Information Structures in Optimal Auctions

Transcription

1 Informaton Structures n Optmal Auctons Dr Bergemann y Martn Pesendorfer z January 2007 Abstract A seller wshes to sell an obect to one of multple bdders. The valuatons of the bdders are prvately nown. We consder the ont desgn problem n whch the seller can decde the accuracy by whch bdders learn ther valuaton and to whom to sell at what prce. We establsh that optmal nformaton structures n an optmal aucton exhbt a number of propertes: () nformaton structures can be represented by monotone parttons, () the cardnalty of each partton s nte, () the parttons are asymmetrc across agents. We show that an optmal nformaton structure exsts. Keywords: Optmal Aucton, Prvate Values, Informaton Structures, Parttons. Jel Classfcaton: C72, D44, D82, D83. We would le to than the Edtor, Alessandro Lzzer, an Assocate Edtor and two anonymous referees for ther valuable suggestons. The authors are grateful to Don Brown, Ncolas Hengartner, Bll Sudderth, Marten Wegamp and Steve Wllams for many nformatve dscussons. We than semnar partcpants at Cornell Unversty, Due Unversty, Pompeu Fabra, Rochester Unversty, Rutgers Unversty, Unversty of Calforna at Bereley, Unversty of Illnos, Unversty of Pennsylvana and Unversty of Wsonsn for useful comments. Fnancal support from NSF Grant SES and a Sloan Research Fellowshp, and NSF Grants SES and SES , respectvely, s gratefully acnowledged. y Department of Economcs, Yale Unversty, New Haven, USA, dr.bergemann@yale.edu. z Department of Economcs, London School of Economcs, London, U.K., m.pesendorfer@lse.ac.u. 1

2 1 Introducton The optmal desgn of an aucton has receved consderable attenton n the economcs lterature. Myerson (1981) consttutes the semnal paper n the eld. Myerson shows whch aucton rules acheve the largest revenues to the seller n a sngle obect aucton. Most of the subsequent lterature on mechansm desgn mantans the assumpton that the nformaton held by maret partcpants s gven as exogenous. Lttle s nown about optmal mechansms when the nformaton of the partcpants s allowed to be endogenous. Ths paper consders the optmal aucton desgn problem when the seller can determne bdders nformaton precson. We consder a problem n whch a seller o ers a sngle obect to a number of rs neutral bdders. The seller wshes to maxmze revenues from the sale. Bdders valuatons for the obect are prvate and not nown pror to the bddng. The seller controls the bdders nformaton structures whch generate the bdders prvate nformaton. The nformaton structure determnes the accuracy wth whch buyers learn ther valuatons pror to the aucton. The seller may assgn an nformaton structure that nforms a bdder perfectly or an nformaton structure that gves the bdder only a rough guess about her true value for the obect. The seller s choce of nformaton structure s made pror to the aucton and does not nvolve transfer payments from the bdders. After the choce of nformaton structure by the seller, the bdders then report ther value estmate to a revelaton mechansm whch determnes the probablty of wnnng the obect and a transfer payment for every bdder. We study nformaton structures and revelaton mechansms that maxmze the seller s revenues. The soluton n Myerson (1981) arses n our model as a specal case when the seller nforms the bdders perfectly. We analyze the optmal nformaton and mechansm desgn problem under strong nformatonal assumptons. We assume that the seller has full control n hs choce of the nformaton structure and there s no cost to adopt a partcular nformaton structure. Our set-up allows us to emphasze two opposng e ects that determne the endogenous choce of the precson of nformaton: rst, more nformaton ncreases the e cency of the aucton and thus seller s revenues; second, more nformaton ncreases the rents of the bdders n form of nformaton rents whch lower the seller s revenues. We analyze ths trade-o and 2

3 characterze the propertes of optmal nformaton structures. The model assumes that the optmal nformaton and mechansm desgn s subect to the nterm ncentve and nterm ndvdual ratonalty constrants of the bdders. By mposng the nterm ndvdual ratonalty constrant, each bdder s allowed to assess the value of the transacton condtonal on hs prvate nformaton. In partcular, ths means that the seller cannot request payment for the prvate nformaton, separately from the aucton of the obect tself. The adopton of the nterm ndvdual ratonalty constrants here can be motvated by a temporal dstncton between the adopton of a set of rules governng an aucton or an entre seres of aucton and the actual aucton event. Wthn such a sequencng context, the ndvdual ratonalty constrants arses naturally at the nterm stage. 1 Emprcal applcatons that share features wth some of our assumptons can be gven, but we wsh to emphasze that we are not aware of an applcaton that ts our assumptons precsely. Our study maes strong assumptons and our results may not be drectly applcable for aucton desgn n practce. In lght of the results, we shall dscuss the role of the assumptons n detal n the nal secton. The lnage prncple of Mlgrom & Weber (1982) s related to our wor but obtaned n a dstnct nformatonal setup. In a symmetrc model wth a lated values, they show that the seller can ncrease revenue by releasng nformaton publcly to all bdders. The publc nformaton reduces the wnner s curse and hence the nformaton rent of the wnnng bdder. In contrast, wth prvate values, an ncrease n nformaton to an ndvdual bdder ncreases that bdder s nformaton rent. Whle we consder the choce of nformaton structure by the seller, a related lterature consders the ncentves of the buyers, to obtan more nformaton, e.g. Cremer & Khall (1992), Persco (2000), and Bergemann & Välmä (2002). Our paper s organzed as follows: Secton 2 descrbes the model. Secton 3 consders the example of bdders wth unformly dstrbuted valuatons on the unt nterval. Secton 4 analyzes the optmal nformaton structure when the sgnal space s nte. We show that: () the optmal nformaton structures are parttons, () the optmal parttons are asymmetrc, and () optmal parttons exst. Secton 5 extends the characterzaton results to the class of all measurable nformaton structures (possbly wth n nte and uncountable 1 We than an anonymous referee for suggestng ths pont of vew. 3

