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1 econstor Make Your Publcatons Vsble. A Servce of Wrtschaft Centre zbwlebnz-informatonszentrum Economcs Zheng, Charles Z. Workng Paper Exstence of monotone equlbra n frst-prce auctons wth resale EPRI Workng Paper, No Provded n Cooperaton wth: Economc Polcy Research Insttute EPRI), Department of Economcs, Unversty of Western Ontaro Suggested Ctaton: Zheng, Charles Z. 2014) : Exstence of monotone equlbra n frst-prce auctons wth resale, EPRI Workng Paper, No Ths Verson s avalable at: Standard-Nutzungsbedngungen: De Dokumente auf EconStor dürfen zu egenen wssenschaftlchen Zwecken und zum Prvatgebrauch gespechert und kopert werden. Se dürfen de Dokumente ncht für öffentlche oder kommerzelle Zwecke vervelfältgen, öffentlch ausstellen, öffentlch zugänglch machen, vertreben oder anderwetg nutzen. Sofern de Verfasser de Dokumente unter Open-Content-Lzenzen nsbesondere CC-Lzenzen) zur Verfügung gestellt haben sollten, gelten abwechend von desen Nutzungsbedngungen de n der dort genannten Lzenz gewährten Nutzungsrechte. Terms of use: Documents n EconStor may be saved and coped for your personal and scholarly purposes. You are not to copy documents for publc or commercal purposes, to exhbt the documents publcly, to make them publcly avalable on the nternet, or to dstrbute or otherwse use the documents n publc. If the documents have been made avalable under an Open Content Lcence especally Creatve Commons Lcences), you may exercse further usage rghts as specfed n the ndcated lcence.

2 Exstence of Monotone Equlbra n Frst-Prce Auctons wth Resale by Charles Z. Zheng Workng Paper # June 2014 Economc Polcy Research Insttute EPRI Workng Paper Seres Department of Economcs Department of Poltcal Scence Socal Scence Centre The Unversty of Western Ontaro London, Ontaro, N6A 5C2 Canada Ths workng paper s avalable as a downloadable pdf fle on our webste

3 Exstence of Monotone Equlbra n Frst-Prce Auctons wth Resale Charles Z. Zheng May 26, 2014 Abstract Exstence of a monotone pure-strategy perfect Bayesan equlbrum s proved for a multstage game of frst-prce auctons wth nterbdder resale, wth any fnte number of ex ante dfferent bdders. Endogenous gans at resale complcate the wnner s curse and upset prevous fxed-pont methods to prove exstence of monotone equlbra. Ths paper restructures the fxed-pont approach wth respect to comparatve statcs of the resale mechansms strategcally chosen after the aucton. Despte speculaton possbltes and the dscontnuty-nducng unform te-breakng rule, at our equlbrum any bd that stands a chance to wn s strctly ncreasng n the bdder s use value. The author wshes to thank Macej Kotowsk and Gábor Vrág for comments on a prevous draft, René Krkegaard for a comment at an early stage of the project, Greg Pavlov for suggestons of lterature sources, and semnar partcpants of the Unversty of Zürch. The fnancal support from the Socal Scence and Humantes Research Councl of Canada s gratefully acknowledged. Department of Economcs, Unversty of Western Ontaro, London, Ontaro, charles.zheng@uwo.ca, 1

4 1 Introducton Analyses of economc nsttutons are based on exstence of equlbra of the underlyng games. Among them frst-prce auctons, wdely used n practce, are of partcular theoretcal nterest because of a dscontnuty problem, arsng at tyng bds, that may upset standard arguments of equlbrum exstence. To solve ths dscontnuty problem sophstcated methods based on fxed-pont theorems have been developed, one guaranteeng exstence of monotone purestrategy equlbra due to Athey 1, McAdams 10, Reny and Zamr 16, and Reny 15, and the other for mxed-strategy equlbra, augmented wth endogenous te-breakng rules, due to Jackson, Smon, Swnkels and Zame 5. 1 However, nether method has been appled to dynamc games such as auctons wth resale. 2 Wth resale, foundatonal assumptons need to be reexamned wth respect to the contnuaton play at resale. For example, a man hurdle for the fxed-pont approach to monotone equlbra s the wnner s curse, whch has been handled n the lterature by boundng t wth suffcently strong prmtve assumptons. But resale would endogenze the wnner s curse and renders t unbounded a pror, as a bdder could magnfy the wnner s curse for the rvals by actng as a hgh-bddng speculator so that hs rvals mght want to lose now and buy the good at resale. Ths paper contrbutes to the monotone pure-strategy fxed-pont approach by restructurng t wth respect to comparatve statcs of resale thereby provng exstence of a perfect Bayesan equlbrum, wth strctly ncreasng bd functons, for a two-stage game of a frst-prce aucton wth resale. Let us start by lookng nto the man steps of the monotone fxed-pont approach to see how they may fal gven resale possbltes. After that, the rest of the Introducton wll outlne how these steps are replaced by new arguments based on analyss of resale. lterature, 3 The general dea of ths fxed-pont approach, datng back to the general equlbrum s to approxmate the orgnal economy by some sequence of fnte economes where equlbra exst and then prove that a lmt pont of the sequence of such approxmaton equlbra s an equlbrum of the orgnal one. For auctons, the man mpedment to such passng-to-lmt arguments s a dscontnuty problem caused by the possblty of tes. For 1 Kotowsk 6 has a recent applcaton of the fxed-pont methods n auctons wth budget constrants. 2 The conceptual awkwardness of the no-resale assumpton has been noted by Zheng 19 and Hafalr and Krshna 4. The possblty of resource msallocaton, whch may occur at equlbrum n frst-prce auctons among ex ante dfferent bdders gven the no-resale assumpton, nduces bdders to attempt resale. 3 For example, Werner 18 and Magll and Qunz 9. 2

