Robustly Optimal Auctions with Unknown Resale Opportunities

Transcription

1 Robustly Optmal Auctons wth Unknown Resale Opportuntes Gabrel Carroll Ilya Segal Department of Economcs, Stanford Unversty, Stanford, CA August 22, 2016 Abstract We study robust revenue maxmzaton by the desgner of a sngleobject aucton who has Bayesan belefs about bdders ndependent prvate values but s gnorant about post-aucton resale opportuntes (ncludng possble leakage of prvate nformaton). We show the optmalty of a Vckrey aucton wth bdder-spec c reserve prces proposed by Ausubel and Cramton (2004), whch allocates the object e cently provded that at least one of the bdders has bd above hs reserve prce. In ths aucton, truthful bddng and no resale s an ex post equlbrum for any ndvdually ratonal resale procedure. We show optmalty of ths aucton for a worst-case resale procedure n whch the hghest-value bdder learns all other bdders values and has full barganng power. The proof nvolves constructon of Lagrange multplers on the ncentve constrants representng non-local devatons n whch a bdder underbds to lose and then purchases from the aucton s wnner. We are grateful to Peter Cramton, Paul Mlgrom, Glen Weyl, and partcpants of the 2016 Stony Brook Workshop on Complex Auctons and Practce for helpful comments. The authors thank the Smons Insttute for ts hosptalty durng the Fall 2015 Economcs and Computaton Program. Segal acknowledges the support of the Natonal Scence Foundaton grant (SES ). Alex Bloedel provded valuable research assstance. 1

2 1 Introducton It s well known that revenue-maxmzng auctons for settngs wth a pror asymmetrc bdders mplement ne cent allocatons that are based n favor of weaker bdders (Myerson 1981, McAfee and McMllan 1989). On the other hand, real-lfe auctons are often followed by resale. One mght wonder whether the advantage of based auctons s undermned due to a strong bdder s ablty to buy the object from another bdder n resale at a possbly lower prce than what he would have to pay n the aucton. Ths paper examnes what auctons are optmal when resale s possble but the desgner s gnorant about the resale procedure. 1 There exsts a substantal lterature on the desgn of optmal auctons when resale procedures are known, ncludng Zheng (2002), Calzolar and Pavan (2006), and Dworczak (2015). The usual assumpton n ths lterature s that no prvate nformaton s leaked before resale (other than nformaton about aucton bds that the desgner may choose to dsclose). It s easy to see that under ths assumpton, the possblty of resale can only reduce the auctoneer s optmal revenues: ndeed, the equlbrum allocaton of any aucton mechansm followed by resale must be ncentve compatble and therefore would be feasble for the desgner f resale were mpossble. Nevertheless, n spte of the above observaton, t has been found that the optmal aucton n the presence of known resale typcally mplements a based allocaton followed by resale. 2 On the other hand, Ausubel and Cramton (2004) proposed a class of auctons, called Vckrey auctons wth reserve prces, whch nduce an expost Nash equlbrum wth truthful bddng and no resale. These auctons allocate the object e cently provded that at least one of the bdders has beaten hs reserve prce. Whle these auctons have nce propertes, n lght of the lterature mentoned above, t has been unclear why they would be optmal for the auctoneer. Ausubel and Cramton (1999, Theorem 4) attempted to derve the op- 1 For expostonal smplcty we focus on revenue maxmzaton, but the analyss extends to the auctoneer s pro t maxmzaton when she has a non-zero value for keepng the object, by de nng bdders values to be net of the auctoneer s value. 2 In a smple example due to Zheng (2002), f a bdder s known to have a zero value for the good and the ablty to desgn a revenue-maxmzng resale mechansm, the desgner would optmally sell to ths bdder at the prce equal to the optmal expected revenue wthout resale. 2

3 tmalty of Vckrey auctons wth reserves from the assumpton of e cent resale. However, the foundatons for ths assumpton are unclear. By the theorem of Myerson and Satterthwate (1983), we would generally expect resale to be ne cent unless the partes prvate values are revealed before the resale. However, f resale takes place under symmetrc nformaton, then typcally, an auctoneer who can antcpate the resale procedure would not want to allocate the object e cently; nstead she would agan want to run an aucton that nduces resale n equlbrum. For an extreme example, suppose that the auctoneer knows that one partcular bdder has a zero value, but wll have full nformaton and full barganng power n resale. Then the optmal aucton would sell to that bdder at the prce equal to the expected maxmal bdder value for the object, lettng the auctoneer extract full surplus. Ths paper provdes the mssng foundatons. We derve the robust optmalty of sngle-object Vckrey auctons wth reserves when the desgner s gnorant about the resale procedure (ncludng possble prvate nformaton revelaton before resale). Namely, we show that such auctons maxmze the desgner s worst-case expected revenue, where the expectaton s taken over buyers ndependent prvate values and the worst case s over the possble resale procedures. Our concluson s conceptually smlar to the robust optmalty of strategy-proof auctons when the desgner s gnorant about bdders belefs about each other s values (Chung and Ely 2007) or about each other s strateges (Yamashta 2015). Namely, whle n those cases revenue maxmzaton that s robust to bdder strategzng makes t optmal to use strategy-proof mechansms, n our case revenue maxmzaton that s robust to resale makes t optmal to use resale-proof mechansms. However, the proof technques are qute d erent. Also, observe that Vckrey auctons wth reserve prces are robust to bdders belefs about each other s values and ther belefs about the resale procedure, hence we obtan those addtonal robustness bene ts for free. We begn wth the smple case n whch the auctoneer s restrcted to always sell the object (Secton 3). For ths case, we show that the smple Vckrey aucton (second-prce sealed-bd aucton) wth no reserve prce s optmal. To do ths, we guess a worst-case resale procedure, n whch the hghest-value bdder learns other bdders values and has full barganng power n resale. Wth ths resale procedure, n any aucton that always sells the object, the hghest-value bdder would be able to extract at least hs margnal contrbuton to the total surplus by bddng low to let another bdder wn and then buyng from the wnner. Gven that the desgner s 3

