α f ),α, s ) ] dg (s )df ) g = π,α)+ E α A g 1 F ) )]},α ) u 1, s )P g g f ),α, s )dg (s. The second equalty above s due to Fubn s Theorem. Thus ]} N N Eπ,α )=,α)+ E α A =1 =1π g )E s p g,sg ) 1 F ) u 1, s ). g g f ) The total expected surplus from the two-stage mechansm s ]} TS=E α A g )E s p g,sg )u, s ) g c. g The seller s expected revenue s thus gven by g ER = TS g N Eπ,α ) =1 ( = E α A g )E s p g,sg ) u, s ) 1 F ) ]} ) N u 1, s ) g c π,α), (7) g f ) =1 where A g ) s the shortlstng rule and p g,sg ) s the second-stage allocaton rule. To maxmze ER subject to IC and IR (ndvdual ratonalty), the seller sets π,α)=0 for all = 1,2,..., N;.e., no rent should be gven to the buyer wth the lowest possble (pre-entry) type. Defne the vrtual value adjusted by the second-stage sgnal as follows: w, s )= u, s ) 1 F ) u 1, s ). (8) f ) From the expresson of the expected revenue, we can derve the optmal allocaton rules n both stages 10

11 as follows. At the second stage, gven the revealed α and the shortlsted group g, s g, 21,s g 1 f = argmax j g w j, s j )} and w, s ) 0 )= g, g. (9) 0 otherwse p g So as also dentfed by Esö and Szentes, the asset should be awarded to the bdder wth the hghest non-negatve vrtual value adjusted by the second-stage sgnal, whch s a generalzaton of the optmal allocaton rule n Myerson (1981). Our analyss shows that the generalzed Myerson allocaton rule s robust to settngs wth costly entry. By Lemma 3, a buyer s expected payoff does not depend on the entry cost, whch mples that the seller bears all the entry costs (ndrectly) n equlbrum. As such, costly entry wll affect the fnal allocaton only through ts effect on the entry rght allocaton rule. Defne the expected vrtual surplus (the vrtual value less the entry cost) as follows: w g )= E s g p g,s g )w, s ) g c ]. Then at the frst stage, contngent on the revealed α, the optmal shortlstng rule s as follows: 22 A g 1 f g= argmax g w g )} and w g ) 0 )= g. (10) 0 otherwse The optmal shortlstng rule admts the set of bdders that gves rse to the maxmal expected vrtual surplus. Alternatvely, the optmal shortlstng rule admts the bdders n descendng order of ther margnal contrbuton to the expected vrtual surplus the bdder wth the hghest contrbuton frst, the bdder wth the second-hghest contrbuton second, etc. provded that ther margnal contrbuton s postve. Smlarly to Esö and Szentes, followng Assumptons 1 and 2, we can establsh the followng propertes of the optmal second-stage allocaton rule: 23 Corollary 1. () p g P g,s g ) ncreases n both α and s, g, g, α, and s g, whch mples that,α, s ) ncreases n both α and s, g, α ; () If α > ˆα, s < ŝ and u, s )= u( ˆα, ŝ ), then p g,α, s,s g ) p g ( ˆα,α, ŝ,s g ), whch mples P g,α, s ) P g ( ˆα,α, ŝ ), g, α. Property () above suggests that whenever α > ˆα, s < ŝ and u, s )= u( ˆα, ŝ ), the optmal allocaton rule favors the truth-tellng par, s ). Gven α, let s ) be defned such that w, s ))=0. To dentfy propertes of the optmal shortlst- 21 Tes occur wth probablty zero and are hence gnored. 22 Agan tes occur wth probablty zero and are hence gnored. 23 Assumpton 2 s used to show property (). 11

12 ng rule, we defne a truncated random varable as follows: w + w, s ) f w, s ) 0 or equvalently s s ), s )=. 