5 nstance, n a three-bdder case depcted by Fgure 1, each bdder plays an equlbrum bddng strategy β m, a nondecreasng functon from hs type t to a bd, n the approxmaton aucton game ndexed by m; when the sequence β1 m, β2 m, β3 m ) m=1 converges to ts lmt, a nonvanshng mass of bds, submtted by bdder 1 of types n a 1, z 1 and bdder 2 of types n a 2, z 2, are clustered wthn an nterval collapsng nto the pont x whle bdder 3 s types that bd wthn the cluster vansh nto a pont z 3 ). The crucal stage of the fxed-pont approach s to demonstrate a contradcton to the approxmaton equlbra by argung that some types of at least one of the bdders, say some elements n a 2, z 2, strctly prefer to devate from ther β2 m -bds wthn the cluster at x to a bd say x slghtly above the cluster. Ths no-te argument, due to Athey 1 and now standard wthn the fxed-pont lterature, bd 0 a 1 z 1 t 1 x x β m 1 β m 2 a 2 z 2 t 2 β m 3 0 a 3 z 3 t 3 Fgure 1: A tyng stuaton can be summarzed nto two steps, llustrated here from bdder 2 s vewpont: 4. One needs to prove that, as bdder 2 s type ncreases from a 2 to z 2, hs preference to wnnng strctly ncreases and eventually, wth suffcently hgh types, he strctly prefers to wn condtonal on the wnnng event that he can wn wth the β2 m -bds wthn the cluster at x, whch roughly corresponds to the event t 1, t 3 ) 0, a 1 0, z 3.. For the desred contradcton t suffces to show that the types obtaned n the prevous step strctly prefer to devate to x from ther β2 m -bds wthn the cluster at x. Ths was done by provng that ther expected net gans from wnnng cannot decrease when 4 The two steps correspond to Clams 1 and 2 n the Appendx of Athey 1. 3

6 they consder only the event n whch the devaton s pvotal,.e., that bdder 2 cannot prefer less to wn when the condtoned event moves from the wnnng event t 1, t 3 ) 0, a 1 0, z 3 up to the pvotal event t 1, t 3 ) a 1, z 1 0, z 3. To see the troubles, consder an ndependent prvate values model where t s bdder s use value of the good for sale. Step ) can fal because a bdder wth hgh types, say the elements of a 2, z 2 n Fgure 1, may eventually acqure and consume the good whether he wns t now or buys t later at resale. Then the type t 2 n bdder 2 s payoff as a wnner s canceled out by the t 2 n hs payoff as a loser, so hs net gan from wnnng does not ncrease n t 2, and a 2, z 2 need not contan a type that strctly prefers to wn, contrary to Step ). To consder a case where Step ) s unsalvageable, suppose wthn ths paragraph that, n Fgure 1, bdder 1 s bds wthn the cluster at x are above bdder 2 s wthn the cluster, so that bdder 1 wns when they both bd n the cluster. Thus, the wnnng event for bdder 2, when he bds wthn the cluster, corresponds to t 1, t 3 ) 0, a 1 0, z 3, whle the pvotal event for bdder 2 s devaton from the cluster to x corresponds to t 1, t 3 ) a 1, z 1 0, z 3. Athey s Step ) would argue that bdder 2 s preference to wn does not decrease when the condtoned event moves from the wnnng event to the pvotal one. Gven resale, however, the opposte can be true. For nstance, let the probablty of a 3, z 3 be so large that, condtonal on the wnnng event 0, a 1 0, z 3, f bdder 2 loses then wth a large probablty he buys the good from the types a 3, z 3 of bdder 3. By contrast, condtonal on the pvotal event a 1, z 1 0, z 3, f bdder 2 loses, he buys the good from bdder 1 wth types n a 1, z 1. Snce a 3, z 3 s hgher than a 1, z 1 n strong-set order, the resale prce offered to bdder 2, n expectaton, s hgher n the wnnng event where bdder 3 s the reseller) than n the pvotal event where bdder 1 s the reseller). Thus, when the condtoned event moves up to the pvotal one, bdder 2 s expected payoff from losng, or roughly speakng the wnner s curse, becomes hgher. On the other hand, bdder 2 s payoff from wnnng s nvarant to hs rvals types because, from Fgure 1, a 2 > z 3 > z 1 and hence f he wns then he wll consume the good to obtan ts use value t 2. Consequently, when he takes nto account that hs devaton s pvotal, bdder 2 prefers strctly less to wn, contrary to Step ). The fundamental reason why Athey s no-te argument does not work here s that a monotoncty assumpton n the lterature may fal gven resale. The assumpton stpulates that a bdder s ex post net payoff from wnnng s nondecreasng n hs rvals types e.g., A.1. of Reny and Zamr). Wth resale, by contrast, a wnner s payoff may fal to be 4

7 nondecreasng n hs rvals types because the optmal resale mechansm may resell the good to a subsdzed bdder who pays a lower prce than someone else, so the wnner s ex post resale revenue could decrease when a subsdzed bdder s type rses to buy the good from hm at resale. A loser s payoff may fal to be nonncreasng because a loser s gan from tradng wth reseller j may be larger than that wth reseller k. Thus, when j has a slghtly hgher type to become the reseller nstead of k, ths bdder s ex post payoff ncreases. Hence the ex post net gan from wnnng may fal to be nondecreasng n the rvals types. In addton to the no-te argument, two other mportant condtons, whch dd not appear dffcult n the receved lterature, become problematc gven resale. One s sngle crossng, crucal to guarantee exstence of the aforementoned approxmaton equlbra. The other s payoff securty, whch s needed to delver the passng-to-lmt result n the lterature. The sngle-crossng condton says that f a bdder prefers a hgh bd to a low one then the preference remans so when hs type gets hgher. The lterature obtaned ths condton by assumng t for every possble profle of realzed types e.g., A.1.v of Reny and Zamr). Wth resale, the assumpton fals when an ncrease of a bdder s type turns hm from a speculator to a consumer, wth suffcently hgh types of hs rvals. 5 The payoff-securty condton says that bddng slghtly above an atom of the rvals bds does not make a bdder worse-off than bddng at the atom. In the lterature, verfcaton of ths condton s smply Step ), 6 whch as llustrated above can fal wth resale. Ths paper s devoted to overcomng these challenges that resale presents to the fxedpont approach. To capture the endogenous nature of resale, we assume that the resale mechansm s a reseller-optmal aucton à la Myerson 13 based on post-aucton belefs. Athey s crtcal steps are restructured wth respect to new comparatve statcs propertes of the Myerson resale aucton, wth ntal bds or post-aucton belefs beng the parameters. The exstence proof starts by establshng an ncreasng-dfference theorem Theorem 1), whch through ts sngle-crossng mplcaton ensures exstence of the aforementoned approxmaton equlbra. It s based on two comparatve statcs propertes of the Myerson resale mechansm Propostons 1 and 2). Then comes the crtcal step, the no-te argument, 5 Whle the hgher bd brngs about hgher revenues for the speculator-type snce he charges hgher resale prces due to the hgher posteror about the wllngness-to-pay of hs clentele, the consumer-type, who benefts from none of such revenue effect, strctly prefers the lower bd, whch costs hm less. Ths also upsets a slghtly weaker sngle-crossng assumpton proposed by Quah and Strulovc 14, Th. 4c), p28. 6 For example, the dsplayed formula A.5) n Reny and Zamr 16. 5