4 unable to reduce bdders nformaton rents below ther expected margnal contrbutons to the total surplus, she could do no better than the smple Vckrey aucton wth no reserve. Snce ths aucton sustans truthful bddng as an ex post equlbrum under any resale procedure, t s robustly optmal wth unknown resale. We proceed to the more complex settng n whch the desgner can wthhold the object, for smplcty startng wth the two-bdder case (Secton 4). We contnue wth the worst-case resale procedure guessed n Secton 3, and derve bdders reduced-form utltes from aucton allocatons under ths resale procedure. Note that these reduced-form utltes exhbt both externaltes and nterdependent values, snce a bdder who does not wn cares whether the other bdder wns and, f so, what the other bdder s value s. Nevertheless, the usual envelope-theorem approach to local ( rst-order) ncentve compatblty constrants yelds a smple expresson for bdders nformaton rents, whch allows us to express the expected revenue as the expectaton of an approprately de ned vrtual surplus. If we were to gnore all other ncentve constrants and solve the resultng relaxed problem by maxmzng the vrtual surplus state-by-state, the soluton would always allocate the object between the bdders e cently. Intutvely, sellng to the ne cent bdder would yeld nformaton rents both to hm and to the e cent bdder (who would buy t n resale), whch s domnated by sellng to the e cent bdder, elmnatng the ne cent bdder s nformaton rents. The soluton to the relaxed problem sells to the e cent bdder f and only f hs value exceeds the optmal reserve prce for hm. Unfortunately, the soluton to the relaxed problem volates non-local ncentve constrants: a reducton n the bd of the strong bdder would sometmes gve the object to the weak bdder rather than leave t unsold, gvng the strong bdder an ncentve to underbd and then buy n resale. We guess that the soluton to the auctoneer s full problem s the Vckrey aucton wth reserves descrbed by Ausubel and Cramton (1999, 2004). To establsh that ths s ndeed a soluton, we construct Lagrange multplers on non-local downward ncentve constrants such that maxmzaton of the Lagrangan yelds the soluton. Snce there s an ncentve constrant for each type and each possble msreport, and a double contnuum of such ncentve constrants s bndng, our Lagrange multplers are de ned by a measure over ths double contnuum. In the two-bdder case, we construct a product measure that works for our purpose: maxmzaton of the Lagrangan (whch can be wrtten as the expectaton of a mod ed vrtual surplus, whch can 4

5 be maxmzed state-by-state) yelds the soluton. Fnally, n Secton 5 we extend the approach to the case of many bdders. Some addtonal complcatons arse because t becomes necessary to consder bndng non-local downward ncentve constrants both from a gven type (when ths type s the hghest-value bdder but may underreport to buy from another bdder n resale) and to the same type (when ths type s reported by some hgher type so as to concede the object to another bdder and buy t from hm n resale). Ths necesstates a somewhat more complex constructon of Lagrange multplers. Under tradtonal regularty assumptons on value dstrbutons we can construct nonnegatve Lagrange multplers that yeld a Vckrey aucton wth reserves as an optmal aucton. Snce ths aucton sustans truthful bddng as an ex post equlbrum under any resale procedure, t s robustly optmal wth unknown resale. Our approach also yelds an teratve constructon of the optmal bdderspec c reserve prces n n-bdder Vckrey auctons wth reserves. We llustrate ths (n Secton 6) for the case where bdders values are dstrbuted unformly wth d erent upper lmts. In ths case, the optmal reserve prce to the kth bdder s obtaned by solvng a kth-degree equaton, whch cannot be done analytcally for k > 4 but s easly done numercally. 2 Setup There are n bdders. Bdder s value for the object s, whch s dstrbuted accordng to a c.d.f. F wth a contnuous strctly postve densty f on support [0; 1]. 3 1 F We wrte ( ) = ( ) f ( for the tradtonal vrtual value ) of type. Values are ndependent across bdders. We wrte = ( 1 ; : : : ; n ) for the pro le of values. The space of (possbly randomzed) allocatons s X = fx 2 [0; 1] n : P x 1g, where x 2 [0; 1] s the probablty of allocatng the object to bdder. To descrbe the general space of mechansms wth resale, n general we need to thnk about how the mechansm n uences resale. For example, the nformaton dsclosed by the mechansm wll n general a ect the outcome of resale. However, for smplcty we now restrct attenton to resale proce- 3 The assumpton that the dstrbutons have a common support s made for expostonal smplcty: n Secton 6 below we argue that the results extend to cases n whch the supports upper lmts d er. The assumpton of contnuous densty s also made for expostonal smplcty, but t wll follow from assumpton (A1) made n Secton 5 below. 5

6 dures n whch all prvate nformaton s revealed before resale (but after the aucton), and the resale outcome depends only on the allocaton spec ed by the mechansm but not on any other features of the mechansm. For such resale procedures, the expected post-resale payo of bdder net of payment spec ed n the mechansm can be wrtten as a reduced-form functon v (x;) of the allocaton x 2 X spec ed by the mechansm and the bdders value pro le, wth the total reduced-form payo s not exceedng the maxmal total surplus avalable n resale: P v (x;) max P x : (1) We also requre the resale procedure to be ndvdually ratonal: v (x; ) x (2) for each. We wll specfy a worst-case resale procedure of ths form, but the optmal mechansms we dentfy wll prove to be robust to a broader class of resale procedures (descrbed n Secton 7 below). If the desgner knows the resale procedure and t s descrbed by reducedform resale payo s, then we can appeal to the revelaton prncple and focus on mechansms where each bdder drectly reports. Thus an aucton s a par of measurable functons (; ), where : [0; 1] n! X spec es the (possbly probablstc) allocaton rule; : [0; 1] n! R n spec es the payments. An aucton must satsfy the usual ncentve compatblty and ndvdual ratonalty constrants: E ~ [v (( ; ~ ); ; ~ ) ( ; ~ )] E ~ [v ((^ ; ~ ); ; ~ ) (^ ; ~ )] for all ; ^ ; (3) E ~ [v (( ; ~ ); ; ~ ) ( ; ~ )] 0 for all : (4) The seller s expected revenue maxmzaton problem s then # max :[0; 1] n! X :[0; 1] n! R n E ~ " X ( ~ ) subject to (3)-(4). (5) 6

7 3 The Must-Sell Case In ths secton, we restrct attenton to auctons that must sell the object wth probablty 1. (For example, ths could be motvated by the seller s prohbtvely hgh cost of keepng the object.) Recall from Myerson (1981) and McAfee and McMllan (1989) that f there s no resale, and f each bdder s vrtual value functon s ncreasng, the optmal aucton allocates the object to the bdder wth the hghest vrtual value ( ). Thus, f bdders have d erent dstrbutons and therefore d erent vrtual value functons, the optmal aucton msallocates the object. In partcular, the aucton s based towards weaker bdders, whch are those wth hgher vrtual value functons. Intutvely, optmal msallocaton trades o reducton of bdders nformaton rents aganst the reducton of the expected total surplus. Now we turn to the analyss of auctons wth resale, and make a guess about a worst-case resale procedure: that the hghest-value bdder learns the values of everybody else and has full barganng power. Let () denote the bdder wth the hghest value at pro le, breakng tes n favor of earlernumbered bdders (the choce of tebreak s nconsequental). Then, each bdder s reduced-form payo s x v (x; ) = + P j6= ( j ) x j f = (); x otherwse. (6) Intutvely, ths s a worst-case resale procedure because t gves bdders maxmal nformaton rents: by lettng another bdder wn and then buyng from the wnner whenever ths s e cent, bdder would get an expected nformaton rent equaln to at least hs expected o margnal contrbuton to the total surplus, E ~ hmax ~ max j6= ~ j ; 0. 4 Gven ths, the expected revenue s maxmzed by the Vckrey aucton wth no reserve, whch acheves the lower bound on bdders nformaton rents and at the same tme maxmzes total surplus wthout any resale. To complete the argument we only need to establsh that a bdder can always submt a bd that lets another bdder wn wth probablty 1. Note 4 Note that the same argument would work f the hghest-value bdder = (), when he fals to wn the object, would buy t back from the aucton s wnner at the prce equal to the second-hghest value max j6= j. Ths would be the approprate model f the resale procedure nvolved Bertrand competton among the aucton s losers to buy the object from the wnner. 7