4 sgnals) and shows that the above characterzaton results reman to hold, n partcular the optmal nformaton structure remans a nte monotone partton. Secton 6 concludes and dscusses the lmts of our analyss. 2 Model 2.1 Utlty A seller has a sngle obect for sale. There are I potental bdders for the aucton, ndexed by 2 f1; ::::; Ig. Each agent has a compact set V = [0; 1] of possble valuatons for the obect, where a generc element s denoted by v 2 V, and V = I V = [0; 1] I. =1 We occasonally adopt the notaton v = (v ; v ). The valuaton v s ndependently dstrbuted wth pror dstrbuton functon F (v ). The pror dstrbuton functon F (v ) s common nowledge. The assocated densty functon f (v ) s postve on V. The utlty of the (wnnng) agent s quaslnear and gven by u (v ; t ) = v t, where t s a monetary transfer. 2.2 Informaton Structure The sgnal space s denoted by S [0; 1]. The space S can ether be countable, nte or n nte, or uncountable. Let (V S ; B (V S )) be a measurable space, where B (V S ) s the class of Borel sets of V S. An nformaton structure for agent s gven by a par S, hs ; F (v ; s ), where S s the space of sgnal realzatons and F (v ; s ) s a ont probablty dstrbuton over the space of valuatons V and the space of sgnals S. 2 We refer to ths class of nformaton structures as (Borel) measurable nformaton structures. 2 By assumpton, the sgnal of agent s ndependent of agent s valuaton, for 6=. If agent s sgnal were to depend on agent s valuaton, then full rent extracton s possble, see Cremer & McLean (1988). 4

5 The dstrbuton and the nformaton structure for all agents are denoted by omttng the subscrpt ; or F (v; s) and S, respectvely. The ont probablty dstrbuton s de ned n the usual way by F (v ; s ), Pr (ev v ; es s ). The margnal dstrbutons of F (v ; s ) are denoted wth mnor abuse of notaton by F (v ) and F (s ) respectvely. For F (v ; s ) to be part of an nformaton structure requres the margnal dstrbuton wth respect to v to be equal to the pror dstrbuton over v. The condtonal dstrbuton functons derved from the ont dstrbuton functon are de ned n the usual way: R v 0 F (v s ), df (; s ) R 1 0 df (; s ) ; and smlarly, R s 0 F (s v ), df (v ; ) R 1 0 df (v ; ) : The auctoneer can choose an arbtrary nformaton structure S for every bdder subect only to the restrcton that the margnal dstrbuton equals the pror dstrbuton of v. The cost of every nformaton structure s dentcal and set equal to zero. The choce of S s common nowledge among the bdders. At the nterm stage every agent observes prvately a sgnal s rather than her true valuaton v of the obect. Gven the sgnal s and the nformaton structure S each bdder forms an estmate about her true valuaton of the obect. The expected value of v condtonal on observng s s de ned and gven by w (s ), E [v s ] = Z 1 0 v df (v s ) : Every nformaton structure S generates a dstrbuton functon G (w ) over posteror expectatons gven by Z G (w ) = fs :w (s )w g df (s ). We denote by W the support of the dstrbuton functon G (). Observe that the pror dstrbuton F () and the posteror dstrbuton over expected values G () need not concde. For future dscussons t s helpful to llustrate some spec c nformaton structures. The nformaton structure S yelds perfect nformaton f F (v ) = G (v ) for all v 2 V. 5

6 In ths case, the condtonal dstrbuton F (s v ) has to satsfy 8 < 0 f s < s (v ) ; F (s v ) = : 1 f s s (v ) ; (1) where s (v ) s an nvertble functon. An nformaton structure whch sats es (1) wthout necessarly satsfyng the nvertblty condton s called parttonal. An nformaton structure s called dscrete f S s countable and nte f S s nte. After the choce of the nformaton structures S by the auctoneer, the nduced dstrbuton of the agent s (expected) valuatons s gven by G (w ) rather than F (v ). The sgnal s and the correspondng expected valuaton w (s ) reman prvate sgnals for every agent and the auctoneer stll has to elct nformaton by respectng the truthtellng condtons. 2.3 Mechansm The seller selects the nformaton structures of the bdders and a revelaton mechansm. The obectve of the seller s to maxmze hs expected revenue subect to the nterm partcpaton and nterm ncentve constrants of the agents. By the revelaton prncple we may restrct attenton to the drect revelaton mechansm. The drect revelaton mechansm conssts of a tuple (W ; t ; q ) I =1 wth transfer payment of bdder : t : I W! R; =1 and the probablty of wnnng the obect for bdder : q : I W! [0; 1]: =1 We sometmes wrte T (w ) for the expected transfer payment, T (w ), E w t (w ; ); where the expectaton s taen over w = (w 1 ; :::; w 1 ; w +1 ; :::; w I ). Smlarly, Q (w ) denotes the expected probablty of wnnng, Q (w ), E w q (w ; ): 6

7 The nterm utlty of bdder wth an expected valuaton w and announced valuaton bw s: U (w ; bw ) = w Q ( bw ) T ( bw ). The mechansm has to satsfy the nterm partcpaton constrants: U (w ), U (w ; w ) 0, for all w 2 W ; and the nterm ncentve constrants: U (w ) U (w ; bw ) ; for all w ; bw 2 W : A mechansm that sats es both, the nterm partcpaton constrants and the nterm ncentve constrants, s called ncentve compatble. The tmng of the events s graphcally summarzed below: x? nformaton structure S; (s ) I =1 realzed ( bw ) I =1 reported mechansm (W ; q ; t ) I =1 ; (w ) I =1 observed (q () ; t ()) I =1 assgned determned x? x?! Fgure 0: Tme lne of events We note that the transfers and the nformaton structures are determned smultaneously for all bdders. In partcular, we do not consder sequental mechansms n whch the nformaton structure for some agents may be determned after some nformaton has already been revealed about a certan subset of bdders. 3 Examples Ths secton llustrates propertes of optmal nformaton structures for some specal cases. Frst, we loo at sngle and two-bdder auctons. We llustrate the unconstraned optmal nformaton structure. Then, we llustrate the constraned optmal nformaton structure when the seller s choce s restrcted to () dentcal nformaton structures across bdders 7