8 to prove that tes do not occur at a lmt pont of a sequence of such approxmatng equlbra Theorem 2). Wth ts counterpart n the receved lterature hndered by resale, our no-te argument s complcated and reles on new propertes of endogenous resale uncovered n ths paper. Then a passng-to-lmt argument delvers the exstence theorem Theorem 3). The frst step of our no-te argument s to prove that, f a te at the lmt occurs then there exsts a domnant bdder whose probablty of wnnng the te converges to one Lemma 8). In Fgure 1, for nstance, the nfmum a 1 of bdder 1 s types that bd wthn the cluster at x s less than all elements of a 2, z 2, bdder 2 s types bddng n the cluster. Consequently, wth types beng use values of the good, condtonal on the pvotal event t 2, t 3 ) a 2, z 2 0, z 3 of the bd ncrease from the cluster to x, bdder 1 would have zero gan from tradng wth the reseller player 2. I.e., bdder 1 would suffer zero wnner s curse wth the bd ncrease. On the other hand, the bd ncrease generates a revenue effect by addng a mass of hgh types a 2, z 2 to bdder 1 s clentele thereby ncreasng hs expected resale revenue by a postve amount Lemma 9, due to a property of the optmal resale mechansm proved n A.2.1). 7 Thus, bdder 1 wth types nearby a 1 would strctly prefer to devate unless wthn the cluster hs bds are almost exclusvely on the top layer so that he mostly outbds the tyng rvals. Hence bdder 1 s the domnant bdder. To derve a contradcton from the supposed occurrence of a te, our next step s to prove that some bdder who s supposed to bd just below the domnant rval wthn the tyng cluster, such as bdder 2 n Fgure 1, strctly prefers to devate to a bd slghtly above the cluster. The proof, from to 5.2.5, s nontrval because the wnner s curse for bdder 2 s not neglgble. Contrary to the case of bdder 1, even the nfmum a 2 of the atom-bddng types of bdder 2 can gan from buyng the good at resale from some atom-bddng types of bdder 1, as a 2 > a 1. Ths nontrval wnner s curse s handled n two substeps. Frst, we prove that f a 2, z 2 contans some suffcently hgh types then for such types of bdder 2 the wnner s curse s more than outweghed by the wnner s blessng payoff from wnnng condtonal on the pvotal event). Then he strctly prefers the devaton 5.2.3, due to a 7 Note that the revenue effect s null n no-resale models. In other words, notwthstandng zero wnner s curse, Athey s no-te argument stll cannot be replcated to prove that bdder 1 strctly prefers the hgher bd. Even f her Step ) works, so that bdder 1 s preference to wn strctly ncreases n hs type on a 1, z 1 condtonal on hs wnnng event, hs preference may stll be reversed when the condtoned event swtches to the pvotal event. That s because hs ex post payoff from wnnng may fal to be nondecreasng n hs rvals types, as explaned above regardng the monotoncty assumpton. 6

9 property of the optmal resale mechansm proved n A.3). Second, n the other case, we fnd some types n a 2, z 2 for whom the wnner s curse s nearly balanced by the wnner s blessng. Ths s done by deducng the vablty of bdder 2 s devaton from the proftablty of bdder 1 s on-path acton despte nformaton asymmetry between them 5.2.4). 8 Then the revenue effect of the devaton, as n the case for bdder 1 n the prevous paragraph, mples bdder 2 s strct ncentve to devate 5.2.5), whch delvers the no-te theorem. In the receved lterature, a no-te theorem would have suffced the passng-to-lmt argument, as the aforementoned payoff-securty condton s mpled by smply repeatng Step ) n Athey s argument. Not so wth resale, because as explaned prevously the monotoncty assumpton may fal. Wth the monotoncty assumpton, Athey s Step ) s accomplshed wthout relyng on any equlbrum condton. Wthout ths assumpton, our no-te argument reles on the condton that the devant bdder 2 s supposed to bd at the cluster accordng to the approxmaton equlbra so that the devaton to x n Fgure 1 costs hm only an nfntesmal ncrease of payment). But such an equlbrum condton s not avalable when the payoff-securty condton s beng consdered. To avod ths problem I assume that the reserve prce of the ntal aucton s zero. Then the no-te theorem mples that the approxmaton equlbra at the lmt allow for only nconsequental atoms, whch stand no chance to wn Lemma 14). To complete the passngto-lmt argument, therefore, t suffces to handle such atoms. Here complcatons can occur when a bdder can change the nconsequentalty of an atom wth a unlateral devaton, whch could cause dscontnuty at the lmt. Ths problem s solved by Lemma 15. Then the exstence proof s complete. Ths exstence theorem s more general than prevous results n frst-prce auctons wth resale n that t allows for any fnte number of dfferently dstrbuted bdders whle the prevous lterature assumed ether two bdders or at most two knds of bdders ex ante, wth bdders of the same knd drawn from the same dstrbuton. Notwthstandng some remarkable results n ths lterature, such as Garratt and Tröger 2 n mxed strateges and Hafalr and Krshna 4, Lebrun 7, 8 and Vrág 17 n pure strateges, the two-dstrbuton 8 The deducton, consstng of Lemmas 11 and 12, s based on two nontrval facts. Frst, bdder 2 can nearly mmc bdder 1 s optmal resale mechansm n the event of the te, largely due to the fact that bdder 1 s the domnant rval. Second, the expected revenue produced by a fxed Myerson aucton does not decrease when the weght of a bdder s type s pushed upward Lemma 22, proved here despte the fact that the ex post revenue generated by a Myerson aucton need not be nondecreasng n a bdder s type). 7