8 that bddng 0 may not accomplsh ths because the seller only needs to sell wth probablty 1, and so can wthhold the object f any bdder bds 0. Yet the argument can be mod ed to obtan the desred result: Proposton 1. If the hghest bdder learns all other values and the resale payo s are gven by (6) then the Vckrey aucton wth no reserve prce s optmal among all auctons that sell wth probablty 1. Proof. Take any aucton mechansm (; ) satsfyng (3)-(4), and let U ( ) = E ~ [v (( ; ~ ); ; ~ ) ( ; ~ )] (7) be P the nterm expected payo enjoyed by when hs type s. Snce j j( ~ ) = 1 wth probablty 1, there exsts arbtrarly small ^ > 0 such that P j j(^ ; ~ ) = 1 wth probablty 1. By ncentve compatblty (3), for all, U ( ) E ~ [v ((^ ; ~ ); ) (^ ; ~ )] E ~ max max ~ j ; 0 ^ ; j6= snce v (x; ) max f max j6= j ; 0g whenever P j x j() = 1 and E ~ [ (^ ; ~ )] ^ by ndvdual ratonalty (4). Therefore, we can wrte " # " # X X E ~ ( ~ ) = E ~ v (( ~ ); ~ X h ) E ~ U ~ h X E ~ ~ ( ~ ) E ~ max ~ max ~ j ; 0 + X j6= Snce the rst two terms descrbe the expected revenue n the Vckrey aucton, and ^ > 0 can be chosen to be arbtrarly small, the result obtans. ^ : Next, note that the Vckrey aucton sustans truthful bddng and no resale as an ex post equlbrum for any resale procedure. Indeed, note when bdder devates downward and loses, he cannot buy the object n resale for below the wnner s value, max j6= j, whch s what he would pay for wnnng the object n an aucton, and conversely when bdder devates upward and wns, he cannot resell the object for more than the hghest loser s value, 8

9 max j6= j, whch s the prce he would have to pay to wn the aucton. In both cases he does no better than bddng hs true value. (A more general verson of ths result, nspred by Ausubel and Cramton (2004), s stated n Secton 7 below.) Ths shows that the Vckrey aucton attans the same expected revenue n equlbrum (namely, the expectaton of the second-hghest value) regardless of the resale procedure; and no other aucton can guarantee hgher expected revenue, by Proposton 1. Thus, the Vckrey aucton solves the seller s maxmn problem: t s a robustly optmal aucton wth unknown resale. 5 4 Can-keep case: Two bdders We now turn to our man focus: the optmal aucton when the seller can wthhold the object. In ths secton we manly study the problem wth two bdders, although some of the ntermedate deas we develop are useful for any number of bdders, and wll be stated n ths generalty. In the next secton, we fully extend the result to any number of bdders. As n the prevous secton, we conjecture that the worst-case resale procedure has the followng form: after the aucton, nature reveals all the bdders values to each other; and the hghest-value bdder gets all the barganng power n resale, whch yelds reduced-form resale payo s gven by (6). So we can set up the mechansm desgn problem assumng ths spec c resale procedure, and solve for the optmal mechansm. It wll then turn out that, for ths mechansm, truthtellng (followed by no resale) s an ex-post equlbrum regardless of the resale procedure. Therefore, the mechansm we derve actually solves the robust optmzaton problem. 4.1 Analyss: the relaxed problem We begn by studyng the relaxed problem for program (5) subject to (3) (4), whch replaces (3) wth the local rst-order ncentve compatblty constrants. For ths purpose, note that by the envelope theorem of Mlgrom and Segal (2002, Corollary 1), ncentve compatblty (3) mples that U s 5 The argument above can be extended to mult-unt auctons. It can also be extended to auctons wth correlated values, provded that we also requre robustness to nformaton dsclosure before the aucton. Thus, the proposton above can be stated for all such cases. 9

10 absolutely contnuous and ts dervatve s gven almost everywhere by U 0 ( ) = E ~ [v 0 (( ; ~ ); ; ~ )]: Here v 0 denotes the dervatve of v wth respect to, whch s de ned except when there are tes n values (a probablty-zero case), and takes the form v(x; 0 ) = x + 1 = () P j6= x j. Snce U (0) = 0 at the optmal mechansm (type 0 s partcpaton constrants are bndng), the nterm expected payment of bdder gven type s E ~ [ ( ; ~ )] = E ~ [v (( ; ~ ); ; ~ )] Z 0 E ~ hv((^ 0 ; ~ ); ^ ; ~ ) d^ : (8) The usual ntegraton by parts allows us to rewrte the objectve (5) as " # " # X X E ~ ( ~ ) = E ~ v (( ~ ); ~ X 1 F ( ) ) v 0 f ( ) (( ~ ); ~ ) : (9) The standard next step s to solve a relaxed problem: maxmze (9) over all allocaton rules, wthout worryng about (3). Ths can be done by maxmzng the vrtual surplus (the expresson nsde brackets) pontwse. The soluton must be a determnstc mechansm (wth probablty 1). At any pro le, f we allocate to the hgh-value bdder = (), the vrtual surplus s 1 F ( ) = ( ): f ( ) If we allocate to some other bdder j 6=, then the v 0 terms are nonzero both for = and for = j, so we get 1 F ( ) f ( ) 1 F j ( j ) : f j ( j ) Ths s less than the vrtual surplus from allocatng to. Hence, t s never optmal to allocate ne cently: we ether allocate to the hghest-value bdder or not at all. Intutvely, allocatng to some non-hgh-value bdder j at a type pro le concedes nformatonal rents to hgher types of j (who can acqure the good by pretendng to be j, and then possbly resell t) as well as for hgher types of (who can buy the good back from j); whereas allocatng 10

