Vickrey Auctions with Reserve Pricing*

Transcription

1 Economc Theory Aprl 24. Reprnted n Charalambos Alprants et al. eds.) Assets Belefs and Equlbra n Economc Dynamcs Berln: Sprnger-Verlag Vckrey Auctons wth Reserve Prcng Lawrence M. Ausubel and Peter Cramton Department of Economcs Unversty of Maryland College Park MD USA e-mal: ausubel@econ.umd.edu and cramton@umd.edu) Receved: December 3 22; revsed verson: May 5 23 Summary. We generalze the Vckrey aucton to allow for reserve prcng n a mult-unt aucton wth nterdependent values. In the Vckrey aucton wth reserve prcng the seller determnes the quantty to be made avalable as a functon of the bdders reports of prvate nformaton and then effcently allocates ths quantty among the bdders. Truthful bddng s a domnant strategy wth prvate values and an ex post equlbrum wth nterdependent values. If the aucton s followed by resale then truthful bddng remans an equlbrum n the aucton-plus-resale game. In settngs wth perfect resale the Vckrey aucton wth reserve prcng maxmzes seller revenues. Keywords and Phrases: Auctons Vckrey auctons Mult-unt auctons Reserve prce Resale JEL Classfcaton Numbers: D44 C78 D82 The authors gratefully acknowledge the generous support of Natonal Scence Foundaton Grants SES SES and IIS We apprecate valuable comments from Ilya Segal. Specal thanks go to Mordeca Kurz who served as Larry s dssertaton advsor and who ntroduced both authors to the economcs professon back at IMSSS at Stanford. Congratulatons and best wshes are extended to Mordeca and hs famly on the happy occason of the publcaton of ths Festschrft n Mordeca s honor. 1

2 Vckrey Auctons wth Reserve Prcng Lawrence M. Ausubel and Peter Cramton 1 Introducton A Vckrey aucton has the dstnct advantage of assgnng goods effcently puttng the goods n the hands of those who value them most. However one crtque of a Vckrey aucton s that t may yeld low revenues for the seller. Indeed Vckrey expressed ths concern n hs semnal artcle Vckrey 1961). When competton s weak and the bdders are asymmetrc revenues from a Vckrey aucton may be small. A vvd example was the 199 New Zealand sale of spectrum lcenses by second-prce aucton. In one case the wnner bd $1 but pad only $6; n another the wnner bd $7 but pad only $5 McMllan 1994). Reserve prcng s a smple and effectve devce to avod such dsasters. The seller may charge the reserve prce or reduce the quantty sold f the bds are too low. Reserve prcng s also an effectve devce for mtgatng colluson snce t lmts the maxmum gan colluson can reap. Reserve prcng s especally mportant n auctons such as electrcty auctons spectrum auctons or Treasury auctons where partcpants bd for multple tems. Then the largest market partcpant may be so large that removng ths bdder may lead to no excess demand. In a Vckrey aucton prces are based on the opportunty cost of wnnng; that s a wnner pays the value that the goods would have n ther best use wthout the wnner. If a bdder s wnnngs are greater than the excess demand n the aucton wth the bdder removed then some of the Vckrey prces are undefned or zero). In auctons to supply electrcty durng peak perods t s common for the capacty of the largest generator to be far greater than the excess capacty n the system. In such a settng a Vckrey aucton must nvolve reserve prcng. We generalze the Vckrey aucton to allow for reserve prcng n a mult-unt aucton wth nterdependent values. In the Vckrey aucton wth reserve prcng the seller determnes the quantty to be made avalable as a functon of the bdders reports of ther prvate nformaton and then effcently allocates ths quantty among the bdders. We prove n Theorem 1 that truthful bddng by all bdders s an ex post equlbrum n a model wth nterdependent values a bdder s value also depends on the prvate nformaton of other bdders) and that truthful bddng s a domnant strategy n a model wth prvate values a bdder s value depends only on ts own prvate nformaton).. Thus reserve prcng does not nterfere wth many of the desrable features of a Vckrey aucton. An mportant motvaton for ths artcle s the possblty of resale after an aucton. The optmal auctons lterature requres the seller to msassgn tems that s to put the tems n hands other than those who value them the most wth postve probablty except n symmetrc models). However the seller s ablty to do ths may be undermned when resale cannot be prevented; bdders wll antcpate the resale 2