0 otherwse Note that condtonal on α, w + s are ndependent across g. Let S g ;α ) denote buyer s margnal contrbuton to the expected vrtual surplus, g, then S g ;α )= S g ) S g ), g, αg, where α g = αg \α } and The followng two propertes are obvous: S g )=E s g max g w+, s )}, g, α g. (1) S g ;α ) ncreases wth α, and decreases wth α j, j, g, g. (2) S g ;α ) S g ;α ), α, g, g g. The revenue-optmal shortlstng rule can be alternatvely descrbed as follows. For gven α, we can rank all α from the hghest to the lowest. The seller admts bdders one by one n descendng order of α s as long as the bdder s margnal contrbuton to the expected vrtual surplus s nonnegatve,.e. S g ;α ) c= S g ) S g ) c 0, where g denotes the group of bdders wth the hghest g types before entry. Two propertes follow mmedately from the optmal shortlstng rule A g : Corollary 2. () Gven α, f bdder wth α s shortlsted, then she would also be shortlsted wth a hgher type α (> α ); () Suppose s shortlsted gven α, then bdder would reman beng shortlsted as long as α s hgher than a threshold ˆα ). As α ncreases, the shortlsted group weakly shrnks. As α ncreases from ˆα ), the bdders n g )\} would be excluded one by one (wth the lowest type orgnally shortlsted beng excluded frst). We are now ready to show that the optmal fnal good allocaton and entry rght allocaton rules (9) and (10) are truthfully mplementable by some well constructed payment rules n both stages. Theorem 1. Under Assumptons 1 and 2, the optmal fnal good allocaton and entry rght allocaton rules (9) and (10) are IR and IC mplementable. Proof. u, s ) ncreases wth s and by Assumpton 1, u 1, s ) (weakly) decreases wth s. Ths mples that w, s ) ncreases wth s. By the fnal good allocaton rule (9), the wnnng probablty P g, s ) s weakly ncreasng n s. By Lemma 1, the second-stage mechansm s ncentve compatble (gven α and g). Thus, gven the truthfully revealed α and shortlsted group g, a second-stage payment rule, 12

13 p g say, t g,s g ), g, g, can be constructed to truthfully mplement the second-stage allocaton rule,s g ), g, g whle mantanng the second-stage IR constrants (to partcpate n the secondstage mechansm),.e. π g,α ;s, s ) 0 on equlbrum path. Ths resembles the Myerson (1981) settng wth asymmetrc bdders. We use π g, ˆα ;α ) to denote the second-stage expected payoff to buyer of type α f she announces ˆα and s shortlsted n group g, gven that everyone else announces α truthfully at the frst stage. π g, ˆα ;α ) s well defned gven Lemma 2. Therefore, when buyer of type α announces ˆα whle others reveal α truthfully, her frst-stage expected payoff can be wrtten as follows: π, ˆα )=E α g A g ( ˆα,α ) π g, ˆα ;α ) c] x ( ˆα,α ) }, where x s the frst-stage payment rule. Next, we wll show that the optmal shortlstng rule (10) s truthfully mplementable by a properly chosen frst-stage payment rule x, together wth the second-stage payment rules t g chosen above. Note that by (5), we have π,α )=E α g A g,α ) π g,α ;α ) c] x,α ) }. (11) Construct the frst-stage payment rule as follows: x ) = A g,α ) π g,α ;α ) c] g α u 1 (y, s ) Eα A g (y,α )P g (y,α, s ) ] dg (s )d y (12) α g Substtutng (12) nto (11), we can verfy that π,α )= α α u 1 (y, s ) Eα A g (y,α )P g (y,α, s ) ] dg (s )d y, g whch s precsely equaton (6) wth π,α)=0 (the optmalty requrement). Note that π,α ) 0, so IR s satsfed n the frst stage. Suppose that all buyers except report ther types α truthfully. Consder buyer wth α contemplatng to msreport to ˆα < α. The devaton payoff s =π, ˆα ) π,α )=π, ˆα ) π ( ˆα, ˆα )]+π ( ˆα, ˆα ) π,α )]. 13

14 Snce (6) s satsfed by the constructon of x ), we have π ( ˆα, ˆα ) π,α )= α ˆα u 1 (y, s ) Eα A g (y,α )P g (y,α, s ) ] dg (s )d y. g Recall the defntons of π, ˆα ) above, we have from Lemma 2 that π, ˆα ) π ( ˆα, ˆα )= Therefore, we have = α + ˆα α α ˆα E α A g (y,α ) g ˆα E α u 1 (y, s ) Eα A g ( ˆα,α )P g ( ˆα,α,σ (y, ˆα, s )) ] dg (s )d y. g u 1 (y, s )P g g A g ( ˆα,α ) A g (y,α )] ( ˆα,α,σ (y, ˆα, s )) P g (y,α, s )]dg (s )d y u 1 (y, s )P g ( ˆα,α,σ (y, ˆα, s ))dg (s )d y. (13) From Corollary 1 (), we have P g ( ˆα,α,σ (y, ˆα, s )) P g (y,α, s ) 0, whch mples that the frst term n s negatve. We now consder the second term n when y > ˆα. By Corollary 2, the