Optimal Information Disclosure in Auctions: The Handicap Auction

Transcription

1 Optmal Informaton Dsclosure n Auctons: Te Handcap Aucton Péter Eső Nortwestern Unversty, Kellogg MEDS Balázs Szentes Unversty of Ccago, Department of Economcs December 2002 Abstract We analyze te stuaton were a monopolst s sellng an ndvsble good to rsk neutral buyers wo only ave an ntal estmate of ter prvate valuatons. Te seller can release (but cannot observe) sgnals, wc refne te buyers estmates. We sow tat n te expected revenue maxmzng mecansm, te seller allows te buyers to learn ter valuatons wt te gest possble precson, and er expected revenue s te same as f se could observe te addtonal sgnals. We also sow tat ts mecansm can be mplemented by wat we call a andcap aucton. In te frst round of ts aucton, eac buyer prvately buys a prce premum from a menu publsed by te seller (a smaller premum costs more), ten te seller releases te addtonal sgnals. In te second round, te buyers play a second-prce aucton, were te wnner pays te sum of s premum and te second gest non-negatve bd. Keywords: Optmal Aucton, Prvate Values, Informaton Dsclosure JEL Classfcaton: C72, D44, D82, D83 We tank Km-Sau Cung, Edde Dekel, Drew Fudenberg, Peter Klbanoff, Pl Reny, Jm Scummer, Nancy Stokey, semnar partcpants at Nortwestern Unversty, te Unversty of Ccago, te Unversty of Pennsylvana, te Vrgna Polytecncal Insttue, and te 2002 Mdwest Teory meetngs at Notre Dame for comments, and Norbert Maer for researc assstance Serdan Rd, Evanston, IL Emal: eso@nortwestern.edu E. 59t St, Ccago, IL Emal: szentes@uccago.edu. 1

2 1 Introducton In many examples of te monopolst s sellng problem (optmal auctons), 1 te seller as consderable control over te accuracy of te buyers nformaton concernng ter own valuatons. Very often, te seller can decde weter te buyers can access nformaton tat refnes ter valuatons; owever, se eter cannot observe tese sgnals, or at least, se s unaware of ter sgnfcance to te buyers. For example, te seller of an ol feld or a pantng can determne te number and nature of te tests te buyers can carry out prvately (wtout te seller observng te results). Anoter example (due to Bergemann and Pesendorfer, 2002) s were te seller of a company as detaled nformaton regardng te company s assets (e.g., ts clent lst), but does not know ow well tese assets complement te assets of te potental buyers. Here, te seller can coose te extent to wc se wll dsclose nformaton about te frm s assets to te buyers. Sometmes, te buyers valuatons become naturally more precse over tme as te uncertanty of te good s value resolves, and te seller can decde ow long to wat wt te sale. Wen te buyers nformaton acquston s controlled by te seller, tat process can also be optmzed by te mecansm desgner. In ts paper, we explore te revenue maxmzng mecansm for te sale of an ndvsble good n a model were te buyers ntally only ave an estmate of ter prvate valuatons. Te valuaton estmates are refned by sgnals (added to te ntal estmates) tat te seller can costlessly release but cannot drectly observe. Ts model captures te common teme of te motvatng examples: te seller controls, altoug cannot learn, prvate nformaton tat te buyers care about. Our man result s tat n te revenue-maxmzng mecansm, te seller wll allow te buyers to learn ter valuatons as precsely as possble, and tat er expected revenue wll be as g as f se could observe te addtonal sgnals. 2 Tat s, te buyers wll not enjoy addtonal nformatonal rents from learnng ter valuatons more accurately wen te access to addtonal nformaton s controlled by te seller. Besdes tese surprsng fndngs, an added teoretcal nterest of our model s tat te 1 Early semnal contrbutons nclude Myerson (1981), Harrs and Ravv (1981), Rley and Samuelson (1981), and Maskn and Rley (1984). 2 In ts ypotetcal stuaton te buyers may or may not observe (but n eter case can ex post verfy) te addtonal sgnals. 2

3 standard revelaton prncple cannot be appled, yet we are able to caracterze te optmal mecansm. We also exbt a smple mecansm, dubbed te andcap aucton, wcmple- ments te revenue-maxmzng outcome. Ts aucton conssts of two rounds. In te frst round, eac buyer buys a prce premum from a menu provded by te seller (a smaller premum costs more). Ten te seller releases, wtout observng, as muc nformaton as se can. In te second round, te buyers bd n a second-prce aucton, were te wnner s requred to pay s premum over te second gest non-negatve bd. We call te wole mecansm a andcap aucton because buyers compete under unequal condtons n te second round: a bdder wt a smaller premum as a clear advantage. 3 For a sngle buyer, te andcap aucton smplfes to a menu of buy-optons (a scedule consstng of opton fees as a functon of te strke prce), were te buyer gets to observe te addtonal sgnal after payng for te opton of s coce. Our model also nests te classcal (ndependent prvate values) aucton desgn problem as a specal case, were te addtonal sgnals are dentcally zero. In ts case, te andcap aucton mplements te outcome of te optmal aucton of Myerson (1981) and Rley and Samuelson (1981). Several papers ave studed ssues related to ow buyers learn ter valuatons n auctons, and wat consequences tat bears on te seller s revenue, bot from a postve and a normatve pont of vew. One strand of te lterature, see Persco (2000), Compte and Jeel (2001) and te references teren, focuses on te buyers ncentves to acqure nformaton n dfferent aucton formats. Our approac s dfferentntatwewantto desgn a revenue-maxmzng mecansm n wc te seller as te opportunty to costlessly release (wtout observng) nformaton to te buyers. In our model, t s te seller (not te buyers) wo controls ow muc nformaton te buyers acqure. Informaton dsclosure by te seller as been studed n te context of te wnner s curse and te lnkage prncple by Mlgrom and Weber (1982). Tey nvestgate weter n tradtonal auctons te seller sould commt to dsclose publc sgnals tat are afflated wt te buyers valuatons. Tey fnd tat te seller gans from commttng to 3 Te andcap aucton can also be mplemented as a mecansm were, n te frst round, eac bdder buys a dscount (larger dscounts cost more), and ten partcpates n a second prce aucton wt a postve reservaton prce, were te wnner s dscount s appled towards s payment. 3