10 assumpton has been crucal to ther dfferental equatons method. Nevertheless, the exstence theorem s stll restrcted by the aforementoned assumpton of zero reserve prce, as well as several other assumptons such as the prvacy of a loser s bd n the ntal aucton, common nfmum for bdders pror supports, and a reseller s power to choose resale mechansms. These assumptons, however, are common n the current auctonresale lterature such as those cted above as well as Zheng 19 and Garratt, Tröger and Zheng 3. 9 Now that the exstence proof has shown t feasble to extend the fxed-pont approach beyond ts prevous confnes of no-resale sngle-stage models, nvestgatons of ts further expanson, ncludng dspensablty of these assumptons, are at hand. 2 The Model 2.1 The Aucton-Resale Game There are two perods, a fnte set I of bdders, and an ndvsble good. For each I, bdder s type, or use value of the good, s ndependently drawn from a commonly known dstrbuton F, wth the realzed value prvately known to. In perod one, every bdder submts as hs bd an element of l B, where l < 0 denotes the losng bd, amountng to nonpartcpaton n the perod-one aucton, and B r, ) s the set of serous bds admssble for bdder, wth reserve prce r 0 for all bdders. Tes are broken randomly and unformly wth equal probabltes. If the good s sold then, after the wnner s selected, the hghest bd and the wnner s dentty are announced publcly, wth nothng else dsclosed, 10 and the wnner pays for the good at the prce equal to hs wnnng bd. Then perod two starts and the perod-one wnner chooses a sellng mechansm that offers resale to the other bdders n I, called losng bdders. A sellng mechansm s any game form to be played by the losng bdders. After the players have acted gven ths mechansm, the entre game ends. Every bdder s assumed rsk-neutral n hs payoff, defned to be hs use value, f he s the fnal owner of the good, plus the net monetary transfer he receves from others. Dscountng 9 Zheng 19 dd not assume common nfmum of the prors but made some other assumptons. Hafalr and Krshna 4 and Lebrun 7, 8 consdered some other dsclosure polces and weaker barganng power of the reseller based on the two-dstrbuton assumpton and take-t-or-leave offers as the resale mechansm. 10 If the acton of a losng bdder s also dsclosed, pure-strategy equlbrum s unlkely to exst unless the loser gets to choose the resale mechansm. 8

11 s assumed away for smplcty. Assume for every bdder the pror F has dfferentable and strctly postve densty f on ts support T := 0, t, wth pror vrtual utlty t 1 F t ))/f t ) havng strctly postve dervatve wth respect to t on T. Denote T := Π k I\ T k and T := Π k I T k. 11 A profle β ) I of bd functons, wth β : T l B for each I, s sad monotone f and only f β s a weakly ncreasng functon for each I,.e., everyone s perod-one bd s weakly ncreasng n hs use value of the good. 2.2 Boldfaced Symbols for Random Varables Denote bdder s type by t as the random varable and t as the realzed value. Denote t := t k ) k I\ and t := t k ) k I\ as the random vector and the realzaton for the type profle across rvals of. Analogously, denote t := t, t ) := t k ) k I, t := t, t ) := t k ) k I, t,j) := t k ) k I\,j and t,j) := t k ) k I\,j. Denote Egx) for the expected value of any functon g of the random varable or random vector x, wth the random varable/vector boldfaced, based on the pror dstrbutons. Denote Egx) E for the expected value condtonal on event E, 1E for the ndcator functon of event E, and PrE := E 1E. 3 The Endogenous Payoff Functons We shall derve a bdder s expected payoff n the aucton-resale game from a contnuaton equlbrum at the resale stage, whch mplements a reseller-optmal aucton à la Myerson Contnuaton Equlbrum at Resale Atoms and Inverse Images of Bds If β : T R s a weakly ncreasng functon, denote for any b β 0) β 1 b) := t T : β t ) = b, β 1,nf b) := supt T : β t ) < b, 1) β 1,sup b) := supt T : β t ) b. 2) 11 The assumpton that bdders have a common nfmum of ther pror supports s used n Lemmas 15 and 23. The postve-dervatve assumpton of pror vrtual utltes s slghtly stronger than the usual one that requres only strct monotoncty. The strengthenng s needed n Lemmas 12 and 15. 9

12 We adopt the conventon of lettng sup S := nf S := 0 when a subset S of T s empty. Note that f β 1 b) then β 1,nf b) = nf β 1 b) and β 1,sup b) = sup β 1 b). For any bdder, an atom of β means a bd b B such that β 1 b) s a nondegenerate nterval,.e., β 1,nf b) < β 1,sup b). An atom of β, wth β := β j ) j, means an atom of β j for some j I \. Lkewse, an atom of β := β j ) j I means an atom of β j for some j I Publc Hstores and Posteror Belefs If bdder wns wth bd b n perod one so b > l,.e., b B ) then, b ) denotes the commonly known publc hstory. Gven any publc hstory, b ), wth every losng bdder k k ) havng played accordng to β k, the posteror dstrbuton F k, b, β) of t k s derved from Bayes s rule based on the observaton that k has been defeated ether because β k t k ) < b or because β k t k ) = b and k dd not wn the te-breakng lottery. Lemma 1 For any publc hstory, b ), any monotone profle β, and any k, the densty f k, b, β) of F k, b, β) s fnte and strctly postve on ts support 0, β 1 k,sup b ) ; f b s not an atom of β k then f k, b, β) s contnuous on ths posteror support; else f k, b, β) s contnuous at all but one pont n the posteror support. Proof Appendx C Posteror Vrtual Utltes For each losng bdder k I \ n publc hstory, b ), defne V k,b,β : T k R by t k 1 F kt k,b,β) f V k,b,βt k ) := V k t k b, β) := k t k,b f t,β) k β 1 k,sup b ) β 1 k,sup b ) f t k β 1 k,sup b ), 3) and defne the posteror vrtual utlty functon for losng bdder k to be ether V k,b,β f b s not an atom of β k, or the roned verson of V k,b,β accordng to Myerson s 13 procedure f b s an atom of β k. By the prevous and the next lemmas, V k,b,β fals to be monotone and hence ronng s needed precsely when the wnnng bd b s an atom of β k. Denote k s posteror vrtual utlty by V k,,b,βt k ) or V k t k, b, β) When the wnnng bd b s an atom of β k, the posteror dstrbuton of t k depends on by Eq. 75). Hence the notaton for the wnner n the roned posteror vrtual utlty functon V k,,b,β cannot be dropped. 10