optmal shortlstng rule mples that gven α, when buyer s admtted wth a hgher α, she must be admtted to a group wth a weakly smaller sze. If y and ˆα are admtted n the same group, then A g ( ˆα,α )= A g (y,α ) and ths term n s zero. We now turn to the case where g ( ˆα,α ) g (y,α ) }. Note that A g (,α ) s 1 for the shortlsted group, and 0 for all other groups. Therefore, A g ( ˆα,α ) A g (y,α )]u 1 (y, s )P g g = u 1 (y, s )P g ( ˆα,α ) 0, ( ˆα,α,σ (y, ˆα, s )) ( ˆα,α,σ (y, ˆα, s )) P g (y,α ) ( ˆα,α,σ (y, ˆα, s ))] whch mples that the second term n s negatve. Snce g ( ˆα,α ) g (y,α ) }, we must have P g ( ˆα,α ) ( ˆα,α,σ (y, ˆα, s )) P g (y,α ) ( ˆα,α,σ (y, ˆα, s )),.e. entrant wns wth a smaller probablty f a strctly bgger group s shortlsted. A smlar argument can be used to rule out devaton to ˆα > α. It s worth notng that Assumptons 1 and 2 are suffcent but not necessary for the optmal entry rule to be truthfully mplementable: the necessary and suffcent condton s that defned n (13) s non-postve, whch s also the ntegral monotoncty condton characterzed by Pavan, Segal, and Tokka (2014). Example 3. Assumptons 1 and 2 are not necessary for Theorem 1 to hold. One can verfy that for the 14

15 case wth H α (v)= v α, where v 0,1], Assumpton 1 fals, but Corollary 1 holds. In ths case, v=s 1/α u, s). Thus ] ] w, s)= s 1/α 1 F) 1 1 F) 1 1+ f ) α logs1/α = u, s) 1+ logu, s). f ) α When there s only one potental bdder, the second-stage allocaton rule can be mplemented va a take-t-or-leave t offer wth a prce P) = s) 1/α, where s) s defned by w, s)) = 0. Note that } s)=exp α 2 f ), whch decreases wth α. Thus P ) < 0. Defne w) = E s max0,w, s)}. The 1 F) optmal shortlstng rule s gven by A)=1 f and only f α α, where w )=0. We next derve the frst-stage payment rule. Note that 1 ] π, ˆα)= A( ˆα) max0, s 1/α P( ˆα)}ds c x( ˆα), α, ˆα α. 0 The FOC requres that A)=1. We thus have π, ˆα) ˆα ˆα=α = 0. Takng ˆα>α α, we have s( ˆα)<s) s ). Note A( ˆα)= π, ˆα) π,α)= s) s( ˆα) Snce s) s( ˆα) s1/α P( ˆα)]ds/ ˆα ˆα=α = 0, we have 1 s 1/α P( ˆα)]ds+ P) P( ˆα)]ds+x) x( ˆα)]. s) π, ˆα) ˆα ˆα=α= P )(1 s)) x )=0. Thus x )= P )(1 s)), together wth boundary condton x )= 1 0 max0, s1/α P )}ds c, jontly determnes the payment functon x) for the frst stage. 3.2 Implementaton of Optmal Mechansms When u, s ) s lnear n α,.e., when u, s )= u 1 α + r(s ) for some constant u 1 and functon r, we wll demonstrate that the optmal mechansm can be mplemented va a two-stage aucton, wth the frst stage beng an aucton for both entry rghts and prce prema and the second stage beng a second-prce or Englsh aucton for the fnal good. Ths two-stage aucton can be regarded as a handcap aucton ntroduced n Esö and Szentes, augmented by an addtonal aucton at the entry stage. 24 More specfcally, our two-stage aucton works as follows. The frst stage s an all-pay aucton, where bdders need to pay what they bd, regardless of beng awarded entry rghts or not. Suppose buyer, knowng her type α, bds an amount b, = 1,2,..., N. After all the frst-stage bds are collected, underlyng types wll be recovered based on a recovery functon, x 1, such that buyer s perceved type α s x 1 (b ), = 1,2,..., N. Gven the recovered type profle α } N =1, the entry rghts are mplemented 24 The assumpton that u 1 s constant s satsfed n both Examples 1 and 2. 15