4 full dsclosure, because tat reduces te buyers fear of overbddng, tereby ncreasng ter bds and ence te seller s revenue. Our problem dffers from ts classc one n many aspects. Most mportantly, n our settng, te sgnals tat te seller can release are prvate (not publc) sgnals, n te sense tat eac sgnal affects te valuaton of a sngle buyer and can be dsclosed to tat buyer only. Te seller wll gan from te release of nformaton (wc se does not even observe) not because of te lnkage prncple, but because te nformaton can potentally mprove effcency, and se can carge for te access. Our motvaton s closer to tat of Bergemann and Pesendorfer (2002). Tey consder te task of desgnng an nformaton dsclosure polcy for te seller tat allows to extract te most revenue n a subsequent aucton. Ter problem s very dfferent from ours n tat te seller s not allowed to carge for te release of nformaton. Ter model also dffers from ours n tat te buyers do not ave prvate nformaton at te begnnng of te game. Under tese assumptons, Bergemann and Pesendorfer (2002) sow tat te nformaton structure tat allows te seller to desgn te aucton wt te largest expected revenue s necessarly mperfect: n ts structure, buyers are only allowed to learn wc element of a fnte partton ter valuaton falls nto. In contrast, n our paper, we desgn te expected revenue maxmzng mecansm were te nformaton structure and te rules of transacton togeter are cosen optmally. Te dfference may frst seem subtle, t s mportant neverteless. Wat we assume s tat te seller can ntegrate te rules of nformaton acquston nto te mecansm used for te sale tself. For example, n our model, te seller can carge te buyers for gettng more and more accurate sgnals (peraps n several rounds); te buyers could even be asked to bd for obtanng more nformaton. Te dea tat sellng te access to nformaton may be advantageous for te seller can be easly llustrated by an example. Suppose tat tere are two buyers wo are bot unaware of ter valuaton (drawn ndependently from te same dstrbuton), wc te seller can allow tem to learn. Ten consder te followng mecansm. Te seller carges bot buyers an entry fee, wc equals alf of te expected dfference of te maxmum and mnmum of two ndependent draws of te value-dstrbuton. In excange, se allows te buyers to observe ter valuatons (after tey ave pad te entry fee), and makes tem play an ordnary second-prce aucton. Te second-prce aucton wll be effcent, and te buyers ex-ante expected proft exactly equals te 4

5 upfront entry fee. Te seller ends up appropratng te entre surplus by cargng te buyers for observng ter valuatons. 4 Ts smple soluton te seller commttng to te effcent allocaton, revealng te addtonal sgnals, and cargng an entry fee equal to te expected effcency gans only works wen te buyers do not ave prvate nformaton to start wt. Oterwse (for example, f te buyers prvately observe sgnals, but ter valuatons also depend on oter sgnals tat tey may see at te seller s dscreton), te auctoneer, as we wll sow, does not want to commt to an effcent aucton n te contnuaton, so te prevously proposed mecansm does not work. We ave to fnd a more sopstcated aucton, and ts s exactly wat we wll do n te remander of te paper. Te paper s structured as follows. In te next secton, we outlne te model and ntroduce te necessary notaton. In Secton 3, we frst derve te revenue maxmzng aucton for te case wen te seller can observe te addtonal sgnals tat refne te buyers valuatons. Ten, we sow tat te same allocaton and expected revenue can also be attaned by a andcap aucton, even f te seller cannot drectly observe te addtonal sgnals. Te results are llustrated by a numercal example. We conclude and remark on extensons n Secton 4. 2 Te Model Assume tat tere are n potental buyers for an ndvsble good. Te seller s valuaton for te good s zero. Te valuaton of buyer {1,...,n} s te sum of two random varables, v (called type ) and s (called sock ), wc are dstrbuted ndependently (of eac oter and across ) accordng to cumulatve dstrbuton functons F and G, respectvely. 5 We assume tat te support of F s [0, 1], on wc f = F 0 exsts, and tat ts dstrbuton exbts a monotone azard rate, tat s, (1 F )/f s weakly decreasng on [0, 1]. We also assume tat te dstrbuton of te socks s atomless, owever, we do not make any restrcton on te support of te G s. We wll use v to denote te vector of types and s to denote te vector of socks. We wll also 4 Tsexample(wentebuyersavenontalprvate nformaton) as been studed ndependently by Gerskov (2002), wo also obtaned te same result. 5 Te reader may fnd t elpful to tnk of te sock as a nose wt zero mean, so te buyer s type s s expected valuaton for te good. However, we wll not make ts assumpton n te formal model. 5

6 use te usual sortand notaton for te vector of types of buyers oter tan, v, and let s denote (s j ) j6=. Te realzaton of v s observed by buyer. Altoug neter te seller nor buyer can drectly observe te sock, te seller as te ablty to generate sgnals condtonal on s, wc only buyer wll observe. In partcular, we assume tat te seller can allow buyer to observe s sock, s, wtout te seller learnng ts value. 6 All partes are rsk neutral. Te seller s objectve s to maxmze er (expected) revenue. Buyer s utlty s te negatve of s payment to te seller, plus, n case e wns, te value of te object, v +s. Every buyer as an outsde opton of zero utlty. Te seller can desgn any (ndrect) mecansm, wc can consst of several rounds of communcaton between te partes (.e., sendng of messages accordng to rules specfed by te seller). Te seller can also release sgnals (wtout observng tem). Transfers of te good and money may also occur as a functon of te story. Te set of all ndrect mecansms s rater complex, and te standard revelaton prncple cannot be appled. However, ts ssue s avoded by te approac tat we take n te next secton. 3 Results Our man result s te caracterzaton of te expected revenue maxmzng mecansm n te model ntroduced n Secton 2. We wll sow tat an optmal mecansm exsts, wc can be practcally mplemented as a andcap aucton (for a descrpton, see te Introducton or Subsecton 3.2 below). We wll also sow tat ts mecansm aceves te same expected revenue as f te seller could observe te realzatons of te socks. In oter words, wle te buyers stll enjoy nformaton rents from ter types, all ter rents from observng te socks can be approprated by te seller. In Subsecton 3.1, we start wt te dervaton of te optmal mecansm wen te seller can observe te socks (wle te buyers cannot) after avng commtted to an ndrect mecansm. In Subsecton 3.2, we sow tat n our model, te same expected revenue can also be generated by te seller wtout observng te socks. 6 In many applcatons, te seller may not be able to generate just any random sgnal correlated wt s. Terefore, all we assume s tat te seller can sow s to buyer f se so decdes. 6