3 market and adjust ther bds accordngly. In Ausubel and Cramton 1999) we prove that whenever resale markets are fully effcent a seller cannot ncrease revenues by msassgnng the tems among the bdders. The revenue-maxmzng aucton s smply for the seller to decde on an optmal quantty to sell based on the bdders reports and to assgn these unts effcently among the bdders. Thus faced wth a perfect resale market the best that the seller can do s to wthhold some of the supply but then to sell the remanng supply effcently. Ths mmedately rases the queston of how to construct a mechansm that lmts the quantty sold but allocates effcently whatever quantty s sold. The Vckrey aucton wth reserve prcng defned n the current artcle performs precsely ths job and thus does exactly what s requred for an optmzng seller facng a perfect resale market. 1 In partcular we prove n Theorem 2 that truthful bddng n the Vckrey aucton wth reserve prcng and hence an effcent assgnment of the goods that are sold) s an ex post equlbrum of the two-stage game consstng of the aucton followed by resale. Indeed for ths result we do not need the resale procedure to be perfect. Truthful bddng wll be an ex post equlbrum whenever the resale game s such that no bdder expects to get more than 1% of the gans from trade that t brngs to the table. The current artcle s related to three strands of lterature. Frst a number of artcles extend the Vckrey aucton to settngs where bdders have nterdependent values. Crémer and McLean 1985) construct a mechansm through whch the full surplus can be extracted from bdders and as a step along the way they construct a mechansm for dscrete types that yelds an effcent assgnment as an ex post equlbrum. Maskn 1992) defned a modfed second-prce aucton whch yelds an effcent assgnment n a sngle-good settng wth nterdependent values. Ausubel 1999) extends Maskn s approach by defnng a generalzed Vckrey aucton for multple dentcal tems wth nterdependent values. Dasgupta and Maskn 2) Jehel and Moldovanu 21) and Perry and Reny 22) also defne aucton mechansms that for the case of multple dentcal objects are outcome-equvalent to the generalzed Vckrey aucton. None of these papers explore reserve prcng or the mplcatons of resale markets. The second strand of lterature consders mult-unt auctons wth varable supply. Back and Zender 21) show that n a unform-prce aucton the seller can elmnate low-prce equlbra Back and Zender 1993) by restrctng supply after the bds are n. Lengwler 1999) n a model allowng two possble prce levels consders the effects of varable supply on seller revenues n both unform-prce and pay-your-bd auctons. McAdams 22) also examnes varatons on the unform-prce aucton n whch the auctoneer s 1 The current artcle does not concern tself wth the determnaton of the optmal quantty based on bdders reports and nstead takes as gven the quantty as a functon of bdders reports. For a treatment of the problem of determnng the optmal quantty see Ausubel and Cramton 1999). 3