8 and () dentcal parttons consstng of equally szed ntervals. The examples llustrate that the seller prefers sparse nformaton and treat bdders asymmetrcally. Second, we depart from the two-bdder model and depct propertes of the numercal soluton to an aucton wth many bdders when the valuatons are drawn from the unform dstrbuton. Agan, we depct the unconstraned and constraned symmetrc soluton. Sngle Bdder Aucton: Consder rst the case wth a sngle bdder. The nformaton structure n whch the seller assgns a perfectly nformatve nformaton structure to the bdder s analyzed n Myerson (1981). Myerson establshes that the seller can extract at most the vrtual valuatons n any ncentve compatble sellng mechansm. The vrtual valuaton of a bdder of type wth valuaton v equals the valuaton of the bdder mnus the ncentve cost, v 1 F (v) f(v) Notce, that the ncentve cost s postve and remans postve even f the seller assgns an nformaton structure whch nforms partally only. In contrast consder the stuaton n whch the seller chooses to assgn an unnformatve nformaton structure to the buyer. Wthout any nformaton, a bdder s wllng to pay up to the ex ante expected valuaton of the bdder to receve the obect. In ths case, the seller can extract all the expected surplus. It s therefore mmedate that assgnng an unnformatve nformaton structure s optmal n a sngle bdder aucton. The seller can post a prce equal to the ex ante expected valuaton. Ths posted prce scheme extracts the total surplus and s e cent. Moreover, f the seller were to assgn an nformaton structure that nforms the bdder, the seller would be worse o because he ncurs an ncentve cost expressed by the vrtual utlty. Two-bdder Aucton: Suppose now we were to add a second bdder to the aucton wth an dentcal pror dstrbuton. The polcy to dsclose no nformaton does not reman optmal wth two bdders. To see ths, notce that assgnng an unnformatve nformaton structure extracts at most the ex ante expected valuaton of the wnnng bdder. But wth symmetrc bdders, the revenue for the auctoneer would then be the same as n the case of a sngle bdder. In a two-bdder aucton there s a smple scheme that acheves more rent by 8

9 explotng the ncrease n the number of bdders. The scheme has the followng feature: The seller assgns an unnformatve nformaton structure to the rst bdder as n the case of a sngle bdder aucton, but assgns a bnary nformaton structure to the second bdder. A bnary nformaton structure permts the bdder to determne whether the valuaton s above or below a certan threshold. The optmal threshold s exactly equal the ex-ante expected value of the obect. The scheme then wors as follows: Intally, the seller o ers the obect to the second bdder at a prce equal to the condtonal expected valuaton n the event that the valuaton s above the threshold. If the second bdder reects the o er, then the seller o ers the obect to the rst bdder at a prce equal to the ex ante expected valuaton. The total revenues to the seller under ths scheme exceed the ex ante expected valuaton of a bdder. Thus, the revenues under ths scheme are hgher than under a scheme n whch the seller assgns an unnformatve nformaton structure. We observe that as before, the seller leaves no nformatonal rent to the bdders. However, the allocaton s not necessarly e cent anymore, as t could be that the rst bdder has n fact a hgher valuaton for the obect than the second bdder wth the bnary partton. However, the coarse nformaton structure doesn t allow the seller to mae ths contngent decson. In fact, t can be shown that the descrbed nformaton structure maxmzes the revenues to the seller wth two bdders and unformly dstrbuted valuatons. Ignorng elements n the nformaton structure whch are assocated wth zero wnnng probablty events, as we do throughout ths paper, ensures that the descrbed nformaton structure s the unque soluton. If attenton s restrcted to the class of nformaton structures wth nte parttons, then ths result follows mmedately from the rst and second order condtons for optmally chosen parttons. Our results n the subsequent sectons establsh that the descrbed scheme wth two bdders s ndeed optmal for the unform dstrbuton under general nformaton structures even permttng non-parttonal and non- nte nformaton structures. For nonunform pror dstrbutons the optmal nformaton structure may change as both, the locaton of the boundary ponts n the partton and the number of elements n the partton, depend on the dstrbutonal assumpton. The scheme wth two bdders has a number of features that are worth emphaszng. Frst, even f bdders have ntally symmetrc pror dstrbutons of valuatons, they are optmally 9

10 assgned asymmetrc nformaton structures. The rst bdder receves no nformaton, whle the second bdder learns wether the valuaton s above or below the ex-ante mean. Second, the seller does not gve an nformatonal rent to buyers. Both bdders are o ered the obect at a xed prce that they can accept or reect. Symmetrc Informaton Structures: Suppose the auctoneer were constraned to o er dentcal nformaton structures to bdders. Wth two bdders and unformly dstrbuted valuatons the bnary nature of the nformatonal structure remans optmal, but the locaton of the boundary pont n the partton s altered by the symmetry restrcton. It s now optmal to set the boundary pont n the partton at one thrd and to o er the bdders the obect at a xed prce of two thrd. If bdders valuatons do not exceed one thrd the seller retans the obect. The event of no award can occur because the cost of nformaton revelaton s hgh and o sets the gans from a sale when valuatons are low. Multple Bdders: A natural queston s whether the features of the optmal nformaton structure for two bdders wth unformly dstrbuted valuatons extend to more general settngs. We address ths queston n the subsequent sectons. Before we start our formal analyss we llustrate graphcally optmal nformaton structures wth many bdders. 3 The followng gure depcts propertes of optmal nformaton structures wth unform dstrbuted valuatons as we vary the number of bdders. The dotted lne llustrates the boundary ponts for constraned symmetrc parttons. The sold lne llustrates the boundary ponts for unconstraned (asymmetrc) partton for the bdder wth the largest nteror boundary pont. Insert Fgure 1 here As can be seen n the gure the number of boundary ponts ncreases monotone wth the number of bdders partcpatng n the aucton. However, the ncrease s only very gradual. For the optmal (asymmetrc) nformaton structure, we count three elements n the partton wth three to sx bdders, four elements wth seven to fteen bdders and ve elements wth sxteen or more bdders. The boundary ponts of the parttons for the constraned optmal (symmetrc) nformaton structure loo very smlar to the unconstraned soluton. We count three elements n the partton wth four to eght bdders, four elements wth nne to 3 The numercal calculatons were mplemented usng the software pacage GAUSS. 10