16 accordng to the optmal entry rule (10), and a prce premum s determned for each shortlsted buyer accordng to the followng premum schedule: p( α ) = u 1 (1 F( α ))/f ( α ). Both the recovery functon x 1 and the premum determnaton rule p are made publc at the outset of the game, whch reman common knowledge throughout the aucton process. Upon beng admtted, each entrant bdder wll ncur the nformaton acquston cost and partcpate n the second-round bddng. The second stage s a tradtonal second-prce or Englsh aucton wth a zero reserve prce, but the wnner s requred to pay her premum over the prce. 25 Ths mechansm s referred to as the handcap aucton n Esö and Szentes, snce the buyers compete under unequal condtons: a bdder wth a smaller premum has an advantage. In our settng, the handcap aucton s modfed so that the optmal entry rule s also mplemented after the frst-round bddng. In Esö and Szentes, buyers pay fees regardless of wnnng the fnal good or not; n our settng, buyers pay b s regardless of beng admtted to the fnal sale or not. For ths reason, the frst-stage aucton s a varant of the all-pay aucton. In the second-stage aucton, t s a (weakly) domnant strategy for entrant bdder wth a prce premum p (determned from the frst stage) to bd u, s ) p. Assumng that all the entrant buyers follow ths weakly domnant strategy n the second stage, the mechansm can be represented by a par of functons, p : α,α] R + and x : α,α] R for = 1,2,..., N, where p ) s the prce premum for a buyer who bds an amount of b = x ). Theorem 2. If u 1 s constant then the optmal mechansm of Theorem 1 can be mplemented va a twostage aucton descrbed above wth the recovery functon x 1 and prce premum functon p defned as follows: p ) = 1 F ) u 1, (14) f ) x ) = E α, s ),0} c ] A g,α )E s g maxw, s ) w g g α E α A g,α )E s g α g u 1 (y, s )1 } w(y,s )>w g,s g ) where w, s )= u, s ) p ) and w g, s )=max j, j g wj, s j ),0 }. Proof. See Appendx. ] d y, (15) The mplementaton s establshed by showng that x ( ) as defned n (15) consttutes a symmetrc (strctly) ncreasng equlbrum bd functon n the (reduced) all-pay aucton game, wth the second stage beng replaced by ts assocated equlbrum payoffs. A major (and tedous) step n the proof of Theorem 2 s to establsh that x ) as defned n (15) s strctly ncreasng for α α,α], where α,α) s the mnmum type that could possbly be allocated wth an entry rght n equlbrum. 26 Thus the recovery 25 Should there be only one entrant, the prce premum for ths sole entrant becomes the effectve reserve prce. 26 α s defned such that } E s max u,s ) 1 F ) f u 1,0 = c. ) 16

17 functon x 1 ( ) s well defned over α,α] and a (truncated) profle of pre-entry types can be recovered from ther bds. 27 Optmal entry can then be mplemented based on the recovered type profle accordng to (10). Upon beng selected n a group, say, g, everyone wll follow the (weakly) domnant strateges n the second round bddng (to bd ther value less the prce premum), so buyer g wth pre-entry type α wll wn the asset f and only f u, s ) 1 F ) u 1 max 0, max u j, s j ) 1 F j) u 1 ]}. f ) j g, j f j ) Hence the optmal allocaton rule (9) can ndeed be mplemented, provded that bddng accordng to x ( ) consttutes a symmetrc equlbrum n the (reduced) frst-stage aucton game, whch s establshed n the second step of the proof. When u 1 s not a constant, n partcular, f u 1 s a functon of s, then one s optmal prce premum also depends on her second-stage sgnal. As such, one s second-stage bd also affects the (total) prce to pay should she wn the object even under a second-prce aucton. A drect consequence s that t s no longer a (weakly) domnant strategy for bdder to bd w, s ). To avod such an nconvenence, as n Esö and Szentes, we also focus on the case n whch u 1 s a constant for aucton mplementaton. 28 Snce x ( ) s ncreasng whle p( ) s decreasng, the prce premum s decreasng n the frst-stage bds. Thus a buyer wth a hgher pre-entry type bds hgher n the frst round, whch results n a hgher probablty to be admtted and a lower prce premum. The equlbrum (entry) fee x has an ntutve nterpretaton. As can be seen from (15), one s entry fee equals her expected proft from entry less her nformatonal rent due to her prvate nformaton about her type α. So the addtonal nformaton from the second-stage sgnals does not contrbute to buyers rents: the seller approprates all rents from entry by chargng each buyer an upfront entry fee equal to her value of entry (or equvalently, the value of addtonal nformaton). 