Vickrey Auctions with Reserve Pricing

Transcription

1 Vckrey Auctons wth Reserve Prcng Lawrence M. Ausubel and Peter Cramton Unversty of Maryland 28 June 1999 Prelmnary and Incomplete Abstract We generalze the Vckrey aucton to allow for reserve prcng n a multple tem aucton wth nterdependent values. By wthholdng quantty n some crcumstances the seller can mprove revenues or mtgate colluson. In the Vckrey aucton wth reserve prcng the seller determnes the quantty to be made avalable as a functon of the bdders 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 where resale exhausts all the gans from trade among the bdders the Vckrey aucton wth reserve prcng maxmzes seller revenues. JEL No.: D44 Auctons) Keywords: Auctons Vckrey Auctons Multple Item Auctons Resale Send comments to: Professors Lawrence M. Ausubel or Peter Cramton Department of Economcs Unversty of Maryland College Park MD ausubel@econ.umd.edu peter@cramton.umd.edu 31) ) The authors gratefully acknowledge the support of the Natonal Scence Foundaton. 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 paper 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 restrcts the quantty sold f the bds are too low and charges reserve prces n other cases. Reserve prcng s also an effectve devce for mtgatng colluson. 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. 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 multple tem 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 prvate nformaton and then effcently allocates ths quantty among the bdders. Truthful bddng s a domnant strategy wth prvate values a bdder s value depends only on ts own prvate nformaton) and an ex post equlbrum wth nterdependent values a bdder s value also depends on the prvate nformaton of other bdders). Reserve prcng does not damage the desrable features of a Vckrey aucton. An mportant motvaton for assgnng goods effcently s the possblty of resale Ausubel and Cramton 1999). Although n an optmal aucton the seller typcally has an ncentve to msassgn goods ths ncentve s undermned when the seller cannot prevent resale. Bdders antcpate the resale market and adjust ther bds accordngly. Here we show that n an aucton followed by resale truthful bddng 2

3 remans an ex post equlbrum n the aucton-plus-resale game so long as the resale game satsfes a natural extenson of ndvdual ratonalty. When resale markets are perfect so that all gans from trade among the bdders are exhausted n the resale market then an upper bound on seller revenues s gven by the resale-constraned aucton program Ausubel and Cramton 1999). In ths program the seller can wthhold quantty but s constraned to assgn effcently the quantty sold. Here we show that the Vckrey aucton wth reserve prcng attans the upper bound on payoffs gven by the resale-constraned aucton program. Faced wth a perfect resale market the Vckrey aucton wth reserve prcng maxmzes seller revenues. Ths paper s related to two strands of lterature. Frst a number of papers extend the Vckrey aucton to settngs where bdders have nterdependent values. Maskn 1992) defned a modfed second-prce aucton whch yelds an effcent outcome n a sngle-good settng wth nterdependent values. Ausubel 1997 Appendx B) extends Maskn's approach by defnng a generalzed Vckrey aucton for multple dentcal tems wth nterdependent values. Dasgupta and Maskn 1998) and more recently Perry and Reny 1998) also defne an aucton mechansm that for the case of multple dentcal objects s outcomeequvalent to the generalzed Vckrey aucton. None of these papers explore reserve prcng or the mplcatons of resale markets. The second strand of lterature consders multple unt auctons wth varable supply. Back and Zender 1999) 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. Nether of these papers consder Vckrey prcng or resale. 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 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 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 3

4 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. A bdder s value V tq ) for the quantty q depends on ts own type t 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 tq) = v ). tydy 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 ) v tq ) s strctly ncreasng n t v tq ) s weakly ncreasng n t j j ) and v tq ) s weakly decreasng n q. Value regularty. 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. Value regularty mples that f a fxed quantty s assgned effcently among the bdders that bdder s quantty q t) may be chosen to be weakly ncreasng n t. Value regularty holds f an ncrease n bdder s type rases s margnal value at least as much as that of any other bdder. Three specal cases of the general model are 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 ). q 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 value regularty 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. 4