7 3.1 Te Optmal Mecansm Wen te Seller Can Observe te Socks Let us assume, n ts subsecton only, tat te seller alone can observe (and verfy to a trd party) te realzatons of te socks after avng commtted to an ndrect mecansm. Te Revelaton Prncple apples, ence we can restrct our attenton to mecansms were te buyers report ter types, and te seller determnes te allocaton and transfers as a functon of te types and te socks. We wll analyze trutful equlbra of drect mecansms tat consst of an allocaton rule, x (v,v,s,s ) for all, and an (expected) transfer sceme, t (v,v,s,s ) for all. Here,x (v,v,s,s ) s te probablty tat buyer receves te good, and t (v,v,s,s ) s te transfer tat e expects to pay, gven te reported types and te realzaton of te socks. We wll use te tools of Bayesan mecansm desgn to fnd te optmal (expected revenue maxmzng) aucton. Te result wll provde an upper bound on te expected revenue te seller can aceve n te case wen se cannot observe te socks drectly, wc s gong to be te subject of Subsecton 3.2. If buyer wt type v reports type ˆv ten s expected payoff wll be π (v, ˆv )=E v,s [x (ˆv,v,s,s )(v + s ) t (ˆv,v,s,s )], (1) were E stands for expectaton. Let X (v )=E v,s[x (v,v,s,s )], and ntroduce Π (v )=π (v,v ) for te ndrect proft functon. Ten, (1) can be rewrtten as π (v, ˆv ) = E v,s [x (ˆv,v,s,s )(ˆv + s + v ˆv ) t (ˆv,v,s,s )] (2) = Π (ˆv )+(v ˆv )X (ˆv ). Incentve compatblty of te mecansm means tat, for all v, ˆv [0, 1], weave π (v,v ) π (v, ˆv ), tat s, (1) s maxmzed n ˆv at ˆv = v. Usng (2), we can rewrte ncentve compatblty as Π (v ) Π (ˆv ) (v ˆv )X (ˆv ), v, ˆv [0, 1], and =1,..., n. (3) In te followng Lemma, we apply standard arguments (see Myerson, 1981) for caracterzng ncentve compatble mecansms. 7

8 Lemma 1 Assume tat, after avng commtted to a sellng mecansm, te seller can observe te realzatons of te socks. A drect mecansm s ncentve compatble f and only f, for all =1,...,n and v [0, 1], X s weakly ncreasng, and Π (v )=Π (0) + Z v 0 X (ν)dν. (4) Proof. By (3) and ts counterpart were te roles of v and ˆv are reversed, (v ˆv )X (ˆv ) Π (v ) Π (ˆv ) (v ˆv )X (v ). Ts nequalty mples tat X s weakly ncreasng and terefore s ntegrable, and so equaton (4) follows. Now we prove tat equaton (4) and X weakly ncreasng are suffcent for ncentve compatblty. If v ˆv ten from (4) wle f ˆv v ten Π (v ) = Π (ˆv )+ R v ˆv X (ν)dν Π (ˆv )+ R v ˆv X (ˆv )dν = Π (ˆv )+(v ˆv )X (ˆv ), Π (v ) = Π (ˆv ) R ˆv v X (ν)dν Ts establses tat (3) olds. Π (ˆv ) R ˆv v X (ˆv )dν = Π (ˆv )+(v ˆv )X (ˆv ). Now we turn to te problem of determnng te revenue-maxmzng mecansm. Usng (4), we can wrte te expectaton (over all types) of buyer s surplus as Z 1 0 Π (v )f (v )dv = Π (0) + = Π (0) + Z 1 Z v 0 0 Z 1 Z 1 0 Z 1 ν X (ν)dνf (v )dv X (ν)f (v )dv dν = Π (0) + X (ν)(1 F (ν))dν 0 = Π (0) + E v,s x (v,v,s,s ) 1 F (v ) f (v ). 8

9 On te second lne, we appled Fubn s Teorem. On te trd lne, we substtuted 1 F (ν) for R 1 f ν (v )dv. Fnally, we plugged n te defnton of X. Te seller s expected revenue equals te dfference between te expected socal surplus and te sum of te ex ante expectaton of te buyers surpluses, nx µ E v,s v + s 1 F (v ) x (v,v,s,s ) f (v ) =1 nx Π (0). (5) Te mecansm desgn problem s to maxmze (5) by coosng te ntegraton constants and te vector of trade probabltes subject to te ncentve compatblty constrants. Tat s, by Lemma 1, te problem s to coose for all, Π (0) and for all and (v,v,s,s ), x (v,v,s,s ),sotatx s weakly ncreasng and (5) s maxmzed. Te followng proposton caracterzes te soluton to ts problem. =1 Proposton 1 Assume tat, after avng commtted to a sellng mecansm, te seller can observe te realzatons of te socks. In te expected revenue maxmzng mecansm, Π (0) = 0, and te allocaton rule s, x (v,v,s,s )= ( 1/ M f M and v + s 1 F (v ) f (v ) > 0 0 oterwse, (6) were M = ½ j : v j + s j 1 F ½ j(v j ) = max f j (v j ) k=1,...,n v k + s k 1 F k(v k ) f k (v k ) ¾¾, 0. Te seller s expected revenue from ts mecansm s, ½ E v,s max j=1,...,n v j + s j 1 F j(v j ) f j (v j ) ¾, 0. (7) In oter words, te seller sets te allocaton rule so tat te buyer wt te largest non-negatve sock-adjusted vrtual valuaton, v j +s j (1 F j (v j ))/f j (v j ), wll wn. Te transfers, and terefore te seller s proft, are determned by te ncentve compatblty constrants. Te proof (below) s standard. Proof. Te proposed allocaton rule, (6), togeter wt Π (0) = 0 for all, pont- 9