11 thrteen bdders and ve elements wth fourteen or more bdders. Fgure 1 llustrates further that n general t s not the case that the seller leaves no nformatonal rent to the bdder. Wth three or more bdders, bnary parttons are no longer optmal and as the auctoneer has to reward agents to report truthfully, he wll have to ncur ncentve costs. As the number of bdders ncreases, the nformaton structure becomes ner. The ntuton s that wth more competton the ncentve costs due to the nformatonal rents are lower and the revenue gans from mprovng allocatve e cency due to more nformaton become more mportant, as the number of bdders ncreases. 4 Optmal Informaton Structure Wth Fnte Sgnals Motvated by the examples, ths secton descrbes the optmal aucton and optmal nformaton structure gven a nte number of sgnals. In the next secton, we then show that the characterzaton and optmalty of nte nformaton structure perssts wth an arbtrary number of sgnals, nte or n nte. Subsecton 4.1 characterzes the seller s expected revenues and optmal mechansm for a gven nte nformaton structure and hence nte types. In Subsecton 4.2 we start the analyss of the optmal nformaton structure by dervng several features of the vrtual utltes of the bdders. Subsecton 4.3 uses the revenue structure of the optmal aucton to show that the nformaton structure has to be a partton and that an optmal nformaton structure for a gven nte number of sgnals exsts. 4.1 Optmal Aucton Desgn Wth Fnte Types Motvated by the examples, ths secton characterzes the seller s expected revenues and optmal mechansm for a gven nte nformaton structure and hence nte types. At ths stage we are merely nterested n characterzng the expected revenues of the auctoneer from bdder. For a gven dstrbuton G (w ), we denote the nte set of mass ponts by w 1 ; :::; w K, and for every w, g, G w G w 1 > 0; wth g beng the postve probablty of mass pont w. For notatonal ease, we shall denote the value of the dstrbuton functon G () at w 11 smply as G, and lewse refer to the

12 nterm probablty of wnnng at w as Q and the nterm transfer at w as T. Lemma 1 descrbes the revenues the auctoneer receves from bdder wth a gven pror dstrbuton G () and a gven expected probablty of wnnng Q (). Lemma 1 (Revenues) The expected revenues from bdder n an ncentve compatble mechansm are: " # KX 1 G R (G ; Q ), w w +1 w eq Q g U w 1, (2) =1 subect to Q () beng non-decreasng, Q e Q Q+1 and U w 1 0. Proof. The proofs for all results are provded n the appendx. The smlarty wth the case of postve densty analyzed n Myerson (1981) s mmedate. The mod caton due to the dscreteness appears n the obvous places. The densty g s now replaced wth the postve probablty g. The local change dw = 1 s beng replaced by the dscrete change between w and w +1, or w +1 w. There are two ndetermnaces n the expresson of revenues (2). Frst, as n the contnuous analogue, the utlty for the lowest type, U w 1, s an arbtrary non-negatve number. Second, the probablty Q e h s an arbtrary number n Q ; Q+1. The second ndetermnacy arses due to the dscreteness of types and s absent n the contnuous analogue. Wth dscrete types, the utlty ncrement for a bdder of type w g Q w attrbutable to the (hypothetcal) gan of mmcng the adacent lower type can be weghed wth probablty Q or Q+1. In fact, any probablty Q e h contaned n Q ; Q+1 yelds ncentve compatble revenues. Henceforth, we select e Q = Q and U w 1 = 0. Ths choce maxmzes the seller s revenue for gven (G ; Q ). Snce we see the nformaton structure and mechansm that maxmzes seller s revenues, we can mae ths selecton wthout loss of generalty. Ths leads us to the followng expresson for seller s revenues: R (G ; Q ) = KX =1 w w +1 w 1 g G Q g. The assocated nterm transfers of agent satsfy the ncremental relatonshp: T +1 = T + Q Q w +1, (3)

13 and the expected revenues from agent can alternatvely be represented as " KX X # R (G ; Q ) = g Q l Q l 1 w l ; (4) wth the conventon that Q 0 = 0. =1 l=1 The revenues of the auctoneer from bdder are characterzed as a functon of the expected probablty of wnnng Q (w ) wth a value w. The nteracton wth the valuaton of the other bdders s represented by expectatons over the valuatons w. Now, we dsaggregate the expresson and consder the dependence on the realzatons of all valuatons explctly. The revenue of the auctoneer from all bdders s gven by: " XK 1 XK I IX R (G; q), q w " # 1 1 ; :::; w I I w w +1 w 1 G IY # g g ; (5) =1 1 =1 I =1 =1 where q (w) 0 and P I =1 q (w) 1. The optmal aucton s then gven by the probablty vector q (w) = (q 1 (w) ; :::; q I (w)) whch maxmzes the expected revenue (5). De ne the vrtual utlty wth dscrete types by:, w w +1 If the vrtual utltes are monotone, then the optmzaton problem can be solved pontwse,.e. for any type realzaton w = w 1 1 ; w 2 2 ; :::; w I I by solvng max fq (w)g I =1 IX =1 q w " 1 1 ; :::; w I I w w 1 g w +1 G. w 1 # G g subect only to the famlar restrcton that q (w) 0 and P I =1 q (w) 1. Ths pontwse optmzaton becomes possble as the monotoncty of vrtual utltes guarantees the monotoncty of the nterm wnnng probabltes Q as a functon of w. We can now readly descrbe some propertes of the optmal aucton. Corollary 1 Suppose the vrtual utltes are ncreasng for every agent. The optmal aucton s descrbed by: n o 1. max 1 1 ; ::::; I I > 0 ) P I =1 q w 1 1 ; :::; w I I = 1; 13 w