29 Fxng the aucton rules n the second round, t s also clear by the envelope theorem that payoff e- quvalence holds among all the entry mechansms n whch (1) the same entry rule (10) s mplemented, and (2) the buyer wth type α makes zero expected proft. In lght of ths equvalence result, based on x ) we can derve the canddate equlbrum bd functon under any alternatve entry mechansm. If the canddate equlbrum bd functon so derved s strctly ncreasng, then the nverse of the equlbrum bd functon can serve as the recovery functon for the mplementaton of optmal entry. For example, we can consder a dscrmnatory-prce aucton, where only bdders who are awarded entry rghts need to pay, and they pay what they bd. Usng the payoff equvalence, the canddate equlbrum bd functons That s, α s the mnmal type that one can possbly be shortlsted (as a sole entrant). We assume α,α). 27 We assume that buyers wth types below α wll stay away from bddng. But that should not affect the mplementaton of optmal entry as those buyers should not be admtted anyway. 28 The aucton mplementaton n the envronment where u 1 depends on s s an open queston. 29 Ths seems to be a robust predcton n optmal dynamc mechansm desgn (see, for example, Courty and L, 2000 and Pavan, Segal, and Tokka, 2014). As ponted out n Esö and Szentes, the value of addtonal nformaton s not well defned, however, as t depends on the specfc rules n the second-round aucton. 17

18 under the dscrmnatory-prce aucton s gven by x D )= x )/Pr α g,α ) }, where g,α ) s the set of shortlsted bdders determned by the optmal entry rule (10), gven the reported type profle,α ). A dscrmnatory-prce aucton can mplement optmal entry f and only f x D ) so derved s strctly ncreasng. 3.3 Applcatons Our optmal mechansm analyss s general enough to encompass many exstng models n the lterature on auctons wth costly entry. Below we demonstrate how we can apply our general optmal mechansm to specal models prevously studed. 1. Bdders do not have pre-entry types and only learn about ther values after entry (e.g., McAfee and McMllan, 1987; Tan, 1992; and Levn and Smth, 1994). In ths case, u, s ) = s. Hence w, s )= s, whch mples that the optmal aucton s ex post effcent, and the optmal entry s to select a set of bdders that results n the maxmal expected socal surplus. Snce bdders are dentcal before entry, optmal entry s entrely characterzed by n, the optmal number of bdders to be selected. The mplementaton s somewhat smple: the second round s a standard aucton (frst-prce, second-prce, or Englsh aucton no prce premum s nvolved). The frst round (entry stage) s to select exactly n bdders, and whomever selected s requred to pay an upfront entry fee e, whch s set so that no rent s left for the entrants ex ante. 2. Bdders know ther values before entry, and entry s merely a bd preparaton process (wthout value updatng) (e.g. Samuelson,1985; Stegeman, 1996; Campbell, 1998; Menezes and Montero, 2000; Tan and Ylankaya, 2006; Cao and Tan, 2009; and Lu, 2009). In ths settng, u, s )=α, and hence w, s ) = α (1 F ))/f ). It s easly verfed that accordng to Theorem 1, the optmal allocaton rules can be descrbed as follows: the bdder wth the hghest type ) s admtted as the sole entrant to wn the tem, as long as her contrbuton to the vrtual surplus w, s ) c s postve. The optmal mechansm can be mplemented as follows: each buyer pays what she bds n the frst stage (regardless of beng admtted or not), and the only entrant wns the tem at a prce equal to her prce premum determned from her frst-round bd. For an llustraton, below we derve the equlbrum frst-stage bd functon x. Consder a bdder wth type α > α, where α (1 F ))/f ) = c. Suppose, n the (reduced) frst-stage drect game, bdder reports ˆα, a suffcently small devaton from α. Her expected payoff s then gven by π, ˆα ) = α 1 F( ˆα ] ( ) ) c Pr α f ( ˆα ) (1) < ˆα x ( ˆα ) 18