4 able to ncrease or decrease quantty after recevng the bds. None of these papers consder Vckrey prcng or resale. The thrd strand of lterature consders auctons wth resale. Hale ) demonstrates that n auctons followed by resale bdders wll antcpate the resale market and adjust ther bds accordngly. Furthermore Hale 1999) consders Vckrey auctons wth resale n the case of auctonng a sngle tem. Ausubel and Cramton 1999) consder optmal mult-unt auctons wth effcent resale. Here we consder how to mplement these optmal auctons. Zheng 22) and Calzolar and Pavan 22) consder sngletem optmal auctons wth alternatve resale games. 2 Secton 2 presents a general model for the aucton of a dvsble good. Bdders demands for the tems may be nterdependent. Secton 3 defnes the Vckrey aucton wth reserve prcng and demonstrates that truthful bddng s an ex post equlbrum despte the fact that the bddng affects the quantty sold. Secton 4 analyzes an aucton followed by resale. It s shown that the possblty of resale does not dstort the Vckrey aucton wth reserve prcng. Truthful bddng remans an ex post equlbrum despte the presence of a resale market followng the aucton. Secton 5 concludes. 2 The General Dvsble Good Model A seller has a quantty 1 of a dvsble good to sell to n bdders N {1 n}. The seller s valuaton for the good equals zero. Each bdder can consume any quantty q [1]. We can nterpret q as bdder s share of the total quantty. Let q q 1 q n ) and let Q {q q 1} be the set of all feasble assgnments. Each bdder s value for the good depends on the prvate nformaton of all the bdders. Let t T [t max ] be bdder s type s prvate nformaton) t t 1 t n ) T T 1 T n and t t \ t = t 1 t -1 t +1 t n ). A bdder s value V tq ) for the quantty q depends on ts own type t and the other bdders types t. A bdder s utlty s ts value less the amount t pays: V tq ) X. Let v tq ) denote the margnal value for bdder gven the vector t of types and quantty q. Then V t q) = v t y) dy. We assume that v tq ) satsfes the followng assumptons: Contnuty. For all t and q v tq ) s jontly contnuous n tq ). Value monotoncty. For all t and q v tq ) s nonnegatve strctly ncreasng n t and weakly decreasng n q. q 2 The prncpal dfference between the approaches of Ausubel and Cramton 1999) and Zheng 22) s that we assume that the resale market s fully effcent whereas Zheng assumes that the wnner of the aucton has full monopoly power n the resale game. Calzolar and Pavan 22) assume that the resale market comprses a sngle take-t-or-leave-t offer by a seller wth probablty λ) or a buyer wth probablty 1 λ). 4

5 Sngle-crossng property. For all j q q j t and t > t v tq ) > v j tq j ) v t t q ) > v j t t q j ) and v t t q ) < v j t t q j ) v tq ) < v j tq j ). Value monotoncty mples that types are naturally ordered and that the bdders have weakly downward-slopng demand curves. The sngle-crossng property mples that f a fxed quantty s assgned effcently among the bdders then bdder s quantty q t) may be chosen to be weakly ncreasng n t. The sngle-crossng property holds f an ncrease n bdder s type rases bdder s margnal value at least as much as any other bdder s. Three specal cases of the general model are partcularly useful. PRIVATE VALUES. A bdder s value V t q ) only depends on ts own type. COMMON VALUE. The bdders values are the same: V tq ) = V j tq ). INDEPENDENT TYPES. The bdders types are drawn ndependently from the dstrbuton functons F wth postve and fnte densty f on T. The prvate values assumpton enables us to strengthen many of the results. In partcular truthful bddng becomes a domnant strategy rather than smply a best response. Also value monotoncty automatcally mples the sngle-crossng property n the prvate value settng. The common value assumpton often s made n models of ol lease auctons and n models of Treasury and other fnancal auctons. Independent types s needed n the optmal aucton analyss our fnal result). Expected revenues depend on the probablty dstrbuton of types and ndependence s needed for a general revenue equvalence theorem. However most of our analyss s based on ex post arguments whch do not requre any assumptons about the dstrbuton of types. Our startng pont for descrbng a Vckrey aucton wth reserve prcng s to specfy the aggregate quantty qt ) Σ q) t that the seller assgns to the bdders as a functon of the vector of reported types. The descrpton of the Vckrey aucton s only guaranteed to make sense f the aggregate quantty qt ) s weakly ncreasng. We therefore requre Monotonc aggregate quantty. The aggregate quantty rule qt ) s weakly ncreasng n each bdder s type. Ths assumpton together wth the sngle-crossng property guarantees that the quantty qt ) can be assgned effcently among the bdders n such a way that each bdder s quantty q t) s weakly ncreasng n t. 5