13 Lemma 2 There exsts λ > 0 such that, for any publc hstory, b ), any monotone profle β, and any k, f b s not an atom of β k, then: a. for any t k T k, V k,,b,βt k ) = V k,b,βt k ) and, f t 0, β 1 k,sup b ), V k,b,βt k ) = t k F k β 1 k,sup b ) ) F k t k ) ; 4) f k t k ) b. V k,b,β s strctly ncreasng on 0, β 1 k,sup b ), at a rate greater than or equal to λ, and s constant on β 1 k,sup b ), t k ; c. f b > b and b s not an atom of β k, then V k,b,β V k,b,β on 0, β 1 k,sup b ) ; d. V k,b,β s contnuous on T k ; Proof Appendx C Resale Mechansms Gven any publc hstory, b ), by Lemma 1, Myerson s 13 characterzaton of optmal auctons s applcable to the aucton-desgn problem for our reseller. 13 Thus, the mechansm M b, t, β) defned below s optmal for the bdder-turned reseller wth type t T : a. each losng bdder k ndependently submts a report, say t k, of hs type; b. for any t T, resells the good to a bdder k such that V k t k, b, β) = max t, max V j t j, b, β) ; j f there are more than one such bdders then pcks one of them through an equalprobablty lottery; f no such k exsts then keeps the good; c. for any k, f bdder k s resold the good then the payment k delvers to equals p k,,b,βt k ) := nf t k T k : V k,,b,βt k) max t, max V j,,b,βt j ) ; 5) j I\,k f k s not resold the good then k pays zero to. 13 Myerson 13 assumed contnuous densty throughout a bdder s support whle our posteror densty may be dscontnuous at one pont Lemma 1). But ths dfference does not affect Myerson s result. Also see Footnote 9 of Garratt, Tröger and Zheng 3 for an explanaton why Myerson s result s applcable here despte the possblty that the reseller may be prvately nformed of her type. 11

14 Followng drectly from Myerson s result, we have Lemma 3 For any publc hstory, b ), any t T and any monotone profle β, f the posteror belef of t j s F j, b, β) for each j, then t s a contnuaton equlbrum for player to choose M b, t, β) and everyone else to partcpate and be truthful. For any publc hstory, b ), f b s not an atom of β, then Lemma 2 mples that, for any losng bdder k, the posteror vrtual utlty functon V k,,b,β s equal to the strctly ncreasng functon V k,b,β on the posteror support 0, β 1 k,sup b ) of t k, hence for any t k such that bdder k of type t k wns n M b, t, β).e., max t, max j I\,k V j,b,βt j ) β 1 k,sup b )), Eq. 5) s smplfed to, wth V 1 k,b,β denotng the nverse functon of V k,b,β, p k,,b,βt k ) = V 1 k,b,β max 3.2 The Payoff from the Aucton t, The Indcator Functon for Wnnng max j I\,k ) V j,b,βt j ). 6) The unform te-breakng rule corresponds to a random vector ρ ) I subject to two condtons: ) for any realzaton ρ ) I, ρ 1,..., I for any I, and ρ ρ j for any j; and ) any such realzaton has the same probablty. The nterpretaton s that f ρ > ρ j then bdder beats j n the con toss when ther bds are ted. For any realzaton ρ k ) k I of the unform te-breakng lottery, any I, any J I\, and any profle b k ) k J of bds across bdders n J, wrte, b ) ρk ) k I b k ) k J, or brefly, b ) b k ) k J, f and only f b B and b > max k J b k or b = max b k and k arg max k J j J b j : ρ > ρ k. And wrte, b ) b k ) k J f and only f, b ) b k ) k J s not true. For example, 1, b ) β k t k )) k I\ s the ndcator functon for the event that bdder wns, possbly after te-breakng, wth bds b from and β k t k ) from each rval k. 12

15 3.2.2 Ex Post Payoff for a Wnner For any publc hstory, b ) and any t, t ) T T, defne W t b, t, β) to be the payoff for player when wns at the ntal aucton wth bd b and offers resale va the Myerson aucton M b, t, β) accordng to the contnuaton equlbrum specfed n Lemma 3, when rvals of abde by the monotone profle β n perod one and the profle of realzed types across other players happens to be t. That bdder wns wth bd b mples b B. For the case b / B,.e., b = l, defne W t l, t, β) := 0. If a serous bd b.e., b B ) s not an atom of β, one can derve from Lemmas 2 and 3 that, for all t k 0, β 1 k,sup b ) except a set of measure zero and for any t T, W t b, t, β) = t 1 t > max V k t k b, β) 7) k + p j,,b,βt j )1 V j t j b, β) > max t, max V k t k b, β). k /,j j Ex Post Payoff for a Losng Bdder For any dstnct bdders j and any t, t ) = t, t j, t,j) ) T T j T,j) such that β j t j ) B j and β j t j ) β k t k ) for all k I \, j, defne L j t t, β) to be the payoff for player when bdder j wns at the ntal aucton wth bd β j t j ) and offers resale va mechansm M j β j t j ), t j, β) accordng to the contnuaton equlbrum, when everyone s supposed by other players to abde by the monotone profle β n perod one and the profle of realzed types across bdders happens to be t, t j, t,j) ). Note that L j t t, β) s nvarant to s perod-one bd b, due to the fact that reseller j n choosng resale mechansms does not know the bds from the losng bdders. If β j t j ) s not an atom of β j then, as n the prevous case for W, for any j, for all t,j) k /,j 0, β 1 k,sup β jt j )) but a set of measure zero, and for any t T, L j t t, β) = t p,j,βj t j ),βt ) ) 1 V,βj t j ),βt ) > max t j, max V k,β j t j ),βt ). 8) k /,j Before the aucton outcome s announced n perod one, bdder does not know who s the wnner, but he knows that, at any realzed type profle t T, f he loses the aucton then the wnner s selected from I \ wth each rval k I \ bddng β k t k ). Thus, s ex post payoff from losng, gven any realzed type profle t, t ) T, s equal to L t t, β) := Pr j, β j t j )) β k t k )) k I\,j L j t t, β). 9) j 13