14 2. q w 1 1 ; :::; w I I > 0 ) 0 ^ 3. q w 1 1 ; :::; w ; :::; w I I > 0 ) 8w 0 ; 8; > w ; q w 1 1 ; :::; w0 ; :::; w I I = 1. The characterzaton s the exact dscrete type analog to the celebrated optmal aucton result for regular envronments by Myerson (1981) wth a contnuum of types. If the vrtual utltes for a gven dstrbuton functon G were not monotone, then the optmal aucton would be subect to a smlar ronng out procedure as necessary n an optmal aucton wth a contnuum of types. We conclude the secton wth a partal characterzaton of the ronng out procedure for future reference. Corollary 2 The optmal mechansm sats es for all ; +1 wth > +1 : Q = Q Vrtual Utltes We rst argue that the optmal nformaton structure wll always generate vrtual utltes whch are strctly ncreasng. Recall the basc ncentve compatblty condton for any Bayesan mplementable aucton s that the wnnng probablty Q s ncreasng n the valuaton w. The revenue formula (5) on the other hand mples that the wnnng probablty Q s ncreasng n the vrtual utlty of the agent. If the vrtual utltes generated by a gven dstrbuton G were not monotone, then the optmal aucton would be subect to an ronng out. The basc element n the former procedure s to mantan the expected probablty Q constant over a set of types whch covers the non-monotoncty n the vrtual utltes. As the constant probablty essentally mples that the ncentves and revenues are also constant on the set, the queston arses as to whether the auctoneer has any nterest n dstngushng between d erent types n ths set. In fact, as the nformaton structure s chosen by the auctoneer, he may wsh to bundle types to whch dentcal allocatons have to be o ered n any case. In other words, when the auctoneer can choose the nformaton structure for the bdders, the ronng out of non-monotonctes n the vrtual utlty may be acheved by a su cent coarsenng of the nformaton structure rather than through constant wnnng probabltes of the form: Q = Q+1. The consequence of ths argument leads to the next result. 14

15 Lemma 2 (Monotone Vrtual Utltes) The optmal vrtual utltes are strctly ncreasng. By Lemma 2, we can descrbe the set of optmal vrtual utltes for bdder by an ordered set = 1 ; :::; ; :::; K, wth 1 < 2 < ::: < K. The local argument regardng the bene ts of a coarser nformaton structure has some addtonal mplcatons for the structure of the set of vrtual utltes, say. Consder two adacent and postve vrtual utltes by agent and +1. Suppose now that these two vrtual utltes do not bracet any vrtual utlty by a compettor, or more precsely that n o < < +1 ; 2 ; 6= = ;: (6) By Corollary 1, the vrtual valuatons and +1 would then wn aganst the same type realzatons of the compettors and n turn they would receve the obect wth the same probablty: Q = Q +1. But then we can use precsely the argument of Lemma 2 to conclude that a coarser nformaton structure would ncrease the revenues of the auctoneer. Lemma 3 (Adacent and Asymmetrc Vrtual Utltes) 1. For 8; 8 < K : 2. 9; such that 6=. n o < < +1 ; 2 ; 6= 6= ;: A drect consequence of the alternatng structure of the vrtual utltes s the asymmetry of the vrtual utltes ndcated by the second part of Lemma 3. Wth two bdders, the same argument leads mmedately to a stronger result, namely that \ = ;. Wth more than two bdders, our argument does not preclude the possblty that some bdders may have vrtual utltes n common. The asymmetry of the vrtual utltes mples asymmetry of the nformaton structure even f the underlyng dstrbutons over valuatons are symmetrc. For legal or farness reasons, symmetrc treatment of bdders may be a requrement n the aucton. It s worth emphaszng that f we mpose a symmetry requrement on the nformaton structure, then 15

16 the basc propertes of the optmal symmetrc nformaton structure wll qualtatvely reman dentcal to the ones wthout the symmetry requrement Monotone Parttons A parttonal nformaton structure can be represented wthout recourse to a ont dstrbuton over the space of valuatons and sgnals by a partton of the orgnal space V. A partton s a collecton of subsets, wth slght abuse of notaton, denoted by S = S such that for all ; 0 we have S 0 \ S = ; and K[ S = V. =1 The partton s monotone f for any v ; v 0 2 S, v + (1 ) v 0 2 S for all 2 [0; 1]. Theorem 1 (Monotone Partton) 1. For every xed K < 1, an optmal nformaton structure exsts. 2. The optmal nformaton structure s a monotone partton. The result that the optmal nformaton structure s a partton as well as the monotoncty of the partton tself stem from the same elementary argument based on a necessary condton of optmalty. The argument s local n the sense that we hold the nformaton structures and condtonal wnnng probabltes of other bdders constant and loo only at the revenues to the auctoneer from bdder. The focus on the sngle agent allows us to llustrate the result wth a smple dagram, whch represents the ncentve compatble revenues from bdder : The dagram depcts the valuatons w of agent on the x-axs and the nterm probabltes Q on the y-axs. In the dagram every rectangle of surface w Q represents the gross socal surplus generated by type w wth the wnnng probablty Q determned by the optmal aucton. We showed n Secton 4.1 that the nterm ncentve compatble transfers satsfy the relatonshp: T +1 = T + Q +1 Q w +1. (7) 4 The earler example llustrates the smlarty of the optmal nformaton structure wth and wthout the symmetry requrement. 16

17 The horzontal rectangles n Fgure 2a represent the share of margnal surplus from the next hgher type whch goes to the auctoneer and the vertcal rectangle represents the share whch goes to agent. Notably absent from the dagram are the probabltes g of agent and ndeed the nterm transfer payments T are ndependent of g. >From the dagram, we can nfer several general propertes of the optmal aucton. Frst, the socal surplus s ncreasng n w and ths property s shared by the ndrect utlty of the auctoneer and the agent. Second, whle there s genune sharng of the surplus, the sharng rule s not lnear and depends on the wnnng probabltes determned by the optmal aucton. Insert Fgure 2 Here The optmalty of a gven nformaton structure requres that the auctoneer does not wsh to ntroduce further randomzaton nto the nformaton structure. A spec c and local verson of such a randomzaton can be represented as a mass preservng mxture between two adacent expected valuatons, w and w +1, whch s gven by the followng mod caton: and w (") = g " w + "w+1 g ; (8) w +1 (") = g +1 " w +1 + "w g +1, (9) for some " satsfyng, 0 < " g ; g+1. Clearly, we can nd a sgnal structure and ont dstrbuton to generate the expected valuatons for every ". The e ect of a postve " s depcted n Fgure 2b. It ncreases w and the margnal revenue from type, but decreases w +1 and lewse the margnal revenue from type + 1. By mxng, we understand here that we assocate (va the sgnals) low true valuatons wth hgh expected valuatons, and conversely hgh true valuatons wth low expected valuatons. Suppose now that the optmal nformaton structure (and aucton) requres " = 0. In consequence an ncrease n " would decrease the revenues. Wth the local changes as suggested by (8) and (9) the margnal revenue as a functon of " s lnear as can be mmedately nferred from the ncentve compatble revenue representaton: " KX X # g Q l Q l 1 w l, =1 l=1 17