6 3 Vckrey Aucton wth Reserve Prcng The Vckrey aucton wth reserve prcng can be thought of as a three-step procedure. Frst the bdders smultaneously and ndependently report ther types t to the seller and the seller determnes the aggregate quantty qt ) that t wshes to assgn to bdders. Second the seller determnes an effcent assgnment of ths aggregate quantty; that s the seller solves for q t) q t) q t)) that maxmzes 1 n Σ subject to V t q )) t Σ q ) t = q). t When the effcent assgnment s not unque due to flat regons n the aggregate demand curve q ) t s chosen so that t s weakly ncreasng n t. Thrd the seller determnes a payment X ) t for each bdder assocated wth the assgnment of q ) t where q ) t and X ) t must be specfed so that truthful bddng s ncentve compatble and ndvdually ratonal for every type of every bdder. The determnaton of the payment rule s most easly understood n an envronment wth dscrete unts. Hold the reports t of bdders other than bdder fxed and consder the quantty q t t ) assgned to as a functon of t. Let t 1 t ) denote the mnmum type such that s awarded at least one unt let t 2 t ) denote the mnmum type such that s awarded at least two unts etc. More precsely for every = the mnmum type such that bdder s awarded at least k unts. k 1 let t k t ) nf { t : q t t ) k} By hypothess q ) t s weakly ncreasng n t. Therefore by value monotoncty and the sngle-crossng property t t + for all k 1. k k 1 Dscrete payment rule. If bdder s assgned K unts then for every k 1 k K) bdder s charged k a prce of v t t ) t k) for the k th unt. Vckrey prcng s best thought of n terms of opportunty costs. The wnner pays the opportunty cost of ts wnnngs. In a standard Vckrey aucton the opportunty cost s always the value to the other bdder that would receve the good f the wnner dd not partcpate. In a Vckrey aucton wth reserve prcng the opportunty cost can come nstead from the seller. Ths occurs for a good that the seller would wthhold were t not for the wnner s bds. Crtcal to the analyss observe that bdder s value s evaluated at the mnmal type at whch receves the k th unt. Ths specfcaton has the effect of subsumng the proper prcng rule both for the case where the k th unt of bdder comes from another bdder as well as for the case where the k th unt of bdder comes from the seller s reserve. If the k th unt for bdder s assgned to bdder from another bdder j then bdder s charged the other bdder s value v k j t t ) t qj) assumng s type k s just hgh enough to receve k unts as by defnton t t ) s the mnmal type of bdder such that 6

7 k k bdder receves ths unt so v t t ) t k) = v t t ) t q ). Meanwhle f the k th unt for bdder s j j assgned to bdder out of the seller s reserve then the seller s mplct reserve prce for ths unt also equals k k v t t ) t k) snce all types of bdder greater than t t ) are recevng ths unt whle all k types of bdder less than t t ) are not. Returnng to the case of contnuous quantty let q ) t q) t q ) t denote the aggregate quantty allocated to bdders other than bdders N \ ) followng reports t. Furthermore for any quantty y let v t y) denote the margnal value to bdders N \ f the quantty y s allocated effcently among bdders N \. Observe that for any aggregate quantty rule qt ) and for any reports t an effcent assgnment rule q t) satsfes = v t q t)) = v t q t)) for such that < q t) < q t) v t q t)) for such that q t) = v t q t)) for such that q t) q t). 1) Otherwse from contnuty and value monotoncty f q t) q t) < < and v t q t > v t q t then )) )) there exsts ε > such that allocatng q ) t + ε to bdder and q ) t ε to bdders would generate socal mprovement and smlarly f v t q t v t q t )) < )). From Eq. 1) and the sngle-crossng property for any monotonc aggregate quantty rule qt ) there exsts an assocated effcent assgnment rule q ) t that s weakly ncreasng n t. To see ths note that the sngle-crossng property mples that n an effcent assgnment any quantty that must go to when t s reported must stll go to when t > t s reported and any quantty that cannot go to when t > t s reported stll cannot go to when t s reported. Ths would guarantee that f aggregate demand were strctly downward slopng then q ) t would be unquely defned and t would be weakly ncreasng n t. However when the aggregate demand curve has a flat regon and the flat porton ncludes more than one bdder then q ) t s no longer unque and ndeed some effcent assgnment rules may not be monotonc. In ths case the seller must choose a te-breakng rule that s consstent wth a monotonc effcent assgnment. For example n the flat porton of aggregate demand award the good frst to the bdder wth the hgher type and splt the quantty equally among bdders wth the same type. Also observe that although q t) s monotonc q t) need not be contnuous n t so t s useful to defne lmts of q t) from above and below n t : 7