16 3.2.4 Interm Expected Payoff Denote U b, t, β) for type-t bdder s expected payoff n the entre game from bddng b n perod one followed by the contnuaton equlbrum specfed by Lemma 3, provded that everyone else abdes by the monotone profle β at perod one. Thus, U b, t, β) = E 1, b ) β k t k )) k I\ W t b, t, β) b L t t, β)) +E L t t, β), 10) where the boldfaced letters nsde the expectaton operator E denote the random varables. Snce W and L are derved from the contnuaton equlbrum at resale, we obtan a perfect Bayesan equlbrum f the perod-one bd functons best reply one another: Lemma 4 If a monotone profle β ) I of perod-one bd functons consttutes a Nash equlbrum, across almost all bdder-types, wth respect to the nterm expected payoff functons U,, β)) I gven by Eq. 10), then β ) I coupled wth the contnuaton play characterzed n Lemma 3 consttutes a perfect Bayesan equlbrum of the aucton-resale game. 4 Increasng Dfference Based on comparatve statcs of the contnuaton equlbrum, the frst theorem says that the dfference n a bdder s expected payoff due to an ncrease n hs perod-one bd s weakly ncreasng n hs type, provded that tes occur wth zero probablty,.e., I : j I \ : b B : b s not an atom of β j. 11) Theorem 1 ncreasng dfference) For any bdder, any monotone profle β of bd functons satsfyng Eq. 11), and any b, b B l such that b > b, U b, t, β) U b, t, β) s a weakly ncreasng functon of t throughout T. Ths property s due to a relatonshp between perod-one bds and the fnal allocaton after resale Propostons 1 and 2), whch say that hgher perod-one bds mply hgher probabltes of beng the fnal owner of the good. Ths relatonshp mples the ncreasng dfference property through the payoff-equvalence routne n mechansm desgn. Wth notatons and lemmas ntroduced n , the proof of the theorem s completed n 4.4. Eq. 11) s needed to ensure that the posteror vrtual utlty functons are well-behaved. 14

17 4.1 Fnal Allocatons For any bdder, any monotone profle β of bd functons, and any t := t k ) k I T, defne: Q b, t, β) to be the probablty wth whch s the fnal owner n the contnuaton equlbrum Lemma 3) condtonal on the publc hstory, b ), when b B and the realzed type profle s t f b / B,.e., b = l, then defne Q b, t, β) := 0); q j t, β) to be the probablty wth whch s the fnal owner n the contnuaton equlbrum Lemma 3) condtonal on the publc hstory j, β j t j )), when β j t j ) B j and the realzed type profle s t f β j t j ) = l then defne q j t, β) := 0); q t, β) to be the probablty wth whch s the fnal owner when some rval of wns the perod-one aucton and offers resale accordng to the contnuaton equlbrum,.e., q t, β) = Pr j, β j t j )) β k t k )) k I\,j q j t, β). 12) j If b B s not an atom of β, then one can derve from Lemmas 2 Clams a and b) and 3 that, for all t k 0, β 1 k,sup b ) but a set of measure zero and for any t T, Q b, t, β) = 1 t max k I\ V k t k b, β). 13) Analogously, for any t j T j wth β j t j ) B j, f β j t j ) s not an atom of β j then for any j, for all t,j) k /,j 0, β 1 k,sup β jt j )) but a set of measure zero and for any t T, q j t, β) = 1 V t β j t j ), β) max t j, max V k t k β j t j ), β). 14) 4.2 The Envelope Condton k I\,j For any bdder, defne wth boldfaced letters denotng random varables): W b, t, β) := E W t b, t, β), b ) β j t j )) j I\, 15) L b, t, β) := E L t t, β), b ) β j t j )) j I\, 16) Q b, t, β) := E Q b, t, t, β), b ) β j t j )) j I\, 17) q b, t, β) := E q t, t, β), b ) β j t j )) j I\. 18) The next lemma follows from the Mlgrom-Segal envelope theorem

18 Lemma 5 For any I, any b B l, and any monotone profle β, the functons W b,, β) and L b,, β) are absolutely contnuous and, for any t T, W b, t, β) = W b, 0, β) + L b, t, β) = Proof Appendx D. t 0 t 0 Q b, τ, β)dτ, 19) q b, τ, β)dτ. 20) 4.3 Intal Bds and the Fnal Allocaton The comparatve statcs n Propostons 1 and 2 are about ex post probabltes condtonal on the profle of realzed types across all bdders, not to be confused wth expected probabltes. Proposton 1 For any I and any monotone profle β satsfyng Eq. 11), f b > b then Q b, t, β) Q b, t, β) for any t T and almost every t k 0, β 1 k,sup b ). Proof Appendx D. Propostons 1 says that f a bdder wns the ntal aucton then hs probablty of eventually keepng the good cannot be lower had he submtted any hgher bd. The ntuton s that a hgher wnnng bd would make the wnner thnk more hghly about the losng bdders wllngness to pay and hence set hgher reserve prces. Consequently, gven the same realzed types, hs mechansm results n no resale wth a hgher probablty. Proposton 2 For any bdders j and any monotone profle β satsfyng Eq. 11), Q b, t, β) q t, β) for any t T and almost every t T such that b max k β k t k ). 14 Proof Appendx D. Proposton 2 says that a bdder s more lkely to become the fnal owner of the good when he s the reseller than when he s a potental buyer at resale. Ths s smlar to an elementary economcs fact that a monopolst who cannot perfectly dscrmnate ts potental buyers would under-supply ts goods. The monopolst at resale, our reseller would not resell the good wthout a prce markup above her own use value, whle potental buyers are wllng to pay for t at any prce not exceedng ther use values. 14 Proposton 2 extends Lemma 1 of Garratt, Tröger and Zheng 3 to the ex post perspectve. 16