19 = α 1 F( ˆα ] ) c F f ( ˆα ) α ( ˆα ) x ( ˆα ), (16) (1) where α s the hghest type among all the buyers other than. (1) Incentve compatblty mples dπ,α ) dα = F α ). (1) Thus we have π,α )= α 1 F ] ) α c F f ) α ) x )= F (1) α (τ) dτ. (17) α (1) Substtutng ˆα = α nto (16) to obtan the equlbrum expected payoff, and then equatng that wth (17), gves the (symmetrc) equlbrum frst-stage bd functon x )= α 1 F ] ) α c F f ) α ) F (1) α (τ) dτ. (18) α (1) It s easly verfed that x ( ) s strctly ncreasng so types can be recovered from frst-round bds n equlbrum. If a dscrmnatory-prce aucton s conducted nstead, by payoff equvalence the canddate equlbrum bd functon n the frst round s gven by x D )= x ) F α (1) ) = α 1 F ) f ) ] c 1 F α ) (1) α α F α (1) (τ) dτ. (19) It s also easly verfed that x D ( ) s strctly ncreasng. Thus the dscrmnatory-prce aucton works to mplement optmal entry n ths context as well. 3. Each bdder s endowed wth pre-entry type α, and learns an addtonal prvate value component s (e.g., Ye, 2007; Qunt and Hendrcks, 2013). The total value s gven by u, s )=α + s. Hence w, s )=α +s (1 F ))/f ). The optmal second-stage allocaton rule thus requres that the asset be allocated to the entrant bdder wth the hghest vrtual value w, s ) provded that t s nonnegatve. The optmal entry rule requres that bdders be admtted n descendng order of ther pre-entry types, as long as ther contrbuton to the expected vrtual surplus s nonnegatve. To further llustrate the optmal entry rule, we assume that α s dstrbuted unformly over 0,1] and s follows a Bernoull dstrbuton, takng value 1 ( Hgh ) wth probablty q and 0 ( Low ) wth probablty 1 q. Then w, s )=2α + s 1. If only one buyer (the one wth the hghest type α (1) ) s admtted, the expected vrtual surplus s gven by w 1 = E ( 2α (1) + s 1 1 ) c= 2α (1) + q 1 c. So the optmal number of entrants n 1 f 2α (1) + q 1 c 0. For ease of computaton we assume 19

20 that α (1) α (2).5 (so that the vrtual values from the top two bdders are guaranteed to be nonnegatve). If two top buyers are admtted, the expected vrtual surplus s gven by w 2 = E max 2α (1) + s 1 1,2α (2) + s 2 1 }] 2c = E max 2α (1) + s 1,2α (2) + s 2 }] 1 2c = Pr(s 1 = 1) (2α (1) +1 ) +Pr(s 1 = s 2 = 0) 2α (1) +Pr(s 1 = 0, s 2 = 1) (2α (2) +1 ) 1 2c = q (2α (1) +1 ) +(1 q) 2 2α (1) +(1 q)q (2α (2) +1 ) 1 2c So the optmal number of entrants n 2 f the ncremental expected vrtual surplus w=w 2 w 1 = q(1 q) 1 2 ( α (1) α (2) )] c 0. Contnung ths procedure of calculaton, 30 t can be verfed that n n f q(1 q) n ( α (1) α (n) )] c 0, 31 or α (1) α (n) c q(1 q) n 1 ]. (20) Ths condton s ntutve: the admsson of the n-th hghest buyer s more lkely to be justfed f (1) the probablty that she wll turn out to be the wnner n the second round s suffcently hgh; (2) the entry cost s suffcently low; or (3) her type s suffcently close to the hghest type. It s thus clear that the optmal number of entrants, n, s determned by the followng condtons: α (1) α (n ) c q(1 q) n 1 ], α (1) α (n +1)> c q(1 q) n ]. 