8 q tˆ t ) = lm q t t ) and q tˆ t ) = lm q t t ). + ˆ t ˆ t t t We can now defne the generalzed Vckrey aucton wth reserve prcng. DEFINITION. Vckrey aucton wth reserve prcng. Gven any monotonc effcent assgnment rule q t) and for reports t of bdders other than bdder and for any quantty z such that z defne: Followng reports t bdder s assgned { } max q t t ) tˆ t z) = nf t q t t ) z. 2) t q ) t unts and s charged a payment X ) t computed by: q ) t ˆ X ) t = v t t z) t z) dz. 3) Note that the payment formula of Eq. 3) s well defned snce the value monotoncty assumpton assures that for any reports t and for any quantty z we have v tˆ t z) t z) v t). [ q t)] In the Vckrey aucton a bdder pays the opportunty cost of ts wnnng for each ncremental quantty won. Hence the margnal payment made at each quantty z s determned by the bdder s margnal value assumng the bdder makes the lowest possble report consstent wth wnnng a quantty z. Ths margnal value may be determned ether by the opportunty to sell to another bdder or by the opportunty to wthhold the good. In ths way the bdder receves 1 percent of the gans from trade that t brngs to the table. The fact that the bdder receves 1 percent of ts ncremental contrbuton s what gves the bdder the ncentve for truthful bddng. THEOREM 1. For any monotonc aggregate quantty rule q) t and assocated monotonc effcent assgnment rule q t and for any valuaton functons v t q ) satsfyng contnuty value monotoncty ) and the sngle-crossng property the Vckrey aucton wth reserve prcng has truthful bddng as an ex post equlbrum. PROOF. By contnuty value monotoncty and the sngle-crossng property we can choose q ) t to be weakly ncreasng n t. Then tˆ t z) defned by Eq. 2) s weakly ncreasng n z. Substtutng Eq. 3) nto the expresson V tq ) X for bdder s utlty yelds the followng ntegral for bdder s utlty from reportng ts type as t when ts true type s t and the other bdders true and reported types are t : q t t ) U t t) = ) ˆ v t z v t t z) t z) dz. 4) 8

9 Observe that the ntegrand of Eq. 4) s ndependent of t bdder s reported type; t enters nto Eq. 4) only through the upper lmt on the ntegral. Moreover by value monotoncty the ntegrand of Eq. 4) s nonnegatve for all z q ) t and s nonpostve for all z when the upper lmt on the ntegral equals q ) t. Hence U t t) s maxmzed for every t q ) t whch s attaned by truthful bddng. For the specal case of prvate values truthful bddng s a domnant strategy. Then truthful bddng s a best response for any reports by the other bdders. Wthout prvate values the domnant strategy result s lost snce a bdder s value depends on the types of the other bdders and so the bdder cares whether the reports of the others are truthful. Truthful bddng s only a best response f the other bdders are truthful; but t remans a best response after the bdder learns the opposng bdders truthful) reports. Hence the ex post equlbrum property of truthful bddng always holds. 4 Aucton followed by Resale A man motvaton for assgnng goods effcently s the possblty of resale Ausubel and Cramton 1999). Resale undermnes the seller s ncentve to msassgn the goods snce the msassgnment may be undone n the resale market. The bdders antcpate the possblty of resale whch alters ther ncentves and dstorts the bddng n the ntal aucton. Hence an equlbrum n the aucton game typcally s not an equlbrum n the aucton-plus-resale game. Here we wsh to show that a Vckrey aucton wth reserve prcng s not dstorted by the possblty of resale. To prove ths we need to show that a bdder wth type t does not wsh to msreport type t n a Vckrey aucton wth reserve prcng followed by resale. Let Δ t t) denote the optmal quantty of resale between bdder and the coalton N \ f bdder msreports ts type as t when ts true type s t and the other bdders true and reported types are t and let GFT t t) denote the gans from trade avalable va resale between bdder and the coalton N \ f bdder msreports ts type as t when ts true type s t and the other bdders true and reported types are t. LEMMA 1. If bdder msreports ts type as t when ts true type s t and the other bdders true and reported types are t the mnmum) optmal quantty of resale between bdder and the coalton N \ s gven by + > Δ t t) = + < mn{ z v tq t t ) z) v tq t t ) z)} f t t mn{ z v tq t t ) z) v tq t t ) z)} f t t 5) and the gans from trade avalable va resale between bdder and the coalton N \ are gven by 9