10 wse maxmzes (5). We wll prove tat t can be made ncentve compatble usng an approprately cosen transfer sceme. By assumpton, (1 F (v ))/f (v ) s decreasng, terefore te sock-adjusted vrtual valuaton functon, v + s (1 F (v ))/f (v ),sncreasngnv. Tsmplestat for all, v, s,ands, x (v,v,s,s ) x (ˆv,v,s,s ) f and only f v ˆv, and, after takng expectaton wt respect to v and s, X (v ) X (ˆv ) f and only f v ˆv.Tats,X s weakly ncreasng. It remans to sow tat tere exsts a transfer sceme suc tat (4) olds, Π (0) = 0, and all types of all buyers partcpate. Recall tat n a mecansm wt allocaton rule x and transfer sceme t, te expected proft of buyer wt type v s Π (v )=E v,s [(v + s )x (v,v,s,s )] E v,s[t (v,v,s,s )]. Usng x gven n (6), defne buyer s transfer as t (v,v,s,s )= Z v 0 X (ν)dν Eṽ, s [(v + s )x (v, ṽ, s, s )], were X (v ) = Eṽ, s[x (v, ṽ, s, s )]. Observe tat (4) olds, and Π (0) = 0. Fnally, all buyers partcpate because ter outsde opton s zero by assumpton, Π (0) = 0, andπ s ncreasng. Remark 1 It s clear from te proof tat te clam of Proposton 1 remans true even f te monotone azard rate assumpton s volated, but te vrtual valuatons are weakly ncreasng, tat s, f v (1 F (v ))/f (v ) s weakly ncreasng n v for all. In te next subsecton we sow tat te same outcome can also be mplemented by te seller even f se cannot observe te socks, as long as se can allow te buyers to observe tem. 3.2 Te Optmal Mecansm Wen te Seller Cannot Observe te Socks: Te Handcap Aucton Assume tat te seller cannot drectly observe te socks, but se can allow te buyers to learn tem. Clearly, n ts case, te seller cannot do better tan under te assumptons of Subsecton 3.1 (were se could observe te socks after avng commtted to 10

11 a mecansm). Te man result of te present subsecton s tat we exbt a mecansm, called te andcap aucton, wc mplements te same allocaton (wt te same expected revenue) as te revenue maxmzng mecansm of Subsecton 3.1. In general, a andcap aucton conssts of two rounds. In te frst round, eac buyer, knowng s type, cooses a prce premum p for a fee C (p ),werec s a fee-scedule publsed by te seller. Te buyers do not observe te prema cosen by oters. Te second round s a tradtonal aucton, and te wnner s requred to pay s premum over te prce. Between te two rounds, te seller may send messages to te buyers. In our model, te seller wll allow every buyer to learn te realzaton of s sock between te two rounds, and te second round s a second prce (or Engls) aucton wt a zero reservaton prce. We call ts mecansm a andcap aucton because n te second round, te buyers compete under unequal condtons: a bdder wt a smaller premum as a clear advantage. An nterestng feature of our aucton s tat te bdders buy ter premum n te ntal round, wc allows for some form of prce dscrmnaton. We wll come back to te ssue of prce dscrmnaton n Subsecton 3.3. An nterestng alternatve way of formulatng te rules of te andcap aucton would be by usng prce dscounts (or rebates) nstead of prce prema. In ts verson, eac bdder frst as to buy a dscount from a scedule publsed by te seller. Ten te buyers are allowed to learn te realzatons of te socks, and are nvted to bd n a second prce aucton wt a reservaton prce r, were te wnner s dscount s appled towards s payment. Te reader can ceck tat a andcap aucton can be easly transformed nto a mecansm lke ts by settng r suffcently g (larger tan te gest p n te orgnal fee-scedules), and specfyng tat a dscount d = r p s sold for a prce C(p ) n te frst round. In wat follows, owever, we wll use te orgnal form of te andcap aucton. If tere s only a sngle buyer, ten te andcap aucton smplfes to a menu of buy optons: p canbetougtofastestrkeprce,andteupfrontfee,c (p ),s te cost of te opton. In te second round, te buyer can exercse s opton to buy at prce p (tere s no oter bdder, so te second-gest bd s zero), for wc e ntally pad a fee of C (p ). We wll revst te sngle-buyer case later n te context of a numercal example. Frst, we state wat appens n te second round of te andcap aucton. 11