19 4.4 Proof of Theorem 1 By Eqs. 10), 15) and 16), U b, t, β) = E 1 b t W b, t, β) b ) + E 1 b t L b, t, β), 21) where b t s a shorthand for s wnnng event, b ) β k t k )) k I\, and b t ts complement. For any b > b, let U t ) := U b, t, β) U b, t, β). By Eq. 21), U t ) = E 1 b t ) W b, t, β) b E 1 b t ) W b, t, β) b +E 1 b t L b, t, β) E 1 b t L b, t, β). Dfferentate ths equaton wth respect to t and then plug nto the rght-hand sde the envelope equatons 19) and 20) and the equatons 17) and 18) for Q and q to obtan U t ) t = E 1 b t Q b, t, t, β) 1 b t Q b, t, t, β) +E 1 b t q t, t, β) 1 b t q t, t, β). The rght-hand sde, after rearrangements, wth notaton β suppressed, s equal to E 1 b t Q b, t, t ) Q b, t, t )) + E 1 b t, b t Q b, t, t ) q t, t )). =:X =:Y For any t at whch the ndcator functon nsde the ntegral X s nonzero, b max j β j t j ) and hence Proposton 1 apples; for any t at whch the ndcator nsde Y s nonzero, b max j β j t j ) and hence Proposton 2 apples. Thus, both X and Y are nonnegatve. Hence t U t ) 0 for any t nteror to T. Ths, coupled wth the fact that U t ) s absolutely contnuous n t snce U by Eq. 21) s a lnear combnaton of W and L, each absolutely contnuous n t by Lemma 5), mples the monotoncty of U. 5 Equlbra of the Approxmaton Games Based on Theorem 1, f the bd spaces n the ntal aucton are replaced by some dscrete spaces, a monotone equlbrum exsts. To obtan equlbrum n the orgnal game, we shall prove that the equlbrum property of such approxmaton equlbra s passed onto the lmt when the dscrete bd spaces converge to the orgnal one. A crtcal step of the proof s to show that tes occur wth zero probablty at the lmt Theorem 2). As explaned n the Introducton, our no-te argument s sgnfcantly dfferent from that n the lterature. 17

20 5.1 The Approxmaton Games For any m = 1, 2,..., defne an m-approxmaton game by replacng for any bdder the space B of serous bds wth a dscrete set B m such that j = B m B m j =, 22) m < m = B m B m, 23) mn b b : b, b B m ; b b = 2 m, 24) lm m mn B m = r, lm m sup B m =. The man condton s Eq. 22), devsed by Reny and Zamr 16 because ther sngle-crossng condton, lke our ncreasng-dfference theorem, apples only to non-atom bds. 15 The condton ensures that, n any m-approxmaton game, a bdder s serous bd s never an atom of a rval s bd functon. Consequently, a bdder s wnnng event s smplfed:, b ) β k t k )) k I\ b > max β j t j ). 25) j Another consequence s that the posteror vrtual utlty functons are smplfed to Eq. 4) due to Lemma 2.a. More mportantly, Theorem 1 apples, so U b, t, β) has the ncreasng dfference property n any m-approxmaton game. For any m = 1, 2,...,, a profle β m ) I of functons β m : T l B m s an m- equlbrum f and only f, for any bdder and any t T, If, n addton, β m b m B m l : U β m t ), t, β m ) U b m, t, β m ). 26) s weakly ncreasng for every, then the m-equlbrum s sad monotone. The next proposton follows from Kakutan s fxed pont theorem appled to each m-approxmaton game based on the sngle-crossng property mpled by Theorem 1. The proof s the same as Athey s 1, Theorem 1 and hence omtted. Proposton 3 For any m = 1, 2,..., there exsts a monotone m-equlbrum. 15 Not needed here s the other perturbaton devsed by Athey 1 and adopted by Reny and Zamr, that a bdder has to submt the losng bd l when hs type belongs to 0, 1/m). They need the perturbaton to ensure a revealed-preference result. It would be redundant n ths paper because our revealed-preference result s ensured by an upcomng noton of consequentalty, whch s needed anyway for our no-te argument. 18

21 By revealed preference, at any m-equlbrum a bdder never bds more than hs expected payoff as a wnner f he stands a postve probablty of wnnng: Lemma 6 For any m = 1, 2,..., f β m ) I s an m-equlbrum then for any I and any t T such that Pr β m t ) > max k β m k t k) > 0, we have W β m t ), t, β m ) β m t ) 0. Proof Applyng Ineq. 26) to the case b m = l and usng Eqs. 10) and 25), we have W Pr β m t ) > max βk m t k ) β m t ), t, β m ) β m t ) L β m t ), t, β m ) ) 0. k By the hypothess Pr β m t ) > max k β m k t k) > 0, the term n the bracket ) s nonnegatve. Then the concluson of the lemma follows from L β m t ), t, β m ) 0, whch s true because can choose not to partcpate n the resale mechansm. 5.2 Impossblty of Tes at the Lmt Gven a monotone profle β of bd functons, call a serous bd b consequental f Prβ k t k ) b > 0 for every bdder k I, and nconsequental f otherwse. A te of β means a serous bd that s an atom for at least two dstnct bdders accordng to ther bd functons n β. Theorem 2 no te) If a sequence β m ) m=1 of monotone m-equlbra converges pontwse almost everywhere to a monotone profle β, then β admts no consequental te. To prove Theorem 2, suppose to the contrary that β admts a consequental te b. We shall derve a contradcton to the equlbrum property of the sequence β m ) m=1. As a prelmnary, the next lemma provdes a mnute pcture of the clusters of rvalng bds collapsng to the atom b as m. Lemma 7 If a sequence β m ) m=1 of monotone profles converges pontwse a.e. to a monotone profle β and f J s the set of bdders such that a serous bd b s an atom of β j all j J, then there exst subsequence β mn ) n=1 and sequence δ n) n=1 0 such that, wth a := sup t T : β t ) < b, z := sup t T : β t ) b, 27) a n := nf t T : β mn t ) > b δ n, 28) z n := sup t T : β mn t ) < b + δ n 29) 19 for