10 6) Δ t t) GFT t t) = v t q t t ) + z) v t q t t ) z) dz. PROOF. Observe that the ntegrand of Eq. 6) gves the margnal gans of the z th unt transferred from coalton N \ to bdder. By value monotoncty f z < z then v t q t t z v t q t t z ) + ) > ) ) mples v t q t t + z > v t q t t z and ) ) ) ) v t q t t + z > v t q t t z mples ) ) ) ) v t q t t + z > v t q t t z. Thus Δ t t) defned by Eq. 5) provdes the mnmal) upper ) ) ) ) lmt for the ntegral n Eq. 6) whch maxmzes the value of the ntegral. The followng calculaton wll be helpful n what follows: LEMMA 2. For any monotonc aggregate quantty rule q) t and assocated monotonc effcent assgnment rule q ) t for any valuaton functons v tq ) satsfyng contnuty value monotoncty and the sngle-crossng property for any bdder for any true type t for any overreport t > t for any vector t of other bdders reported and true types and for any z such that z Δ t t) v t q t t + z v tˆ t q t t z t q t t z 7) ) ) ) ) ) ). PROOF. Consder any z such that z Δ t t) and defne By the ˆz t tˆ t q t t ) z) t. defnton of ˆ z t for every ˆ z t > t t s the case that q t t ) q t t ) z; therefore v t t q t t q t t + z v t t q t t z for every ˆ z t > t and so takng the lmt as ) ) ) ) ) t t mples that ˆz v tˆ t q tˆ t q t t + z v tˆ t q t t z Note that z + z z ) ) ) ) ). v t q t t + z v t q t t q t t + z v tˆ t q tˆ t q t t + z snce tˆz t z + z ) ) ) ) ) ) ) ) mples ) ˆz qt t + q t t ) and snce ˆ z t t. Combnng nequaltes we conclude that v t q t t + z v tˆ t q t t z as desred. z ) ) ) ) To prove Theorem 2 we need some structure on the resale game. In partcular we need a constrant on how much a msreportng bdder can gan n the resale game. Wth two bdders ndvdual ratonalty s all that s requred. In the resale game a bdder cannot get a surplus that s greater than the avalable gans from trade for to do so the other bdder would have to strctly lose from resale. In ths case the other bdder would smply refuse to partcpate n resale. Wth more than two bdders and nterdependent values we must extend the defnton of ndvdual ratonalty. Ths s because one bdder s msreport n the aucton may create gans from trade among the other bdders. These other bdders then should 1

11 consder the gans from trade they can secure among themselves n decdng whether to partcpate n resale wth the msreportng bdder. Coaltonal Ratonalty. For any ntal allocaton a of the good among bdders for any vector t of types and for any subset S of the set N of bdders let vs at) denote the avalable gans from trade f the bdders n subset S trade only amongst themselves startng at allocaton a and evaluated at types t). Further let s denote the surplus from the resale process realzed by bdder. The resale process s coaltonally ratonal f for every subset S of the set N of bdders the bdders n subset S obtan no more surplus s than they brng to the table: s vn at ) vn \ S at ). 8) S The resale process s coaltonally-ratonal aganst ndvdual bdders f for every element of the set N of bdders bdder obtans no more surplus s than t brngs to the table: s vn at ) vn \ at ). 9) The ntuton behnd ths assumpton s that n the barganng process underlyng resale the bdders n coalton N \ S always have the outsde opton of excludng the bdders n the complementary set S from the barganng and only tradng amongst themselves. Hence the bdders n S cannot deprve the bdders n N \ S of the gans from trade that they could stll obtan by tradng amongst themselves. We should remark that the assumpton of coaltonal ratonalty s qute natural and qute weak. It s mpled for example by the requrement n the defnton of the core that no coalton can mprove upon an allocaton. All we wll need for our resale theorem s the stll-weaker assumpton of coaltonal ratonalty aganst ndvdual bdders. Ths s the requrement that any ndvdual bdder not receve any hgher payoff than ts margnal contrbuton to the set N \ of bdders. Observe that ths s trvally mpled by coaltonal ratonalty. Wth superaddtve values whch s always the case when value reflects potental gans from trade) t s also satsfed by standard soluton concepts such as the Shapley value whch has every bdder recevng ts expected margnal contrbuton to the set S of bdders the expectaton taken over all subsets S N \ ). In the prvate values case the defnton of coaltonal ratonalty reduces to ndvdual ratonalty. Wth prvate values f all bdders except bdder report truthfully n the aucton then observe that n the resale round vn \ at ) = snce the objects dstrbuted to the coalton N \ are already assgned effcently. Thus ndvdual ratonalty s j and feasblty s j vn at ) s vn at ) vn \ at ) whch s coaltonal ratonalty. j N mply that 11