12 Lemma 2 In te second round of te andcap aucton (after eac buyer learns te realzaton of s sock), t s a weakly domnant strategy for buyer wt prce premum p to bd b = v + s p. Proof. Te second round s a second-prce aucton were buyer knows tat wen e wns, e pays te second gest non-negatve bd plus s own premum p. If e submts a bd ˆb >v + s p nstead of b, ten te only occason wen ts bd makes a dfference s wen e wns wt ˆb (wc s terefore non-negatve) and te second gest bd (or zero, wcever s larger), b j, s between b and ˆb.Hsproft s v + s b j p <v + s b p =0,soeendsupworseoff. A smlar argument sows tat can only mss proftable opportuntes by bddng ˆb <b. Terefore bddng b = v + s p s ndeed a weakly domnant strategy. From now on, we assume tat te buyers follow ter weakly domnant strateges n te second round. Ten te andcap aucton can be represented by pars of functons, p :[0, 1] R and c :[0, 1] R, for =1,..., n, werep (v ) s te prce premum tat type v [0, 1] cooses (n equlbrum) for te fee of c (v ) C (p (v )). Inwat follows, let w j = v j + s j p j (v j ),ntroduceanartfcal buyer numbered j =0wt w 0 =0, and denote max j6= {w j, 0} by w max. Incentve compatblty of te andcap aucton {c,p } n =1 means tat type v does not want to devate and coose p (ˆv ) for fee c (ˆv ) n te frst round. If e devates, tenbylemma2,ewllbdv + s p (ˆv ) n te second round. Terefore, f buyer wt type v pretends to ave ˆv n te frst round wle te oter buyers beave trutfully (tat s, for all j 6=, typev j buys premum p j (v j )), ten s payoff s, π (v, ˆv )=E v,s }(v + s p (ˆv ) w max ) c (ˆv ). (8) 1 {v +s p (ˆv ) w max Incentve compatblty of te mecansm means tat v maxmzes π (v, ˆv ) n ˆv.Let Π (v )=π (v,v ) be te buyer s equlbrum proft functon. Introduce Q (v, ˆv )=E v,s 1 {v p (ˆv )+s w max }, (9) te expected probablty tat type v wns te second round after avng cosen premum p (ˆv ) n te frst round, gven tat all oter bdders beave trutfully. 12

13 Lemma 3 A andcap aucton {c,p } n =1 s ncentve compatble f and only f, for all =1,...,n and v [0, 1], Π (v )=Π (0) + Z v and for all v 0,v 00 [0, 1] suc tat v 0 <v <v 00, 0 Q (ν, ν)dν (10) Q (v,v 0 ) Q (v,v ) Q (v,v 00 ). (11) Condton (11) states tat a buyer wt a gven type s weakly more (less) lkely to get te good n equlbrum tan e would be by mtatng a lower (ger) type n te frst round. Essentally, ts means tat lower types sould get ger prema n any ncentve compatble andcap aucton. More precsely, f p s weakly decreasng for all ten (11) olds, and te converse s true f te densty of s as full support on (, + ). Proof. [Necessty] We frst prove tat ncentve compatblty of te andcap aucton mples (10) and (11). Incentve compatblty s equvalent to, for all and ˆv <v, π (v, ˆv ) π (v,v ) and π (ˆv,v ) π (ˆv, ˆv ). (12) In te rest of ts part of te proof, assume ˆv <v.introduce,forallx, y [0, 1], (x, y) =E v,s 1 {w max s +p (y) (x y,x y]}(x + s p (y) w max ). Rewrte π (v, ˆv ) as π (v, ˆv ) = E v,s +E v,s 1 {ˆv +s p (ˆv ) w max 1 {ˆv +s p (ˆv ) w max }(ˆv + s p (ˆv ) w max ) c (ˆv ) (v ˆv ) } +E v,s 1 {v +s p (ˆv ) w max >ˆv +s p (ˆv )}(v + s p (ˆv ) w max = Π (ˆv )+Q (ˆv, ˆv )(v ˆv )+ (v, ˆv ). ) 13

14 By smlar decomposton, π (ˆv,v )=Π (v ) Q (v,v )(v ˆv ) (ˆv,v ). Gven ts, ncentve compatblty of te andcap aucton, (12), s equvalent to, for all and ˆv <v, Q (ˆv, ˆv )+ (v, ˆv ) v ˆv Π (v ) Π (ˆv ) v ˆv Q (v,v )+ (ˆv,v ) v ˆv. Note tat (x, y) 0 f and only f x y, terefore (ˆv,v ) 0 (v, ˆv ). Hence, ncentve compatblty mples Q (ˆv, ˆv ) Π (v ) Π (ˆv ) v ˆv Q (v,v ). (13) From ts, Q (ν, ν) s monotone weakly ncreasng n ν, ence t s ntegrable, and so (10) follows. Next,wesowtat(11)mustold. Assumetatˆv <v. If p (ˆv ) p (v ) ten clearly, Q (ˆv, ˆv ) Q (v, ˆv ) Q (v,v ). Suppose p (ˆv ) <p (v ).Introduce ε (x, y) =E v,s 1 {w max s (x p (x) p (y),x p (x) p (y)]}(x + s p (y) w max ). Rewrte π (v, ˆv ) = E v,s +E v,s +E v,s 1 {v +s p (v ) w max }(v + s p (v ) w max ) c (v ) 1 {v +s p (v ) w max } (p (v ) p (ˆv )) + c (v ) c (ˆv ) (14) 1 {v +s p (ˆv ) w max >v +s p (v )}(v + s p (ˆv ) w max ) = π (v,v )+Q (v,v )(p (v ) p (ˆv )) + c (v ) c (ˆv )+ε (v, ˆv ), π (ˆv,v ) = E v,s +E v,s E v,s 1 {ˆv +s p (ˆv ) w max }(ˆv + s p (ˆv ) w max ) c (ˆv ) 1 {ˆv +s p (ˆv ) w max } (p (ˆv ) p (v )) + c (ˆv ) c (v ) (15) 1 {ˆv +s p (ˆv ) w max >ˆv +s p (v )}(ˆv + s p (v ) w max ) = π (ˆv, ˆv ) Q (ˆv, ˆv )(p (v ) p (ˆv )) + c (ˆv ) c (v ) ε (ˆv,v ). 14