4 DISCUSSION In our precedng analyss, we have focused on the revelaton polcy so that the frst-stage reports are fully revealed to the shortlsted bdders. Due to ths partcular revelaton polcy, one concern s that there mght be some loss of generalty n dentfyng optmal mechansms. To address ths concern, we next dentfy an upper bound for the expected revenue that can be acheved by examnng a relaxed settng by droppng the IC and IR constrants for the shortlsted n the second stage so that all shortlsted bdders must ncur entry costs to learn ther second-stage sgnals as n our orgnal setup, and regardless of ther second-stage sgnals, they must partcpate n the second-stage sellng mechansm and report truthfully ther second stage sgnals. As a result, regardless of the dsclosure polcy of the frst-stage reports, the hghest possble expected revenue achevable n ths relaxed settng should mpose an upper bound for the expected revenue that can be obtaned n our orgnal setup, where the bdders second-stage IC and IR must both be satsfed. A useful observaton s that n the relaxed settng, bdders can only msreport ther frst-stage sgnals, and the shortlsted buyers belefs on buyers frst-stage type profles have no 30 We contnue to consder the case α (1) > α (2)... α (n).5 so that the vrtual value from these buyers wll be postve. 31 The addton of the nth hghest buyer only contrbutes to the expected vrtual surplus when she turns out to be the only one havng a good shot n the second stage (.e., s n = 1, whle s 1 =...= s n 1 = 0). 20

21 mpact on ther second-stage decsons (as shortlsted must enter and truthfully report ther second-stage sgnals). Ths observaton mples that the revelaton polcy of the frst-stage reports s not relevant to the mechansm desgn n the relaxed settng. Consequently, the hghest expected revenue attanable n ths relaxed settng does not depend on the prevalng dsclosure polcy of the frst-stage sgnals. We next proceed to dentfy ths bound. In the relaxed settng, the mechansms are specfed exactly the same as n Secton 2. All potental bdders report ther types α, gvng rse to a reported type profle α. The mechansm specfes the frststage shortlstng rule A g ) and payment rule x,α ). Every shortlsted bdder j ncurs cost c to dscover her second-stage sgnal s j. The second-stage sellng mechansm specfes the wnnng probablty p g,sg ) and payment rule t g,sg ), g, g 2 N. Recall that P g, s ) = E g s p g,sg ) and T g, s ) = E g s t g,sg ). For shortlsted bdder g wth type α, her nterm expected payoff when she reports ˆα and others report truthfully s gven by π, ˆα )= E α g A g ( ˆα,α )E s ((u, s )P g ( ˆα,α, s ) T g ( ˆα,α, s )) c] x ( ˆα,α )}. (21) Applyng the envelope theorem, the IC condton π,α ) π, ˆα ) leads to the followng necessary condton: Therefore, we have dπ,α ) = π, ˆα ) ˆα =α dα α = E α A g,α )E s u 1, s )P g,α, s )]}. g π,α ) α = π,α)+ E α A g (y,α )E s u 1 (y, s )P g (y,α, s )]d y α g α = π,α)+ E α u 1 (y, s ) A g (y,α )P g (y,α, s )dg (s )d y. α g Note that the above expresson s exactly the same as (6), whch mples that the seller s expected revenue must be the same as n (7); n other words, the expected revenue upper bound n the relaxed settng s acheved n our orgnal settng. In ths sense, there s no loss of generalty to derve optmal mechansms by only consderng mechansms that fully reveal the buyers frst-stage reports to all admtted bdders. Another mportant aspect n our analyss s that we model nformaton acquston as entry. An mplcaton s that nformaton acquston s mandatory, n the sense that a bdder s not allowed to bd wthout gong through the due dlgence process. Ths assumpton s due to the specfc nsttutonal setup we are tryng to model. For example, data rooms are usually provded by the sellng party to dsclose a large amount of confdental data to bdders durng the due dlgence process. A typcal data room s a contnually montored space that the bdders and ther advsers wll vst n order to nspect and report on the varous documents and data made avalable. Often only one bdder at a tme wll 21