5 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 qt ) s a weakly ncreasng n each bdder s type. Ths assumpton together wth value regularty guarantees that the quantty qt ) can be assgned effcently among the bdders n such a way that bdder s quantty q t) s weakly ncreasng n t. 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 tq )) t Σ q ) t = qt ). When the effcent assgnment s not unque due to flat regons n the aggregate demand curve determnes a payment q ) t s chosen so that t s weakly ncreasng n t. Thrd the seller X ) t for each bdder assocated wth the assgnment of q ) t where q ) t and X ) t must be specfed so that sncere 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 to as a functon of t. Let t 1 denote the mnmum type such that s awarded at least one unt q t t ) assgned 1 q t t ) 1) let t 2 2 denote the mnmum type such that s awarded at least two unts q t t ) 2) and let t k denote the mnmum type such that s awarded at least k unts q k t t ) k). By hypothess qt ) s weakly ncreasng. Therefore by value monotoncty and value regularty t t + for all k 1. Dscrete payment rule. If bdder s assgned K unts then for every k 1 k K) bdder s charged a prce of v k 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 k k 1 5

6 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 t k t q ) assumng s type s j j just hgh enough to receve k unts as by defnton t k s the mnmal type of bdder such that bdder k k receves ths unt so v t t k) = v t t q ). Meanwhle f the k th unt for bdder s assgned to bdder j j out of the seller s reserve then the seller s mplct reserve prce for ths unt equals v k t t k) snce all types of bdder greater than t k are recevng ths unt whle all types of bdder less than Returnng to the case of contnuous quantty let k t are not. q ) t qt ) q ) t denote the aggregate quantty allocated to bdders other than bdders ) followng reports t. Furthermore for any quantty y let v ty) denote the margnal value to bdders f the quantty y s allocated effcently among bdders. Observe that for any aggregate assgnment rule qt ) and for any reports t an effcent assgnment rule q t) satsfes v tq t q t = v tq )) t = v tq )) t for such that < q ) t < qt ) )) for such that ) = v tq )) t for such that q ) t qt ). 1) Otherwse from contnuty and value monotoncty f q ) t qt ) < < and v tq t > v tq 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 tq t v tq t )) < )). From Eq. 1) and value regularty for any monotonc aggregate quantty qt ) there exsts an effcent assgnment q ) t that s weakly ncreasng n t. To see ths note that value regularty 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 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 q ) t s no longer unque 6

7 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 : q tˆ t ) = lm qt t )and q tˆ t ) = lm qt t ). + ˆ t ˆ t t t We can now defne the generalzed Vckrey aucton wth reserve prcng. Vckrey aucton wth reserve prcng. Gven an effcent assgnment rule q t) and for reports t of bdders other than bdder and for any quantty z such that z q t t ) defne max Followng reports t bdder s assgned { } 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) We have the followng results: THEOREM 1. For any monotonc aggregate assgnment rule qt ) and assocated monotonc effcent assgnment q ) t for any valuaton functons v tq ) satsfyng contnuty value monotoncty and value regularty the Vckrey aucton wth reserve prcng has sncere bddng as a best response to all other bdders bddng sncerely. PROOF. By contnuty value monotoncty and value regularty we can chose 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 ) ) ˆ t t = ) ). v tz v t t z t z dz 4) 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 7

8 nonnegatve for all z upper lmt on the ntegral equals q ) t and s nonpostve for all z q ) t whch s attaned by sncere bddng. q ) t. Hence U t t) s maxmzed when the For the specal case of prvate values sncere bddng s a domnant strategy. Then sncere 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. Sncere bddng s only a best response f the other bdders are truthful. 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 s typcally 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 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 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 6) t ) t GFT t ) t = v ) ) ) ) tq t t + z v tq t t z dz 8

9 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 tq t t z v tq t t z ) + ) > ) ) mples v tq t t + z > v tq t t z and ) ) ) ) v tq t t + z > v tq t t z mples ) ) ) ) v tq t t + z > v tq t t z. Thus 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 assgnment rule qt ) and assocated monotonc effcent assgnment q ) t for any valuaton functons v tq ) satsfyng contnuty value monotoncty and value regularty 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 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 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 qt 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 tq t t + z v t qt 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 our man theorem 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. A bdder cannot get n the resale game 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 consder the gans from trade they can secure among themselves n decdng whether to partcpate n resale wth the msreportng bdder. 9