18 as we eep the condtonal probabltes Q and type probabltes g constant. The argument for a monotone partton s now based on the followng dea. Suppose an optmal nformaton structure s not a monotone partton. Then by the rst-order condtons further mxng would decrease the revenues. But the same condtons also allow us to nfer the converse. Further de-mxng would ncrease the revenues. As every nformaton structure whch s not a monotone partton presents the possblty of some de-mxng between at least two adacent types, ths demonstrates the optmalty of a monotone partton. 5 Optmal Informaton Structure wthout Fnte Sgnals So far we have obtaned a number of qualtatve results for optmal nformaton structures when the sgnal space of each agent contaned at most K elements. In ths secton we establsh that the optmal nformaton structure s ndeed nte and monotone partton n the class of all measurable nformaton structures as de ned n Secton 2. Consder any ncentve compatble mechansm (q; t) and the dstrbuton G over expected valuatons nduced by any arbtrary nformaton structure. Types n the dstrbuton G can have zero densty, postve densty, or postve probablty. Proposton 1 (Approxmaton) Let G be a dstrbuton generated by an arbtrary nformaton structure S and let (q; t) be an ncentve compatble mechansm. For any " > 0 there exsts a dstrbuton functon b G generated by a nte nformaton structure b S and an ncentve compatble mechansm bq; bt such that R bg; bq R (G; q) " Proposton 1 establshes that the set of revenues generated by any ncentve compatble mechansm wth a nte nformaton structure s dense n the set of revenues generated by ncentve compatble mechansm wth an arbtrary nformaton structure. Hence the ncentve compatble revenues generated by an arbtrary nformaton structure can be approxmated arbtrarly well by a nte nformaton structure. 5 5 We would le to than an anonymous referee for suggestng ths contnuty result. 18

19 A smlar local varaton allows us to establsh an mportant property of the condtonal wnnng probabltes Q. Ths property wll play a central role n the argument to demonstrate that a nte nformaton structure s optmal. Lemma 4 (Increasng D erences) The condtonal wnnng probabltes Q satsfy strctly ncreasng d erences. Lemma 4 establshes that the condtonal wnnng probabltes of any bdder have the property of strctly ncreasng d erences, or that Q +1 Q > Q Q 1. Henceforth we shall refer to ths property for smplcty as the convexty of the condtonal wnnng probabltes even though they are de ned over a nte set of ndces. 6 The proof of the above Lemma reles agan on a local argument as we examne the revenue from bdder only. Theorem 2 (Exstence) An optmal nformaton structure n the class of all Borel measurable nformaton structures exsts and t s a nte monotone partton. The proof of Theorem 2 proceeds n three steps: (1) an optmal nformaton structure and assocated revenues exst n the class of nte nformaton structures; (2) the revenues from the optmal nte mechansm are maxmal n the class of all dscrete (possbly non- nte) nformaton structures; and (3) the nte nformaton structure revenues are also maxmal n the class of all measurable nformaton structures. Theorem 2 bulds mmedately on our earler results: By Proposton 1, we can restrct attenton to sequences of mechansms wth nte nformaton structures nstead of arbtrary nformaton structures. By Theorem 1, for every nte K a soluton exsts. If we consder any sequence, then by 6 We chose to rst establsh propertes of the vrtual utltes n Lemma 2 and 3, and then use these propertes to derve the partton property of the nformaton structure and ncreasng d erences of the wnnng probabltes. Alternatvely, we could start by establshng the ncreasng d erence property and then proceed to vrtual utltes and the partton property. 19

20 Lemma 4 any element of the sequence must have convex condtonal wnnng probabltes for every bdder. It follows that the lmtng values have to be convex as well. Consder now the lmt of the condtonal wnnng probablty of the type of agent wth the lowest strctly postve vrtual utlty for every nte K. If the lmt Q 1 s postve, then by Lemma 4 we can conclude that at most 1=Q 1 sgnals can have a postve probablty. In partcular, the convexty of the condtonal wnnng probablty allows us to assert that the condtonal wnnng probabltes of agent have to satsfy for all and Q Q 1 Q 1, and thus the optmal nformaton structure has to be nte. If the lmt Q 1 s zero, then the argument s a lttle more subtle. Essentally, we use the fact of Q 1 = 0 to show that there exsts at least one agent whose lowest type has strctly postve probablty g 1 and strctly postve probablty of wnnng Q 1. Ths n turn allows us to show that at most a nte number of types of agent can wn and wn n partcular aganst w 1. We are thus lead to conclude that the optmal nformaton structure exsts, s nte, and by Theorem 1 t has to be a monotone partton. The ont optmalty of dscrete nformaton structures and convex wnnng probabltes s now llustrated usng the followng nformal reasonng, based on well-nown results for contnuous rather than dscrete types. Consder agan the nterm problem wth a sngle bdder. The socal surplus from type v s gven by v Q (v ). We now that the margnal ndrect utlty of type v n an ncentve compatble mechansm s Q (v ). The resdual margnal gans v Q 0 (v ) consequently belong to the auctoneer. It further follows that the ndrect utlty of the agent s convex as Q (v ) s ncreasng and that the socal surplus as well as the auctoneer s surplus s convex f Q (v ) s not too concave. The auctoneer receves from agent the expected revenue gven by: Z 1 Z v 0 0 r Q 0 (r ) dr df (v ) : (10) The sngle bdder scenaro suppresses the decson as to how large Q (v ) should be. Ths wll be naturally determned by the opportunty cost stemmng from allocatng the obect to the competng bdders. We now pursue the followng thought experment. Suppose 20