11 Fgure 1: Allocaton rule from relaxed problem. 1 means allocate to bdder 1; 2 means allocate to bdder 2. In the remanng regons, the good s not sold. to the hgh-value bdder leaves rents to hm only, and so s preferable for the seller. Furthermore, we should allocate to the hghest-value bdder f and only f hs vrtual value s postve,.e. hs value s above r, where r = 1 (0) s the optmal prce for a monopolst sellng to bdder only. The allocaton rule s shown n Fgure 1 for the case of two bdders. Assume that r 1 > r 2. Transfers consstent wth (8) can be acheved, for example, usng threshold prces: f s allocated the good, he s charged maxfr ; max j6= ( j )g; losers pay nothng. Wth these payments, the aucton would be ncentve-compatble f there were no resale. However, wth resale, t volates ncentve compatblty constrants (3). To see ths, consder bdder 1 s type 1 2 (r 2 ; r 1 ). By tellng the truth, bdder 1 gets a payo of 0, snce he never gets the object (and cannot get t from 2 n resale). However, he could report a lower type ^ 1, n whch case 2 wns the object and then 1 can pro tably buy t back n 11

12 resale f 2 2 (^ 1 ; 1 ). (Ths devaton s llustrated wth a horzontal arrow n Fgure 1.) Thus, n our model, the soluton to the relaxed problem s not ncentve-compatble: we must consder non-local ncentve constrants to nd the correct soluton. 4.2 Vckrey aucton wth reserves Intutvely, to avod ths ncentve to devate and buy back, we mght use an allocaton rule n whch a lower bd never causes the auctoneer to sell the object. (Ausubel and Cramton (1999) call ths property monotoncty n aggregate.) For example, we mght try to x the allocaton rule by llng n the trangular regon r 2 < 2 < 1 < r 1 n Fgure 1, allocatng to bdder 1 n ths regon (based on the above ntuton that we prefer to allocate to the hgh-value bdder or to nobody). Note, however that ths soluton can be mproved: snce bdder 1 has a negatve vrtual value n the lled-n trangle, the seller would rather not sell to hm. By rasng the reserve prce for bdder 2 above r 2, she can shrnk the sze of ths trangle, although dong so also means mssng out on pro table sales to bdder 2. The optmal reserve prce trades o these two e ects. Ths leads to an allocaton rule of the form shown n Fgure 2. Ths aucton belongs to a class of auctons that we formally de ne now. De nton 1 (Ausubel and Cramton 1999, 2004). A Vckrey aucton wth reserves s the aucton mechansm parameterzed by reserve prces p 1 ; : : : ; p n 2 [0; 1], de ned as follows: Allocaton rule: 1 f = () = () and j > p j for some j; 0 otherwse. Payments: 8 < max fp ; max j6= j g f () = 1 and j p j for all j 6= ; () = max j6= j f : () = 1 and j > p j for some j 6= ; 0 otherwse. In words, f at least one bdder beats hs reserve prce p, then the good s allocated to the hgh-value bdder, otherwse, the good s left unsold. 12

13 Fgure 2: Allocaton rule for Vckrey aucton wth reserves. 1 means allocate to bdder 1; 2 means allocate to bdder 2. In the remanng regons, the good s not sold. 13

14 Importantly, snce the reserve prces are asymmetrc, a bdder can wn the good wthout meetng hs reserve prce p f another bdder j wth a lower reserve has met hs reserve p j. The wnner s payments n the aucton s constructed to be hs threshold prce the mnmal bd that would have allowed hm to wn. By the standard argument, ths ensures that the aucton s strategy-proof wthout resale. More mportantly for us, as noted by Ausubel and Cramton (2004), n Vckrey auctons wth reserves t s an ex post equlbrum for bdders to bd truthfully even f bdders beleve resale wll occur, for any belefs about the (ndvdually ratonal) resale procedure and for any pro le of values. To see that the possblty of resale does not create any advantageous devatons, note that when a devaton causes another bdder to wn, the devator would need to pay at least the wnner s value to buy the object back, but he could have nstead acqured the object at ths prce by submttng a hgh bd n the aucton. On the other hand, f a bdder s devaton causes hm to wn, he would pay at least the others hghest value for the object, and would not be able to resell t for a hgher prce. It turns out that a Vckrey aucton wth reserves s the correct soluton to our optmzaton problem. In the remander of ths secton we wll sketch the argument for the case of two bdders, leavng the general formal result and proof for the next secton Optmal Reserve Prces We now dscuss how to dentfy the optmal reserve prces. Snce the object s always allocated to the hgher-value bdder, the formula (9), expressng 6 The workng paper by Ausubel and Cramton (1999) consdered a standard aucton wth the constrant that the good should only be sold to the hghest-value bdder or not at all (but wthout explctly modelng resale), and derved the soluton shown n Fgure 1. They also observed that ths aucton nvtes devatons f there s resale. Ther paper then consdered the problem of the optmal aucton subject to monotoncty n aggregate, as well as sellng only to the hghest bdder, and clamed (wthout proof) the Vckrey aucton wth reserves as the soluton n a specal case wth two bdders. Here, n contrast, we derve the propertes of monotoncty n aggregate and hghest-bdder-only from the maxmn problem of a seller who s concerned wth revenue, rather than assumng those propertes. We also prove that ths aucton format s optmal, both wth two bdders and more generally. 14

15 revenue as the ntegral of vrtual surplus, can be smpl ed to n o E ~ h1 ~ p for some ( ~ ) (~ ( ~ ) ) : (10) Assume the optmal reserves satsfy p 1 > p 2 (ths wll turn out to be true, gven our assumpton that r 1 > r 2 ). The e ect of bdder 1 s reserve prce p 1 on the expected revenue occurs only when 2 < p 2. Condtonal on 2 n ths range, the expected revenue s gven by R 1 (p 1 ) p 1 (1 F 1 (p 1 )) (the expected revenue on bdder 1 as a functon of the prce charged to hm). By assumpton, ths s maxmzed by settng p 1 = r 1. As for the optmal reserve prce for bdder 2, consder the e ects of rasng bdder 2 s reserve prce p 2 slghtly to p 2 + ". Ths change would have two rst-order e ects, both n the slver 2 2 (p 2 ; p 2 + "), the probablty of whch s f 2 (p 2 ) ". The rst e ect s to ncrease the expected revenue on bdder 1 from R 1 (p 2 ) to ts maxmal value R 1 (r 1 ). The second e ect s to reduce the expected revenue on bdder 2 when 1 < p 2 (thus wth probablty F 1 (p 2 )), by not sellng to hm when 2 2 (p 2 ; p 2 +"), by an amount equal to hs vrtual value 2 (p 2 ). The rst-order condton for p 2 equalzes these two e ects: F 1 (p 2 ) 2 (p 2 ) = R 1 (r 1 ) R 1 (p 2 ): (11) Note that (11) has a soluton p 2 2 [r 2 ; r 1 ] provded that 2 s nondecreasng. Indeed, at p 2 = r 2, the left-hand sde s zero and the rght-hand sde s nonnegatve, whle at p 2 = r 1, the left-hand sde s nonnegatve and the rght-hand sde s zero. Hence, exstence obtans by the Intermedate Value Theorem. We can also see that the soluton s unque when 2 s strctly ncreasng (whch makes the left-hand sde strctly ncreasng) and R 1 s concave (whch makes the rght-hand sde nonncreasng on [r 2 ; r 1 ]). 4.4 Optmalty of Vckrey wth Reserves To show that Vckrey wth reserves s optmal, we need to make actve use of the non-local ncentve constrants, snce usng only the local ncentve constrants gave us the ncorrect soluton n Fgure 1. For the two-bdder case we wll only need to use the non-local ncentve constrants (3) for bdder 1. Usng the formula (8) for transfers, the constrants can be rewrtten entrely 15