12 We now can prove our second theorem whch concerns the game wth resale. THEOREM 2. For any monotonc aggregate quantty rule q) t and assocated monotonc effcent assgnment rule q t and for any valuaton functons v t q ) satsfyng contnuty value monotoncty ) and the sngle-crossng property truthful bddng followed by no resale s an ex post equlbrum of the two-stage game consstng of the Vckrey aucton wth reserve prcng followed by any resale process that s coaltonally-ratonal aganst ndvdual bdders. PROOF. Let π t t) denote the combned payoff to bdder n the Vckrey aucton and the resale market from msreportng t when ts true type s t and the other bdders reported and true types are t. By coaltonal ratonalty aganst ndvdual bdders π t t) U t t) + GFT t t) snce GFT t t) s defned to be the gans from trade avalable va resale between bdder and the coalton N \. By Eqs. 4) and 6) q ) t π t t) ) ˆ v t z v t t z) t z) dz q t t ) q t) ˆ + v t q t t ) z) v t t q t t ) z) t q t t ) z) dz Δ t t) + v t q t t ) + z) v t q t t ) z) dz. 1) Snce t t t q t t z for all z between and ˆ ) ) q t t ) q t) the second ntegrand of Eq. 1) s weakly negatve. Snce Δ we further have t t) q t t ) q t) q ) t π t t) ) ˆ v t z v t t z) t z) dz Δ t t) ˆ + v t q t t ) z) v t t q t t ) z) t q t t ) z) dz + + Δ t t) v t q t t ) z) v t q t t ) z) dz. 11) But then usng Eq. 4) we can smplfy ths as Δ t t) ˆ π t t) U t t) + v t q t t ) z) v t t q t t ) z) t q t t ) z) + dz. 12) Fnally observe by Lemma 2 that the ntegrand of Eq. 12) s nonpostve for all z such that z Δ t t); consequently the ntegral s nonpostve whenever Δ t t). By the sngle-crossng property and the monotoncty of qt ) t > t mples Δ t t). Ths allows us to conclude that π t t) U t t) for all t > t and for all t. Analogous reasonng apples for all underreports t < t. 12