21 the auctoneer had decded to gve a small nterval of types, say [v ; v ] a xed aggregate probablty, say b Q, wth Z v v Q (v ) df (v ) = b Q; (11) and all he had to decde s how to allocate ths total probablty nsde the nterval. further he were only concerned wth maxmzng the surplus that he can extract from all hgher types, then he should pursue the followng obectve functon: Z v max v Q 0 (v ) df (v ) ; Q (v ) v subect to the constrant (11), to maxmze the ntegral of margnal gans. As the margnal ncrement n Q (v ) s weghed by v, t s then easly seen that the auctoneer would ndeed le to choose a very convex functon for the wnnng probabltes as the margnal ncreases Q 0 (v ) would then receve the largest possble weght. However ths exclusve concern wth the margnal revenue s mtgated n the standard optmal aucton by the concern for the nframargnal revenue as represented by the complete revenue functon (10). However, by controllng the nformaton structure we can bundle types together to (locally) elmnate the nframargnal concern and pursue only the maxmzaton of the margnal revenues. If 6 Dscusson Ths paper reconsdered the desgn of the optmal aucton by mang the nformaton structure an ntegral part of the desgn problem. Notable features of the optmal nformaton structure were the parttonal character, the nteness of the partton and therefore of prvate types as well, and the asymmetry of the nformaton structures. The analyss reveals an mportant trade-o between the mnmzaton of nformaton rent and the maxmzaton of allocatonal e cency. The optmal nformaton structure balances these two con ctng obectves. We would le to emphasze that the current results may not nform us drectly about aucton desgn n practce. 7 Whle we expect the trade-o between nformaton rent and 7 There are many auctons n whch the precson of the nformaton avalable to the buyers s at least partally controlled by the seller. In US o shore wldcat ol tract auctons, the bddng rms are permtted to 21

22 allocatonal e cency to reman mportant, the current analyss maes a number of assumptons whch would have to be weaened to provde a better t wth emprcal observatons. We bre y dscuss the restrctons mposed by the three ey assumptons of the model: () the seller has complete control over the precson of each bdder s sgnal, () each bdder s ntally unnformed and () the seller cannot prce the nformaton drectly (through ex ante payments). In the model, the seller s free to choose from the set of all nformaton structures, and n partcular, the seller can choose to leave the bdder unnformed about her true valuaton. However, n practce the seller may be severely constraned n hs choce of the nformaton structure. For example, the nformaton structure mght be restrcted to a nosy samplng process as n the o shore ol tract auctons, where the choce of nformatveness s determned by the number of samples. In addton, each bdder may have some prvate nformaton and thus leavng the bdder unnformed may not be a feasble nformaton structure. In our analyss, the seller o ers allocatons and prces only after each bdder has receved her prvate sgnal. In partcular, the seller cannot prce the nformaton structure drectly. Ths assumpton mght be ust ed n lght of the observable lac of drect prcng of nformaton n auctons, as n the auctons mentoned n the above footnote. Yet, from a theoretcal pont of vew there mght be a tenson between the ablty to control the nformaton structure and an nablty to prce the nformaton structure. In fact, Eso & Szentes (2007) and Gershov (2002) consder a smlar settng to the one presented here, but allow the seller to prce the nformaton. Gershov (2002) shows that the optmal soluton then conssts of partcpaton fees equal to the expected bdders rent followed by a gather nformaton about the lease value and ther drllng costs pror to the sale usng sesmc nformaton, but no on-ste drllng s allowed. In contrast, n US o shore dranage ol leases, some bdders are ntentonally gven access to superor nformaton by allowng them pror drllng n the area, see Porter (1995). Smlarly, Genesove (1993) reports that n wholesale used car auctons, d erent auctoneers adopt strngly d erent rules as to how potental bdders may nspect a used car before they place a bd on t. Auctons n whch the seller ntentonally lmts the amount of nformaton are sometmes referred to as blnd auctons and documented examples are the lcensng procedure for moton pctures, see Kenney & Klen (1983) and Blumenthal (1988), and the competton of broers for the trade of a large portfolo on behalf of an nsttutonal asset manager, see Kavaecz & Kem (2005) and Foucault & Lovo (2003). 22

23 standard Vcrey aucton. In Eso & Szentes (2007) each bdder receves an ntal prvate sgnal and possbly a second sgnal that can be released by the seller later on. They show that the seller can extract the rent assocated wth the sgnal released by the seller, but cannot extract the rent assocated wth the ntal prvate sgnal. The emprcal absence of a prce for nformaton n auctons suggests that addtonal factors mght be at wor. In a rcher envronment, the optmal nformaton structure wll then have to ncorporate these factors. The basc trade-o analyzed here would then be augmented, but also rendered more complex by the nature of the constrants. 23

24 7 Appendx The appendx contans the proofs to all lemmata, propostons and theorems n the text. Proof of Lemma 1. The proof conssts of two arguments: Frst, we establsh a bound on the utlty d erence of two adacent types, w 1 and w as a functon of the expected probablty of wnnng Q 1 and Q. Second, we use the bound repeatedly to obtan an expresson for the expected transfer payment and thus revenue from bdder. Along the way we shall show that the expected probablty of wnnng Q () s non-decreasng. Incentve compatblty requres that the allocaton fq (w ) ; T (w )g sats es the nterm ncentve and partcpaton constrants. The ncentve constrant for a bdder w mmcng a bdder wth expected valuaton w 1 Smlarly, for bdder w 1 yelds: U U w yelds: = w Q T w Q 1 T 1 : (12) who consders mmcng a bdder wth expected valuaton w w 1 = w 1 Q 1 T 1 w 1 Q T : (13) Now, subtractng (13) from (12) yelds the followng set of nequaltes: w w 1 Q U w U w 1 w w 1 Q 1 ; (14) whch gves bounds on the utlty d erence of two adacent types, w 1 functon of the expected probablty of wnnng Q 1 nequalty n (14) requres that: w w 1 Q Q 1 and w, as a and Q. We observe that the outer 0; whch mples that Q () s non-decreasng. Observe also that the nterm partcpaton constrant mples that U w 1 0. Next, we repeatedly apply the nequalty n (14) to obtan an expresson for the expected transfer payment and ultmately the revenue expresson (2). An ndetermnacy arses as the utlty gan based on mmcng the adacent lower type can be weghed wth the left or the 24