16 n terms of the allocaton rule: S( 1 ; ^ 1 ; ) Z 1 ^1 E ~2 [v 0 1(( 1 ; ~ 2 ); 1 ; ~ 2 )] d 1 E ~2 [v 1 ((^ 1 ; ~ 2 ); 1 ; ~ 2 ) v 1 ((^ 1 ; ~ 2 ); ^ 1 ; ~ 2 )] 0: We account for these constrants usng a Lagrangan approach, by addng extra terms to the objectve functon (9) to penalze volatons of the constrants. Snce there s a contnuum of such constrants, the Lagrange multplers (weghts) on the constrants must be descrbed wth some approprately constructed nonnegatve measure M on [0; 1] [0; 1]. Our Lagrangan then takes the form " # X E ~ v (( ~ ); ~ X ZZ 1 F ) f ( ) v0 (( ~ ); ~ ) + S( 1 ; ^ 1 ; ) dm( 1 ; ^ 1 ): (13) Because ths Lagrangan s a bounded lnear functonal of the allocaton rule, by the Resz Representaton Theorem (e.g., Royden 1988) t can be wrtten n the form # " X E ~ ( ~ ) ( ~ ) : (14) We wll refer to () as the mod ed vrtual value of bdder. It combnes the ordnary vrtual value and the terms comng from (12). Of course, () depends on the choce of measure M. (For the types of measures proposed below, an explct formula for mod ed vrtual values appears n dsplay (34) n the appendx.) Now, we show optmalty of allocaton rule n the canddate soluton by constructng a measure M such that (; M) s a saddle pont of the Lagrangan (13) (see Luenberger (1969), Theorem 2 on p. 221),.e., the followng condtons hold: (a) The allocaton rule maxmzes the Lagrangan gven measure M; (b) The canddate soluton wth allocaton rule sats es all ncentve compatblty constrants; (c) Complementary slackness: M puts zero measure on all constrants (12) that are slack (hold wth strct nequalty) at,.e., the second term n the Lagrangan (13) s zero. (Together wth (b), ths ensures that gven, (13) s mnmzed by measure M.) 16 (12)

17 To see the su cency of condtons (a)-(c) for optmalty of the canddate soluton, note that the revenue from any alternatve ncentve-compatble aucton wth allocaton rule 0 wll be at most the value of the Lagrangan (13) at ( 0 ; M), whch by (a) s at most the value of the Lagrangan at (; M), whch by (c) equals the revenue at. Regardng condton (b), the feasblty of Vckrey auctons wth reserves was already argued nformally, and wll be stated formally n Secton 7 below. Regardng condton (c), note that the constrants (12) that hold wth equalty at the canddate soluton are those wth ^ 1 1 and ether 1 ; ^ 1 p 1 or 1 ; ^ 1 p 1. (The constrants nvolvng upward devatons are slack because they carry the rsk of gettng the good at a prce above value, and the constrants from above p 1 to below p 1 are slack because those devatons may cause the good to be unsold.) To ensure complementary slackness, the support of M must be restrcted to those constrants. But we can restrct the support of M further by observng that our Vckrey aucton d ers from the relaxed allocaton rule n two ways: t sells to bdder 1 more often when 1 2 (p 2 ; p 1 ), and t sells to bdder 2 less often when 1 < p 2. Ths suggests that we should focus on the bndng constrants wth 1 2 (p 2 ; p 1 ) and ^ 1 < p 2. Thus we wll look for a measure M wth support [p 2 ; p 1 ] [0; p 2 ] (and any such measure wll automatcally satsfy complementary slackness). We construct a measure M( 1 ; ^ 1 ) that takes the form of a product of a margnal measure over 1 and another one over ^ 1. We denote the two measures dstrbuton functons by ( 1 ) and ^(^ 1 ), respectvely. The two measures wll be naled down by two nd erence condtons n maxmzng the Lagrangan (13): 1. Ind erence between sellng to bdder 2 and wthholdng the good when 1 < 2 = p 2, whch dctates that bdder 2 s mod ed vrtual value at these pro les should be 0; 2. Ind erence between sellng to bdder 1 and wthholdng the good when 2 = p 2 < 1, whch dctates that bdder 1 s mod ed vrtual value at these pro les should be 0. To see the necessty of these nd erences, note that the nd erence should be broken n ether drecton wth an arbtrarly small change n 2. For condton #1, note that sellng to bdder 2 when 1 < 2 = p 2 a ects the Lagrangan n two ways: addng the vrtual value of agent 2, weghted by 17

18 the densty of = ( 1 ; p 2 ), to the rst part of (13), and nducng nformaton rents 1 p 2 for all types 1 > p 2 of bdder 1, ntegrated over 1 wth measure and weghted by ^ 0 ( 1 ) (the densty of ^), to be subtracted from the second part of (13). Thus, nd erence condton #1 takes the form f 1 ( 1 ) 2 (p 2 ) = ^ 0 ( 1 ) Z p1 p 2 ( 1 p 2 ) d ( 1 ) for all 1 2 [0; p 2 ] : (15) Note, n partcular, that the condton requres the densty of ^( 1 ) to be proportonal to the densty f 1 ( 1 ). Intutvely, ths proportonalty s necessary to ensure that the sale s e ect on tghtenng bdder 1 s non-local ncentve constrants bndng to 1 (whose weght s proportonal to the densty of ^( 1 )) exactly o set the e ect of addng bdder 2 s vrtual value (whose weght s proportonal to the densty f 1 ( 1 ) of bdder 1 s type dstrbuton). By rescalng and ^ by constants, wthout loss of generalty we can let the proportonalty factor be 1, so ^(^ 1 ) = F 1 (^ 1 ). Now we turn to nd erence condton #2. One way of understandng ths condton s to condton on 2 = p 2, so that the allocaton s a functon of 1 only. A seller tryng to maxmze ths condtonal Lagrangan should then be nd erent across many such allocaton rules. In partcular, settng any reserve prce p 2 [p 2 ; p 1 ] creates an alternatve allocaton rule, whch allocates to 1 when 1 > p, and does not allocate at all when 1 p. The condtonal Lagrangan maxmzer should be nd erent between any such allocaton rule and the actual allocaton rule used n the Vckrey aucton. Note that ths observaton depends on nd erence condton #1, to ensure the seller s wllng to wthhold the good when 1 < p 2. Now, magne the condtonal-lagrangan-maxmzng seller startng from a reserve of p 1 and changng to any other reserve p 2 [p 2 ; p 1 ]. The nd erence means that the loss n the expected pro t on bdder 1 (represented by the rst term of the condtonal Lagrangan), R 1 (p 1 ) R 1 (p), should be exactly o set by the newly created weghted slack of the non-local ncentve constrants (the second term of the condtonal Lagrangan). The slack ncentve constrants are now those from 1 2 [p; p 1 ] to ^ 1 < p: ndeed, any such ncentve constrant s now slack by 1 p, snce the devaton would cause the good to be unsold. Integratng over those constrants wth measures and ^ yelds the equaton ^ (p) Z p1 p ( 1 p) d ( 1 ) = R 1 (p 1 ) R 1 (p) : (16) 18