15 By ncentve compatblty, (12), π (v, ˆv ) π (v,v ) 0 π (ˆv, ˆv ) π (ˆv,v ). By (14) and (15), ts s equvalent to Q (v,v )(p (v ) p (ˆv )) + ε (v, ˆv ) c (ˆv ) c (v ) Q (ˆv, ˆv )(p (v ) p (ˆv )) + ε (ˆv,v ). (16) Observe tat snce ˆv <v and p (ˆv ) <p (v ),weaveε (ˆv,v ) 0 ε (v, ˆv ).Ten, ε (v, ˆv )=ε (ˆv,v )=0, oterwse (16) mples Q (v,v ) <Q (ˆv, ˆv ) contradctng (13). By ε (v, ˆv )=0, E v,s 1 {v +s p (ˆv ) w max >v +s p (v )} =0, wc s equvalent to Q (v, ˆv )=Q (v,v ). Terefore, ˆv <v mples Q (v, ˆv ) Q (v,v ),nomatterweterornotp (v ) p (ˆv ),andtefrst nequalty of (11) olds. Smlarly, by ε (ˆv,v )=0, E v,s 1 {ˆv +s p (ˆv ) w max >ˆv +s p (v )} s zero, ence Q (ˆv, ˆv )=Q (ˆv,v ). Terefore, ˆv <v mples Q (ˆv, ˆv ) Q (ˆv,v ),tats,te second nequalty of (11) olds. [Suffcency] We now sow tat (10) and (11) mply tat te andcap aucton s ncentve compatble. Let U (v,s ) be te expected equlbrum proft oftypev wt sock s n te second stage. Clearly, Π (v ) E s [U (v,s )] c (v ). From te ncentve compatblty of te second round t routnely follows (see also equaton 4 and te proof of Lemma 1) tat, for all s ŝ, U (v, ŝ ) U (v,s )= Z ŝ s E v,s 1 {v +σ p (v ) w max } dσ, were E v,s 1 {v +s p (v ) w max } s te probablty tat, n equlbrum, type v observng sock s wns te second round. By Lemma 2, buyer wt type v wo pretends to ave type ˆv n te frst stage and observes s before te second stage wll bd b = v +s p (ˆv ) n te second stage, as f e ad type ˆv and observed s +v ˆv. Hs probablty of wnnng and expected paymentwllbetesameasfeadatype-sockpar(ˆv,s + v ˆv ). Hence, s expected proft n te second round wll be U (ˆv,s + v ˆv ). 15

16 Ts noted, we can rewrte π (v, ˆv ),wtˆv <v,as π (v, ˆv ) = E s [U (ˆv,s + v ˆv )] c (ˆv ) = E s [U (ˆv,s )+U (ˆv,s + v ˆv ) U (ˆv,s )] c (ˆv ) Z s +v ˆv = Π (ˆv )+E s E v,s 1 {ˆv +σ p (ˆv ) w max } dσ = Π (ˆv )+ s Z v ˆv E v,s 0 1 {ˆv +x+s p (ˆv ) w max } dx. In te last lne, we replaced σ by s +x and canged te order of ntegraton. Smlarly, π (ˆv,v )=Π (v ) Z 0 ˆv v E v,s 1 {v +y+s p (v ) w max } dy. Incentve compatblty of te andcap aucton now becomes, for all, ˆv [0, 1), and v (ˆv, 1], Z v ˆv 0 Q (ˆv + x, ˆv )dx Π (v ) Π (ˆv ) Z 0 ˆv v Q (v + y, v )dy. (17) From condton (11), Q (ˆv + x, ˆv ) Q (ˆv + x, ˆv + x) for x [0,v ˆv ].Terefore Z v ˆv 0 Q (ˆv + x, ˆv )dx Z v ˆv 0 Q (ˆv + x, ˆv + x)dx = Π (v ) Π (ˆv ), so te frst nequalty of (17) olds. From (11), Q (v + y, v + y) Q (v + y, v ) for y [ˆv v, 0], so Π (v ) Π (ˆv )= Z 0 Z 0 Q (v + y, v + y)dy Q (v + y, v )dy, ˆv v ˆv v and te second nequalty of (17) olds, too. Terefore, te andcap aucton s ncentve compatble. In Lemma 1, we caracterzed ncentve compatble mecansms under te assumpton tat te seller can observe te socks, wle n Lemma 3, we caracterzed ncentve compatble andcap auctons for te case wen se cannot. Note tat te necessary and suffcent condtons for ncentve compatblty (n partcular, te monotoncty condtons on X and Q, respectvely) are not te same n te two cases. We wll 16

17 comment on te consequences of ts fact n te next subsecton. Lemma 3 can be used to derve te andcap aucton tat maxmzes te objectve of te mecansm desgner. In partcular, we can easly determne te expected revenue maxmzng andcap aucton. In te next proposton we do just tat; moreover, we clam tat ts andcap aucton aceves te same expected revenue as f te seller could observe te realzaton of te socks. Proposton 2 Assume tat te seller cannot observe te realzatons of te socks, altoug se can allow te buyers to observe tem. Te seller can mplement allocaton rule (6) wt expected revenue (7) va a andcap aucton {c,p } n =1, were and c (v ) s defned by p (v )= 1 F (v ), (18) f (v ) c (v )=E v,s 1 {v +s p (ˆv ) max j6= w j }(v + s p (v ) max w j ) Z v 0 j6= E v,s 1{ν+s p (ν) max j6= w j dν. (19) Proof. If, for all j =1,...,n and v j [0, 1], typev j of buyer j purcases a prce premum p j (v j )=(1 F j (v j ))/f j (v j ) n te frst round, ten buyer wllwnnte second round f and only f, for all j, v + s 1 F (v ) f (v ) ½ max v j + s j 1 F j(v j ) f j (v j ) ¾, 0. Ts s so because n te second round, every buyer j bds v j + s j p j (v j ). Hence te allocaton rule s ndeed te same as (6), provded tat all buyers beave trutfully,.e., every buyer j wt type v j cooses p j (v j ) for a fee c j (v j ) defned n (19). We can easly ceck tat te andcap aucton defned by (18) and (19) satsfes te ypoteses of Lemma 3. Frst, p s weakly decreasng by te assumpton of monotone azard rate. Second, te fee-scedule, (19), s equvalent to (10), as Π (v ) E v,s 1 {v +s p (v ) max j6= w j }(v + s p (v ) max j6= w j ) c (v ). 17