25 rght-hand sde probablty, or any number n between. Accountng for ths ndetermnacy, the expresson for the equlbrum utlty equals: U w = U w 1 X + w l w l 1 eq, where Q e h l 1 2 Q l 1 ; Q l accounts for the ndetermnacy. By de nton, T, and the expresson for the expected transfer payment s gven by: U w l=2 = w Q T w = w Q U w 1 X l=2 w l w l 1 eq l 1, whch taes agan the ndetermnacy nto account. The seller s revenues are obtaned by the equvalent of ntegraton by parts for the dscrete probabltes. Dong so, leads to the formula: R (G ; Q ) = " KX w =1 w +1 w 1 g G eq Q # Q g U w 1. Proof of Corollary 1. The characterzaton follows mmedately from pontwse optmzaton of the obectve functon (5) for any realzaton of values w = w 1 1 ; :::; w I I. Proof of Corollary 2. Suppose to the contrary (and by Lemma 1) that Q < Q+1. Then there must exst w such that q w ; w < q w +1 ; w. The ncentve compatblty condtons of all agents except, and n partcular ther condtonal wnnng probabltes reman constant under q () and a mod ed probablty assgnment bq () as long as g q w ; w + g +1 q w +1 ; w = g bq w ; w + g +1 bq w +1 ; w : (15) By the hypothess of > +1, any ^q () such that (15) s mantaned and dsplays q w ; w < bq w ; w must strctly ncrease the revenues of the auctoneer, whch delvers the contradcton. Proof of Lemma 2. Suppose to the contrary and hence that there exsts and +1 that +1. Suppose ntally that ndeed > +1. Then by Corollary 2, t follows 25 such

26 that Q = Q+1. In contrast, consder the revenues from agent f the orgnal nformaton structure were mod ed by sendng a sngle sgnal bs whenever the orgnal nformaton structure emtted the sgnal s or s +1. The so mod ed nformaton structure e ectvely ons the types w and w+1 created type s gven by and ts condtonal expected value s: nto a sngle type, denoted by bw. The probablty of the newly bg = g + g +1 ; g +1 bw = w g + w+1 g + g+1 The d erence n the revenue between the orgnal and mod ed nformaton structure s gven, after some ntal cancellatons, by h R bg ; Q R (G ; Q ) = g bq Q 1 bw h +g +1 bq Q 1 bw Q Q 1 w 8 h KX < bq Q 1 + g l bw + h : l=+2 Q Q 1 w + Q +1 Q The combnaton of w and w +1 : Q Q 1 w Q +1 Q +2 Q w +1 + w +1 bq w +2 Q +2 Q +1 w +2 9 = ; : a ects only the revenue from all types startng at. By constructon, the condtonal wnnng probablty of the new type sats es b Q = Q = Q+1, and thus the d erence smpl es to R bg ; Q R (G ; Q ) = g +g +1 Q +1 Q 1 bw w Q Q 1 KX + but by hypothess, w +1 > w, and hence bw w l=+2 g l bw w n bq Q 1 bw w o ; > 0, and thus each of the three terms are postve, yeldng the desred result. Fnally, n the case that = +1, there are several optmal soluton for Q and Q +1, but snce Q = Q+1 s always guaranteed to be one of them, the same argument goes through for the case of = +1. Proof of Lemma 3. (1.) Suppose to the contrary. Then there exst such that n < < o ; 6= = ;.

27 Observe next that f two adacent vrtual utltes belong to bdder then the probablty of recevng the good has to be dentcal on both ntervals, Q = Q +1 by Lemma 2 and Corollary 1. But by the same argument as Lemma 2, we may then on the mass ponts w and w +1 and the expected revenues for the auctoneer wll strctly ncrease, a contradcton. (2.) Suppose to the contrary and thus = for all ;. Then there exsts an optmal aucton such that for some and some, Q = Q+1. We can now appeal to the same argument as n (1.) to conclude that the revenues of the auctoneers can be strctly ncreased by onng the mass ponts w and w+1, whch destroys the symmetry n the vrtual utltes. Proof of Theorem 1. We rst establsh that there s always a monotone partton whch acheves strctly hgher revenues than any other nte nformaton structure. We then argue that a optmal monotone partton exsts. (2.) A necessary condton for an optmal nformaton structure s that gven the type probabltes g and the wnnng probabltes Q, the auctoneer does not wsh to ntroduce further randomzaton nto the nformaton structure. A local verson of such a randomzaton s a mass preservng mxture between w and w+1. If w and w+1 are canddate types, then a local randomzaton between these two types s gven by the followng mod caton: and w ("), g " w + "w+1 g ; (16) w +1 ("), g +1 " w +1 + "w g +1, (17) for some " satsfyng, 0 < " g ; g+1. We denote the revenue resultng from the mod caton as a functon of " by R (" G ; Q ) for gven G and Q. A necessary condton for the optmalty of the nformaton structure s R 0 (0 G ; Q ) 0. (18) The functon R (" G ; Q ) s lnear n " and the dervatve R 0 (" G ; Q ) can be wrtten as: w +1 w Q Q 1 g +1 1 G 1 g g+1 27 Q +1 Q R 0 (" G ; Q ) = (19) g 1 G 0:

28 By hypothess, w +1 w > 0 and g g+1 R 0 (" G ; Q ) 0, > 0, and t follows that: Q Q 1 g 1 G. (20) Q +1 Q g +1 1 G 1 Next we argue that n fact the necessary condton for optmalty has to be Q R 0 Q 1 (" G ; Q ) < 0, < g 1 G. (21) Q +1 Q g +1 1 G 1 The argument s by contradcton and thus suppose that R 0 (") = 0 over the entre range of ". An mmedate mplcaton s that the auctoneer would then be nd erent between facng types w and w +1 and all convex combnatons represented by (16) and (17). But consder the vrtual utltes of these two types, whch are gven by: and As " ncreases w (") = w (") +1 (") = w +1 (") 1 w +1 (") w (") w +2 w +1 (") g G ; (22) 1 G +1. (23) g +1 (") approaches w+1 (") and n consequence, eventually +1 (") < ("). But by Lemma 2, every nformaton structure wth non-monotone vrtual utltes s strctly domnated by one wth monotone ncreasng vrtual utltes, and hence we have the contradcton. It follows that (21) s a necessary condton for optmalty. We argue now that every nformaton structure whch s not a monotone partton necessarly fals to satsfy condton (21). Suppose therefore that at least one agent has an nformaton structure whch s not a monotone partton. It follows that there must be two adacent expected valuatons w and w +1, where we recall that: R 1 w 0 = v df v s R 1 0 df ; v s and an x 2 (0; 1) such that lower and upper segment of each condtonal dstrbuton has strctly postve probablty, or: Z x 0 df v s ; Z 1 x df v s ; Z x 0 s +1 df v ; 28 Z 1 x s +1 df v > 0.