19 Recall from the above that ^ s supported on [0; p 2 ], and ^ (p) = ^ (p 2 ) = F 1 (p 2 ) for p > p 2. D erentatng both sdes of (16) yelds F 1 (p 2 ) Z p1 p d ( 1 ) = R 0 1 (p). Takng nto account that the ntegral n the above dsplay equals (p 1 ) (p), we obtan (p 1 ) ( 1 ) = R 0 1( 1 )=F 1 (p 2 ) for 1 2 [p 2 ; p 1 ] : Ths equaton descrbes the measure. Note that s nondecreasng, and so the constructed weght measure really s nonnegatve, when functon R 1 s concave. Also, pluggng n ths nto (15) and ntegratng by parts yelds the rst-order condton (11) for p 2. To complete the proof of optmalty of the Vckrey aucton wth reserves, t remans to check condton (a) that the Lagrangan s maxmzed by the canddate soluton. For ths, we can use the Lagrangan expressed by means of mod ed vrtual values, (14). We need to check that the proposed Vckrey aucton wth reserves maxmzes the mod ed vrtual value for every pro le ( 1 ; 2 ), and not just when 2 = p 2 and 1 p 1, whch we have already checked. Ths s some work but s reasonably straghtforward. 5 Can-Keep Case wth Many Bdders We now proceed to the general settng wth n bdders. In ths secton we make the followng assumptons on bdders value dstrbutons (n addton to those made n Secton 2): A1. Both F and 1 F are log-concave for each. A2. 1 () : : : n () for all. Note that the latter part of (A1) s the usual monotone hazard rate assumpton. Both parts of (A1) are sats ed, n partcular, when the densty f s log-concave, whch s sats ed by many standard dstrbutons (Bagnol and Bergstrom 2005). Note also that (A1) s not nested wth the assumpton used n the prevous secton that the revenue functon R () = (1 F ()) 19

20 s concave. Ths leads us to beleve that (A1) s not the weakest possble assumpton, but t yelds the result n a smple way. Assumpton (A2) means that the bdders are unformly ranked from stronger to weaker ; for example, t ensures that n the absence of resale, the optmal aucton would always dscrmnate aganst the stronger bdders (McAfee and McMllan 1989), and the optmal bdder-spec c reserve prces would satsfy r 1 : : : r n. We begn wth a characterzaton of the canddate optmal aucton, whch s a Vckrey aucton wth reserve prces. We characterze the optmal reserve prces p 1 ; : : : ; p n. Then we establsh the optmalty of ths aucton, not just among Vckrey auctons wth reserves, but among all possble auctons. 5.1 Characterzaton of Canddate Optmal Aucton As a rst step toward de nng the optmal reserve prces, de ne R k (p), for any k = 1; : : : ; n and any prce p, to be the expected revenue from a Vckrey aucton n whch only the rst k bdders partcpate and all partcpants face the same reserve prce p. Also de ne R 0 (p) = 0. Now, recursvely de ne a weakly decreasng sequence of reserve prces p k, and a sequence of revenue levels R k for k = 1; : : : ; n, by ntalzng p 0 = 1 and R 0 = 0 and lettng for all k 1, Rk = R k (p k ) + F k (p k ) Rk 1 R k 1 (p k ) (17) = max Rk (p) + F k (p) Rk 1 R k 1 (p) : (18) p2[0;p k 1 ] Inductvely, Rk for k 1 s the revenue from a Vckrey aucton on bdders 1; : : : ; k wth reserves p 1 ; : : : ; p k. To see that t sats es formula (17), compare ths asymmetrc Vckrey aucton to the symmetrc Vckrey wth reserve p k. Notce that the two auctons d er only when bdder k s value s below p k, whch happens wth probablty F k (p k ). In ths case, the former aucton reduces to an aucton on the rst k 1 bdders wth reserves p 1 ; : : : ; p k 1, yeldng expected revenue Rk 1 ; and the latter s an aucton wth symmetrc reserve p k on the rst k 1 bdders, yeldng revenue R k 1 (p k ). (18) requres that p k 2 [0; p k 1 ] s chosen to maxmze expected revenues. Frst we make the followng observaton (proven n the Appendx): Lemma 1. p k > 0 for all k = 1; : : : ; n. 20

21 If the maxmzaton problem (18) has an nteror soluton p k, t has to satsfy the rst-order condton k (p k ) Y j<k F j (p k ) = R k 1 R k 1 (p k ). Intutvely, ncreasng p k by " only matters when k 2 (p k ; p k + ") and j < p j for all j > k, and n that case t has two rst-order e ects: () on the expected revenue from bdder k, when all other bdders values are below p k, and () on the expected revenue from the k 1 strongest bdders, changng t from R k 1 (p k ) (obtaned when bdder k beats hs reserve prce) to Rk 1 (obtaned when bdder k does not beat hs reserve prce). The rst-order condton equalzes those two e ects. The rst-order condton can be rewrtten n the form where the functon H s de ned on p 2 [p n ; 1) by k (p k ) = H (p k ), (19) H (p) = R k 1 R k 1 (p) j<k F j (p) for p 2 [p k ; p k 1 ), k = 1; : : : ; n. (20) Furthermore, the followng lemma (proven n the appendx) shows that the rst-order condton (19) has to hold even f (18) has a corner soluton, and yelds an approach to constructng the functon H and the reserve prces: Lemma 2. Under assumptons (A1)-(A2), the functon H de ned by (20) has the followng propertes: () H s contnuous, () H sats es (19) for each k 1; () For p 2 [p k ; p k 1 ], for each k 2, " 1 H (p) = ( j<k F j (p k 1 )) k 1 (p k 1 ) j<k F j (p) (v) H s nonncreasng. Z pk 1 p ( j<k F j ()) X j<k (21) # f j () j () d ; F j () 21