18 Also note tat Π (0) = 0 for all. By Lemma 3, (18) and (19) defne an ncentve compatble andcap aucton. It remans to sow tat te seller s expected revenue n ts andcap aucton s equal to tat n te optmal mecansm of Proposton 1 (were te seller could observe tesocks). Notetatteexpectedpayoff of buyer wt type v n te mecansm of Proposton 1 s gven by (4) wt allocaton rule (6) and Π (0) = 0. Te expected payoff of buyer wt type v n te proposed andcap aucton s gven by (10) wt premum scedule (18) and Π (0) = 0. Terefore,Π (v )=Π (v ) for all v.sncete allocaton rules n te two mecansms concde, te total socal surplus s te same n bot cases. Te seller s expected revenue s just te dfference of te total surplus and te buyers payoff, terefore, t must also be te same. Remark 2 If te support of eac sock s (, + ) ten, as we remarked before te proof of Lemma 3, p must be weakly decreasng for te andcap aucton to be ncentve compatble. Also observe tat te optmal allocaton rule n Proposton 1 s unque (.e., allocate te good to te buyer wt te gest non-negatve sockadjusted vrtual valuaton). Terefore, f te seller can aceve te same revenue n a andcap aucton wtout observng te socks, ten te allocaton rule must be te same. Hence, te premum n ts andcap aucton must equal te azard rate, (18), and te monotone azard rate assumpton s necessary to guarantee te same revenue. 3.3 Dscusson From te seller s perspectve, te premum fee scedule offered n te frst round of te andcap aucton works as a devce to dscrmnate among buyers wt dfferent value estmates. Wen a buyer decdes to partcpate n te andcap aucton, e knows s type (expected valuaton), wc tells m weter e s more or less lkely to wn. Terefore, n te frst round, a buyer wt a g type cooses a small prce premum for a large fee n order not to pay muc wen e wns. Usng analogous reasonng, low types coose large prce prema, wc are ceaper, but make wnnng more expensve. It s nterestng to observe tat n te optmal andcap aucton, two buyers wt te same actual valuaton (same v + s ) do not ave te same probablty of wnnng. Te buyer wt te larger v wll coose a smaller prce premum, bd ger n te second round, and wll be more lkely to wn. Ts sows tat te aucton does not aceve 18

19 full ex post effcency, even under ex ante symmetry of te bdders and condtonal on te object beng sold. 7 In order to better explan our man result (Proposton 2), consder a setup were te buyers are ex ante symmetrc (te v s are dentcally dstrbuted), and te socks are mean zero random varables. Let us compare te optmal allocaton rule n te case wen te seller can observe te socks (as n Subsecton 3.1) wt tat of te revenue maxmzng aucton wen nobody (neter te seller nor te buyers) can observe tem. In te latter case, te seller sould allocate te good to te buyer wt te largest non-negatve vrtual value-estmate, v (1 F (v ))/f(v ). If te seller can observe te socks, ten, n te optmal mecansm, te good wll be allocated more effcently, as te wnner wll now be te buyer wt te gest non-negatve sock-adjusted vrtual valuaton, v + s (1 F (v ))/f(v ), accordng to equaton (6). 8 Accordng to Proposton 2, te seller, by controllng te release of te socks and wtout actually observng tem, can mplement te same allocaton, and surprsngly, can approprate tencreaseneffcency. 9 One may suggest tat te way te seller can approprate all rents from te addtonal nformaton s tat n te andcap aucton, se essentally carges te buyers a type-dependent up-front fee equal to te value of te nformaton tey are about to receve. Ts ntuton may be appealng, but t overly smplfes te workngs of te mecansm. Frst, te value of te addtonal nformaton to te partcpants s not well-defned because t depends on te rules of te sellng mecansm. Ts value could be dfferent f te seller cose a mecansm dfferent from te andcap aucton. Anoterargumentstatwesowed,tesellermaynotalwaysbeabletoextract all rents for te addtonal nformaton va a andcap aucton. Ts s so f te vrtual value-estmates are monotone ncreasng, but te type-dstrbutons do not exbt 7 In contrast, n te classcal setup wt determnstc valuatons, te optmal aucton of Myerson (1981) and Rley and Samuelson (1981) s effcent condtonal on sale, provded tat te buyers are ex ante symmetrc. 8 It s easy to see tat f v (1 F (v ))/f(v ) <v j (1 F (v j ))/f(v j ), but, by addng te socks to bot sdes te nequalty s reversed, ten v + s >v j + s j. Terefore, an allocaton based on te sock-adjusted vrtual valuatons pontwse mproves effcency. (Ts may not be true f te F s are not dentcal.) 9 If te buyers ex ante type-dstrbutons are not dentcal ten, as te seller gets to observe te sgnals, te effcency of te optmal mecansm may only mprove n ex ante expectaton. Stll, tere wll be some effcency gan, wc wll be fully extracted by te seller even f se cannot observe te addtonal sgnals. 19