13 Fnally consder the problem of a seller that seeks to maxmze revenues but cannot prevent resale. Ausubel and Cramton 1999) show that a seller faced wth a perfect resale market cannot gan by msassgnng goods. The best the seller can hope to do s to assgn the goods effcently perhaps wthholdng quantty. Ths result requres ndependent types so that the optmal aucton program s well specfed and a general revenue equvalence theorem holds. Theorem 2 states that any monotonc aggregate quantty rule and assocated monotonc effcent assgnment rule can be mplemented wth a Vckrey aucton wth reserve prcng. Ths suggests that a revenue-maxmzng seller then can optmze over all monotonc aggregate quantty rules to attan the upper bound on revenues gven by the resale-constraned aucton program n Ausubel and Cramton 1999). Indeed ths s the case provded the Vckrey aucton wth reserve prcng holds the lowest type t = ) of every bdder to a payoff of zero. To see ths note that tˆ t y) = for all t and y [ q t )] so that the lowest type s payment X t ) s exactly equal to the value t gets from q t ). Hence we have: COROLLARY. Wth ndependent types the Vckrey aucton wth reserve prcng attans the upper bound on revenues n the resale-constraned aucton program. 5 Concluson A Vckrey aucton wth reserve prcng has two man advantages. Frst t assgns goods effcently. Effcency s especally mportant n aucton markets wth resale snce the apparent revenue benefts from msassgnment are undermned by resale. Second t allows the seller to wthhold supply and set reserve prces to mprove revenues. The use of reserve prces s especally mportant when competton s weak and the bdders are asymmetrc. It s also mportant n auctons of multple dentcal tems where one or more of the bdders purchases a sgnfcant share of the goods. We have extended the Vckrey aucton to nclude reserve prcng n a mult-unt settng wth nterdependent values. Truthful bddng remans an ex post equlbrum despte the fact that the seller vares the quantty based on the bds. Ths effcent outcome s robust to the possblty of resale. So long as the resale game satsfes a natural extenson of ndvdual ratonalty truthful bddng followed by no resale s an equlbrum n the aucton-plus-resale game. Moreover f resale s effcent then the Vckrey aucton wth approprate reserve prcng s the optmal aucton. No alternatve aucton can yeld hgher revenues. A practcal dffculty of usng Vckrey prcng when auctonng multple tems s that dentcal tems sell for dfferent prces. Worse large wnners tend to pay lower average prces than small wnners. Ths 13

14 fact s an unavodable mplcaton of achevng effcency. Large bdders have a greater ncentve to reduce demands than small bdders. Hence effcent prcng must reward large bdders for bddng ther true demands by lettng large bdders wn the effcent quantty at lower average prces. In contrast unform prcng necessarly leads to an neffcent assgnment Ausubel and Cramton 22) and hence to suboptmal revenues when resale s effcent. Partcpants n many actual markets voce a strong preference for unform prcng Wlson 22). Often the case for unform prcng s made on effcency grounds and the case aganst Vckrey prcng s based on examples of lost revenue. Wth dmnshng margnal valuatons these arguments have lttle mert. On ether effcency or revenue grounds a Vckrey aucton wth reserve prcng should be preferred. References Ausubel L.M.: A mechansm generalzng the Vckrey aucton. Unversty of Maryland Workng Paper 1999) Ausubel L.M. Cramton P.: The optmalty of beng effcent. Unversty of Maryland Workng Paper 1999) Ausubel L.M. Cramton P.: Demand reducton and neffcency n mult-unt auctons. Unversty of Maryland Workng Paper ) Back K. Zender J.F.: Auctons of dvsble goods: On the ratonale for the Treasury experment. Revew of Fnancal Studes ) Back K. Zender J.F.: Auctons of dvsble goods wth endogenous supply. Economcs Letters ) Calzolar G. Pavan A: Monopoly wth resale. Northwestern Unversty Workng Paper 22) Crémer J. McLean R.P.: Optmal sellng strateges under uncertanty for a dscrmnatng monopolst when demands are nterdependent. Econometrca ) Dasgupta P. Maskn E.: Effcent auctons. Quarterly Journal of Economcs ) Hale P.A. Auctons wth resale. Unversty of Wsconsn Workng Paper 1999) Hale P.A. Auctons wth prvate uncertanty and resale opportuntes. Journal of Economc Theory ) Jehel P. Moldovanu B.: Effcent desgn wth nterdependent valuatons. Econometrca ) Lengwler Y.: The multple unt aucton wth varable supply. Economc Theory ) Maskn E.: Auctons and prvatzaton. In Sebert H. ed.): Prvatzaton: Symposum n Honor of Herbert Gersch Tubngen: Mohr Sebeck) ) McAdams D.: Modfyng the unform-prce aucton to elmnate collusve-seemng equlbra. MIT Workng Paper 22) McMllan J.: Sellng spectrum rghts. Journal of Economc Perspectves ) Perry M. Reny P.J. An effcent aucton. Econometrca ) Vckrey W.: Counterspeculaton auctons and compettve sealed tenders. Journal of Fnance ) Wlson R.: Archtecture of power markets. Econometrca ) Zheng C.Z.: Optmal aucton wth resale. Econometrca ) 14