22 Formula (21) can be used to construct the sequence p 1 ; : : : ; p n as follows: Frst let p 1 = r 1, snce ths value solves (18) for k = 1. Then we construct p k for k 2 teratvely as follows: Gven p k 1, (21) descrbes H (p) on the nterval p 2 [p k ; p k 1 ], and then p k can be obtaned by solvng equaton (19). We can see that (19) has a soluton p k 2 [r k ; p k 1 ], by notng that on the one hand, k (r k ) = 0 = H (r 1 ) H (r k ) (usng part (v) of the lemma), and on the other hand, k (p k 1 ) k 1 (p k 1 ) = H (p k 1 ) (usng (A2) and (19) for k 1). Thus, a soluton to (19) exsts by the Intermedate Value Theorem (usng contnuty of H and k ). Furthermore, the soluton to (19) s unque snce k s strctly ncreasng due to assumpton (A1) and H s nonncreasng by part (v) of the lemma. In Secton 6 below we use ths approach to calculate optmal reserve prces when bdders value dstrbutons are unform wth d erent supports. 5.2 Optmalty of the Aucton The maxmzaton n (18) suggests that p k s the optmal reserve for bdder k, takng as gven the reserves p 1 ; : : : ; p k 1 of the stronger bdders and gven the constrant p k p k 1. Now we show that the constrant does not bnd and ths s ndeed the optmal aucton and not just among Vckrey auctons wth reserves, but among all possble auctons: Theorem 1. Under assumptons (A1)-(A2) and the resale procedure descrbed n (6), the Vckrey aucton wth reserves p 1 ; : : : ; p n characterzed above s an optmal aucton for the seller (and the optmal revenue s Rn). 7 A full proof of the theorem s gven n the appendx, but here we descrbe the key steps of the proof and develop some ntuton. Just as n the twobdder case, the theorem s proven by constructng Lagrange multplers on non-local ncentve constrants n such a way that the saddle-pont condtons (a)-(c) lsted n Secton 4 are sats ed. Wth n bdders, we use Lagrange multplers M ( ; ^ ) for the ncentve constrants of bdders = 1; : : : ; n 1. Smlarly to (12), the ncentve constrants can be wrtten as S ( ; ^ ; ) Z ^ E ~ [v 0 (( ; ~ ); ; ~ )] d E ~ [v ((^ ; ~ ); ; ~ ) v ((^ ; ~ ); ^ ; ~ )] 0; 7 Ths aucton s also optmal under the alternatve resale assumpton n footnote 4. 22

23 and the Lagrangan takes the form " # X E ~ v (( ~ ); ~ X 1 F ( ) ) v 0 f ( ) (( ~ ); ~ Xn 1 ZZ ) + S ( ; ^ ; ) dm ( ; ^ ): =1 (22) An mportant d erence from the two-bdder case s that t s no longer possble to restrct attenton to multplers that take a product form ( ) ^ (^ ). The reason s that for each bdder < n there wll be a range of types where the non-local ncentve constrants both for hgher types to mtate, and for to mtate lower types, are bndng. Ths means that n a product form, the supports of the margnal dstrbutons of and ^ would have to overlap, meanng that some upward ncentve constrants are bndng, whch s nconsstent wth complementary slackness because all upward non-local ncentve constrants are slack n our canddate optmal aucton. It turns out that the correct weghts have just a slghtly more general form: a product measure ^ on [0; 1] [0; 1], restrcted to the half-plane ^. The support of wll be [p n ; p ] and the support of ^ wll be [0; p ]. As before, we wll use and ^ to denote the dstrbuton functons of the respectve measures. (The weghts constructed here would also work for bdder = 1 n two-bdder case, establshng the result under assumptons (A1)-(A2) nstead of concavty of R 1.) Now we descrbe nformally the constructon of the dstrbutons and ^, usng mod ed versons of the arguments developed for the two-bdder case. Frst, we argue that the dstrbuton ^ should be proportonal to the bdder s value dstrbuton F restrcted to [0; p ], and so wthout loss we normalze ^ = F. Just as n the two-bdder case, proportonalty s necessary n order to exactly balance the e ect of bdder s non-local ncentve constrants nto all < p on allocatng the object to other bdders aganst those bdders vrtual values, so that the other bdders allocaton n the regon where bdder does not wn and does not beat hs reserve prce s not contngent on. Next, we can derve the dstrbutons by a calculaton that s smlar to that used n the two-bdder case, but a bt trcker. Namely, to derve ( ), where 2 (p k ; p k 1 ) for some k >, we use the observaton that the maxmzer of the Lagrangan (22) should be nd erent between allocatng to the hghest-value bdder and not allocatng at all at any type pro le ( 1 ; : : : ; n ) at whch j = p j for bdders j k and j 2 (p n ; p j ) for bdders j < k. (Ths nd erence must hold, because f any bdder j < k has hs bd 23

24 perturbed slghtly upward, the canddate optmum must allocate to the hgh bdder; f slghtly downward, the good must be left unsold.) In partcular, ths mples that once we condton on j = p j for j k, the condtonal Lagrangan maxmzer s nd erent to changng the reserve prces for the rst k 1 bdders, as long as the new reserve prce for each bdder j les n the nterval [p n ; p j ]. Spec cally, we consder settng a reserve q 2 (p k ; p k 1 ) for bdder and a reserve p 2 [q; p k 1 ) for all other bdders j < k, j 6=. Let R k 1 (p; q) denote the expected revenue from the resultng Vckrey aucton for the k 1 strongest bdders. The Lagrangan nd erence means that the loss n the expected revenue the rst term of the Lagrangan (22), Rk 1 R k 1 (p; q), should be exactly o set by the second term n (22), whch s the weghted slack created n agents non-local ncentve constrants. The created slack n the non-local ncentve constrants of a bdder j < k nvolves constrants from every j above hs new reserve prce to every ^ j below hs new reserve prce when bddng j would gve the object to ths bdder whle bddng ^ j would cause t to be unsold. For bdder, the slack s the bdder s expected payo loss from a devaton from > q to any ^ < q. Ths payo loss arses only when the hghest bd of the other k 1 strongest bdders, ~ max j<k;j6= ~ j, s below both and p, so bdder s bd of makes hm wn whle hs bd of ^ < q causes the good to be unsold. Ths payo loss can be wrtten by accountng both for states n whch bddng makes hm wn and pay q and for states n whch bddng makes hm wn and pay ~ : s ( ; q; p) = F (q) ( q) + E ~ h1 q< ~ <mnfp;g ~ ; where F denotes the c.d.f. of ~. For bdders j 6=, j < k, we wll not derve the slack exactly, but note that t occurs only when < q, and n ths event t s ndependent of q, hence the weghted expected slack takes the form F (q) Q j (p) for some functon Q j. Addng up yelds the equaton Z p ^ (q) s ( ; q; p) d ( )+ X F (q) Q j (p) = Rk 1 0 j<k;j6= R k 1 (p; q). (23) Now we dvde both sdes of (23) by ^ (q) = F (q) and d erentate wth 24