20 monotone azard rates (compare Remarks 1 and 2). Te allocaton rule n te optmal mecansm wen te seller can observe te socks wll be based on te sock-adjusted vrtual value-estmates, but te correspondng premum functons, te p s, would not be weakly decreasng. Terefore, te former allocaton rule s not mplementable va a andcap aucton wen te seller cannot observe te socks. 3.4 Determnng te Optmal Handcap Aucton: ANumercalExample It may be useful to compute a numercal example not only for llustratve purposes, but also, to see ow a seller may be able to compute te parameters of te optmal andcap aucton (te prce premum fee scedule) n a practcal applcaton. Te optmal p (v ) s gven by (18), and c (v ) s gven by (19). Supposng tat p s dfferentable, we can rewrte (19) as c (0) = E v,s 1 {s 1/f (0) max j6= w j }(s 1/f (0) max w j ), (20) j6= c 0 (v ) = p 0 (v ) E v,s Pr[s max w j v + p (v )]. (21) j6= Note tat (21) s just te frst-order condton of v arg maxˆv π (v, ˆv ), were π (v, ˆv ) s gven by (8). We wll consder te followng setup. Te types are dstrbuted ndependently and unformly on [0, 1], and te socks are dstrbuted ndependently accordng to a standard logstc dstrbuton. 10 Frst, assume tat tere s a sngle buyer, tat s, n =1. As we mentoned t earler, te andcap aucton wt a sngle buyer can be tougt of as a menu of buy optons, represented by C 1 (p 1 ),werep 1 s te strke prce and C 1 (p 1 ) s te fee of te opton. In te frst round, te buyer cooses a prce p 1 and pays C 1 (p 1 ); n te second round (after avng observed s 1 ), e as te opton to buy te good at prce p 1. Let us represent te menu of buy optons as a par of functons, c 1 (v 1 ) and p 1 (v 1 ), v 1 [0, 1]. In te unform-logstc example, te expected revenue maxmzng strke 10 Te cdf of te standard logstc dstrbuton s G (s )=e s /(1 + e s ), s (, + ). 20

21 prce-scedule s gven by (18), p 1 (v 1 )=1 v 1. Te equatons caracterzng te fee-scedule, (20) and (21), become, Z c 1 (0) = (s 1 1) 1 (1 + e ds s 1 ) 2 1 =ln(1+e) 1, c 0 1(v 1 ) = 1 e1 2v 1 1+e. 1 2v 1 By ntegraton, we get an explct expresson for c 1 (v 1 ) as e s 1 c 1 (v 1 )= 1 2 ln (1 + e) 1+v ln 1+e 1 2v 1. We can express te cost of te opton as a functon of te strke prce as c 1 = C 1 (p 1 )= 1 2 ln (1 + e) 1+e 2p 1 1 p 1. Ts (downward-slopng) scedule s depcted as te top curve n Fgure 1. If te buyer as a ger estmate ten e wll coose to buy an opton wt a lower strke prce at a ger cost. For example, f te buyer as te lowest estmate, v 1 =0, ten e buys te opton of gettng te good at p 1 =1, wc costs c 1 =ln[(1+e)/e] upfront, and yelds zero net surplus. In contrast, te gest type, v 1 =1,buys a call opton wt zero strke prce at a cost of about Now we turn to te case of many buyers, n>1, n te unform-logstc example. We wll compute te optmal andcap aucton represented by {c,p } n =1. As n te case of n =1, n te revenue-maxmzng mecansm, p (v ) 1 v. Instead of analytcally dervng c (v ) for dfferent numbers of buyers, we carry out a more practcal Monte Carlo smulaton. Wat we descrbe below s also te metod tat a seller could use n order to determne te parameters of an optmal andcap aucton n practce. We take 100,000 random draws from te jont dstrbuton of (s, v ),andcompute w j = v j + s j p j (v j ) for all j. Ten we determne c (0) from (20), were te expectaton s estmated by te sample mean. We compute c (v ) recursvely, c (v +step) =c (v )+step c 0 (v ),werestep=1/100. Te dervatve can be estmated 21

22 c, fee (x 100) p, prce premum (x 100) Fgure 1: Fee scedules n te revenue maxmzng andcap aucton (unform-logstc setup; scedules from top to bottom for n=1, 2, and 5) from (21), rewrtten as c 0 (v )=E s,s,v 1{v +s p (v ) max j6= w j }. From c (v ) and p (v ) we compute C (p ) c (p 1 (p )). Te results of a (typcal) smulaton are sown n Fgure 1. Te top curve sows C (p ) for te case of n =1. Tere are actually two (almost dentcal) curves supermposed on eac oter: one graps te formula tat we derved before, te oter s te result of te Monte Carlo experment. Te curve n te mddle s C (p ) for n =2, and te one n te bottom s C (p ) for n =5.Asnncreases, C (p ) sfts down and flattens out. 22

23 4 Conclusons In ts paper, we analyzed an aucton model were te seller could decde ow accurately te buyers learned ter prvate valuatons. In partcular, n our settng, te buyers only knew an ntal estmate of ter prvate valuaton, and te seller ad te ablty to release (wtout observng) ndependent sgnals tat were added to te buyers estmates to determne ter ex post valuatons. In oter words, te buyers valuatons were ntally uncertan, but te seller could allow tem to resolve ts uncertanty. We derved te expected revenue maxmzng mecansm. In te optmal mecansm, te seller allows te buyers to learn ter valuatons wt te gest precson and obtans te same expected revenue as f se could observe te addtonal sgnals (wc se can release, but cannot drectly observe). Te buyers do not enjoy any addtonal nformaton rents from te sgnals wose dsclosure s controlled by te seller. Te outcome of ts mecansm can be mplemented va a andcap aucton. In te frst pase of ts mecansm, te seller publses a prce premum fee scedule for eac buyer; eac buyer cooses a prce premum and pays te correspondng fee. Ten te seller allows te buyers to learn ter valuatons wt te gest precson. In te second pase, te buyers bd for te good n a second-prce sealed-bd aucton wt a zero reservaton prce, knowng tat te wnner wll pay s premum over te prce. For a sngle buyer, te andcap aucton smplfes to a menu of buy-optons. Interestngly, our man result extends to general adverse selecton models, assown n our related paper Eső and Szentes (2002). In tat paper, we consder a setup were a prncpal controls te precson of te agent s nformaton regardng s own type (productvty, ablty, etc.), by beng able to release, wtout observng, sgnals tat refne te agent s estmate. 11 In te prncpal s optmal contract te agent learns s type wt te gest precson, yet no nformaton rents wll be left wt te agent for te addtonal sgnals. 12 Agan, te one wo controls te flow of nformaton approprates te rents of nformaton. 11 Ts s te case, for example, wen te prncpal s te employer of te agent, and decdes about te extent of te agent s learnng te detals of te task, etc. 12 Te optmal contract, owever, may be a lot more complcated tan te andcap aucton. 23

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