10 Coaltonal Ratonalty. For any ntal allocaton a of unts 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 S always have the outsde opton of excludng the bdders n the complementary set N ~ S from the barganng and only tradng amongst themselves. Hence the bdders n N ~ S cannot deprve the bdders 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 ~ ) = snce the objects dstrbuted to the coalton N ~ are already assgned effcently. Thus coaltonal ratonalty mples s vn at) whch s ndvdual ratonalty. We now can prove our man theorem. THEOREM 2. For any monotonc aggregate assgnment rule qt ) and assocated monotonc effcent assgnment q ) t and for any valuaton functons v tq ) satsfyng contnuty value monotoncty and 1

11 value regularty sncere 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 coaltonallyratonal 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 tz v t t z) t z) dz q t t ) q ) t ˆ + v ) ) ) ) ) ) tq t t z v t t q t t z t q t t z dz t ) t + v ) ) ) ) tq t t + z v tq t t z dz. 1) Snce t tˆ t q t t ) z) for all z between and q t t ) q ) t the second ntegrand of Eq. 1) s weakly negatve. Snce we further have: t ) t q t t ) q ) t q ) t π t ) t ) ˆ v tz v t t z) t z) dz t ) t ˆ + v ) ) ) ) ) ) tq t t z v t t q t t z t q t t z dz + + t ) t v ) ) ) ) tq t t z v tq t t z dz. 11) But then usng Eq. 4) we can smplfy ths as π t ) t ˆ 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 t t); consequently the ntegral s nonpostve whenever t t). By value regularty and the monotoncty of qt ) t > t mples 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. 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 11

12 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 assgnment rule and assocated monotonc effcent assgnment can be mplemented wth a Vckrey aucton wth reserve prcng. Ths suggests that a revenue-maxmzng seller then can optmze over all monotonc aggregate assgnments 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 mportant n aucton markets wth resale snce the 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 multple tem settng wth nterdependent values. Truthful bddng remans an 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 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 12

13 unform prcng necessary leads to an neffcent assgnment Ausubel and Cramton 1998) and hence suboptmal revenues when resale s effcent. Partcpants n many actual markets voce a strong preference for unform prcng Wlson 1999). Often the case for unform prcng s made on effcency grounds and the case aganst Vckrey prcng s based on examples of lost revenue. These arguments have lttle mert. On ether effcency or revenue grounds a Vckrey aucton wth reserve prcng should be preferred. References Ausubel Lawrence M. 1997) An Effcent Ascendng-Bd Aucton for Multple Objects Workng Paper Unversty of Maryland. Ausubel Lawrence M. and Peter Cramton 1998) Demand Reducton and Ineffcency n Mult-Unt Auctons Workng Paper Unversty of Maryland. Ausubel Lawrence M. and Peter Cramton 1999) The Optmalty of Beng Effcent Workng Paper Unversty of Maryland. Back Kerry and Jame F. Zender 1993) Auctons of Dvsble Goods: On the Ratonale for the Treasury Experment Revew of Fnancal Studes Back Kerry and Jame F. Zender 1999) Auctons of Dvsble Goods wth Supply Restrctons Workng Paper Unversty of Arzona. Dasgupta Partha and Erc Maskn 1998) Notes on Effcent Auctons Workng Paper Harvard Unversty. Lengwler Yvan 1999) The Multple Unt Aucton wth Varable Supply Economc Theory forthcomng. Maskn Erc 1992) Auctons and Prvtzaton n Horst Sebert ed) Prvatzaton: Symposum n Honor of Herbert Gersch Tubngen: Mohr Sebeck) McMllan John 1994) Sellng Spectrum Rghts Journal of Economc Perspectves Perry Motty and Phlp J. Reny 1998) An Ex-Post Effcent Mult-Unt Aucton for Agents wth Interdependent Values Workng Paper Hebrew Unversty. Vckrey Wllam 1961) Counterspeculaton Auctons and Compettve Sealed Tenders Journal of Fnance Wlson Robert 1999) Market Archtecture Workng Paper Stanford Unversty. 13