25 respect to q at q = p. On the left-hand sde of (23) we obtan Z p ( ; q; p) p d ( = F (p) d ( ) = F (p) [ (p ) (p)] : q=p p 0 On the rght-hand sde of (23) R k 1 R k 1 (p; q) F (q) q=p = f (p) R k 1 F 2 R k (p) 1 (p; R k 1 (p; q) =@q q=p : F (p) Now note that R k 1 (p; p) = R k 1 (p) and R k 1 (p; q) =@q q=p = F (p) f (p) (p) (ths s the rst-order e ect on the expected vrtual surplus of agent, whle the e ect on the expected vrtual surplus of the other agents s second-order snce t only occurs when q < ~ < ~ < p, the probablty of whch s O (p q) 2 ). Thus we obtan the equaton F (p) [ (p ) (p)] = f (p) R k 1 R k 1 (p) F (p) F (p) F (p) (p) : Dvdng both sdes by F (p) = Q j<k;j6= F j (p) and usng (20) yelds (p ) (p) = f (p) F (p) [H (p) (p)] : (24) Intutvely, the reason functon H appears n the constructon of s that, as suggested by the rst-order condton (19) for optmal reserve prcng, H (p) descrbes the total shadow weght of all bdders non-local ncentve constrants nvolvng devatons to just below p and then buyng from the wnner, whch would be tghtened f we sell to a bdder when hs value s p. One way to nterpret (24) s by wrtng t as (p) H (p) + F (p) f (p) [ (p ) (p)] = 0: In the proof, we show that the left-hand-sde expresson s the mod ed vrtual value of sellng to agent (recall equaton (14)) when hs value s p and he s the hghest bdder. The rst term s the usual vrtual value of agent. The second term, H (p), s the total shadow weght of all agents non-local ncentve constrants that would be tghtened by sellng to bdder when 25

26 hs value s p. The thrd term s the shadow weght of agent s non-local ncentve constrant from values above p nto values below p, whch are all relaxed by sellng to agent when hs value s p (snce ths rases the utlty of all types above p wthout changng the utlty of any types below p). Ths shadow weght s therefore the measure of the event ^ < p <, whch s ^ (p) [ (p ) (p)] (recall that ^ (p) = F (p)), dvded by f (p) so that upon takng take expectaton over bdder s values we obtan the ntegral wth the Lagrange multpler measure. The equaton says that the mod ed vrtual value should be zero, so that the sale to agent can be condtoned on whether another bdder has met hs reserve prce. We can show that (24) ndeed descrbes a nonnegatve measure: Lemma 3. Under assumptons (A1)-(A2), the functon de ned by 8 >< 0 f < p n ; f (p n) ( ) = F (p n) >: [H(p f n) (p n )] ( ) [H( F ( ) ) ( )] f 2 [p n ; p ]; f (p n) [H(p F (p n) n) (p n )] f > p s nondecreasng. Proof. Consder 2 [p n ; p ]. Observe that s nondecreasng by logconcavty of 1 F and H s nonncreasng by Lemma 2(v), hence H ( ) ( ) s nonncreasng and n partcular H ( ) ( ) H(p ) (p ) = 0. s nonncreasng, and puttng together, we see that ( ) s nondecreasng on 2 [p n ; p ]. Fnally, by constructon ( ) = (p n ) for < p n and ( ) = (p ) for > p. Also, log-concavty of F means that f ( ) F ( ) The last step of the proof of Theorem 1, just as n the two-bdder case, s to show that wth the constructed Lagrange multplers, our Vckrey aucton wth reserves maxmzes the mod ed vrtual value (recall (14)) at every type pro le : that s, that the mod ed vrtual value of a non-hghest-value bdder s always lower than that of the hghest-value bdder, and that when every bdder s value s below hs reserve, all bdders mod ed vrtual values are negatve. 6 Example: Unform Dstrbutons In ths secton we apply the results to the settng n whch the value dstrbuton of each bdder s unform on [0; a ], wth d erent upper lmts a. At 26

27 rst glance the analyss s napplcable to ths settng because t has assumed that all dstrbutons have the same support. However, the analyss can be extended to supports wth d erent upper lmts, by formally de nng bdder s vrtual value for > a to be ( ) = to avod the 0/0 dvson n that regon whle ensurng contnuty. Note that wth d erent upper lmts, assumpton (A2) requres that a 1 : : : a n : Wth d erent upper lmts t s possble that the reserve prce p k of bdder k sats es p k a k. Note that n ths case by Lemma 2() we have H (p k ) = k (p k ) = p k = j (p k ) for all j k; and therefore p j = p k a j for all j k. Therefore, n ths case bdder k as well as all the weaker bdders are excluded from the aucton. For the case of unform dstrbutons, formula (21) can calculated as H k (p) = for p 2 [p k ; p k 1 ] : " k 1 pk 1 2 p k p k k 1 # X a j a k 1 + j<k 2 (k 1) p k 1 k 1 Then p k s gven by equaton (19). If bdder k s not excluded from the aucton, then k (p k ) = 2p k a k and (19) can be expressed as " # k 1 pk 2 k p k + 1 X a j a k = 2 k 1 k p k X a j a k 1 : (25) k 1 p k 1 j<k Ths s a kth-degree equaton for p k, so for k 5 t needs to be solved numercally. To check whether bdder k s excluded, t su ces to evaluate the left-hand sde of (25) at p k = a k : snce t s an ncreasng functon of p k, f at p k = a k t exceeds the rght-hand sde of (25) then the equaton s soluton s below a k and bdder k s not excluded, otherwse bdder k s excluded (and so are all the weaker bdders). For example, for two bdders, we have p 1 = a 1 =2 and equaton (25) for k = 2 takes the form p 2 [p 2 + a 1 a 2 ] = p 2 1 = a 2 1=4: Bdder 2 s excluded f a 2 a 1 a 2 1=4, or a 2 a 1 =4. Otherwse, the optmal p 2 s gven by the postve root of the above quadratc equaton: p 2 = 1 q a (a 1 a 2 ) 2 (a 1 a 2 ) : 2 j<k X j<k a j 27