22 for each, we have: J : J : t a n, z n ) : b δ n < β mn t ) < b + δ n, 30) lm n Pr t T \ a n, z n ) : b + δ n β mn t ) b + δ n + 2 mn = 0, 31) I : a = lm n a n, z = lm n z n, 32) k / J : lm n Pr t k T k : b δ n β mn k t k ) b + δ n + 2 mn = 0. 33) Proof Appendx E.1. Wth the δ n ) n=1 n Lemma 7, the collapsng nterval b δ n, b + δ n ) s the range of the β mn -bds for those types of bdder n a n, z n ), says Ineq. 30). Along the subsequence β mn ) n=1, Eq. 31) says that the probablty wth whch the types outsde an, z n ) would bd wthn b δ n, b + δ n ) vanshes, Eq. 32) says that a n, z n ) converges to a, z ), and Eq. 33) says that f βk has no atom at b then the probablty wth whch player k bds n b δ n, b n ), wth b n beng any bdder s lowest grd pont above b + δ n, goes to zero. Gven the subsequence β mn ) n=1 dentfed n Lemma 7, for each n denote β n := β mn. By Eq. 33) and the consequentalty of b, we have k / J : lm n Pr βn k t k ) < b δ n > 0. 34) For any n 1, 2,..., any, any t n T and any bds b n and c n n B mn wth b n > c n, the expected-payoff dfference for a type-t n bdder caused by hs bd ncrease from c n to b n n the m n -equlbrum β mn s U n t n ) := U b n, t n, β n ) U c n, t n, β n ). 35) To prove Theorem 2 by contradcton, t suffces to fnd a bdder and a sequence t n, c n, b n ) n=1 such that lm sup n U n t n ) > 0 and, for any suffcently large n, the β n -nverse-mage of c n s nondegenerate and contans t n. Then for all suffcently large n, U n t n ) > 0 and, wth U n ) contnuous Lemma 5), the strct nequalty extends to a neghborhood of t n, whch contradcts the fact that β n consttutes an m n -equlbrum. To ths end, decompose U n t n ) nto three parts proved n Appendx E.2): U n t n ) = W n t n ) b n + Π n t n ), 36) 20

23 where W W n t n ) := Pr b n > max β k n t k ) b n, t n, β n ) W c n, t n, β n ) ), 37) k b n := b n c n ) Pr b n > max β k n t k ), k W Π n t n ) := Pr b n > max β k n t k ) > c n c n, t n, β n ) c n L n t n ) ), 38) k L n t n ) := E L t t n, β n ) bn > max β k n t k ) > c n. k Eq. 36) says that U n t n ) conssts of the revenue effect W n t n ), payment effect b n, and pvotal effect Π n t n ), whch ncludes L n t n ), the wnner s curse n our context Step 1: Locatng a Devant Bdder Recall the set J of tyng rvals specfed n Lemma 7. Pck an element j J such that k J : a j a k. 39) Wth B mn j dscrete, there exsts c n j := mn βn j t j ) : t j a n j, z n j Lemma 8 lm n Pr c n j < max k J\j βn k t k ) b + δ n = 0. ). 40) Lemma 8 s proved n Appendx E.3. It can be understood from the vewpont of those types of bdder j nearby a j. If the lemma were not true, there would be a mass of rvalng bds wthn b δ n, b + δ n ) that outbd such types of bdder j, and the mass would not vansh along the sequence of the approxmaton equlbra. On one hand, wth valuaton nearly equal to a j and wth Ineq. 39), such types of bdder j would have almost zero gan from buyng the good from these rval-types at resale,.e., the wnner s curse for such types of bdder j to jump over these rval-types s neglgble. On the other hand, f such a lowvalue bdder j outbds these rval-types, he would proft from resellng to them, agan due to Ineq. 39); wth the mass of these rval-types nonvanshng, ths expected proft s bounded away from zero. Both sdes consdered, bdder j wth types nearby a j would devate to a bd slghtly above b δ n, b + δ n ) f Lemma 8 does not hold. For any n = 1, 2,... and any J \ j, wth c n j c n := max βn t ) : t 21 0, βn ) 1,nf cn j ) defned n Eq. 40), let ). 41)

24 For any suffcently large n, 0, ) ) 1 βn,nf cn j ) due to Lemma 8 and the hypothess that b s consequental; wth B mn dscrete, c n exsts. Snce J \ j s fnte, there exsts J \ j wth c nγ nfnte subsequence n γ ) γ=1. For ths, lm γ Pr c nγ = max k J\j c nγ k < max k J\j, βn γ k for all γ n an = 0. t k) < c nγ j Combnng ths wth Lemma 8 and Eq. 41) and relabelng subsequence n γ ) γ=1, we have lm Pr c n n < max β k n t k ) < b + δ n = 0. 42) k J\j Thus, as n, the m n -equlbrum bds from all players other than bdder j vansh from c n, b + δ n ). By c n < c n j, the nterval c n, b + δ n ) s almost exclusvely occuped by the bds from bdder j wth types n a n j, z n j ), whch converges to the nondegenerate a j, z j ) snce b s an atom of β j. Ths coupled wth Eq. 34) consequentalty of b ) mples lm Pr c n n < max β k n t k ) < b + δ n > 0. 43) k I\ By constructon, c n < c n j < b + δ n ; by Eq. 42), the mass of s bds n c n, b + δ n ) vanshes whle, wth J, a nonvanshng mass of s bds remans n b δ n, b +δ n ). Thus, for all large n, c n > b δ n and hence By Eq. 41), the β n -nverse-mage of c n b δ n < c n < c n j < b + δ n. 44) s nondegenerate. To complete the proof by contradcton, t suffces to prove exstence of a sequence t n ) n=1 such that each t n belongs to ths nverse mage and lm sup n U n t n ) > 0, wth U n t n ) the expected-payoff dfference rendered by the devaton from c n to b n := mn b B mn : b b + δ n. 45) To ths end, we calculate the three components of U n t n ) accordng to Eq. 36). Among them, the payment effect b n Ineq. 44) and b n c n s Oδ n ) hence O1/n) by Lemma 7) because of 2 mn + b + δ n c n, whch follows drectly from Eq. 45). Thus, we need only to calculate the revenue effect W n t n ) and pvotal effect Π n t n ) Step 2: The Revenue Effect of the Devaton By a revealed-preference argument, one can prove W n 0 Proposton 4, Appendx A.2.1). The next lemma asserts further that the revenue effect s bounded away from zero f bdder has potental gan of trade wth hs rvals when he wns wth the hgher bd. 22

25 Lemma 9 If t n n t such that 0 < t < max k z k, then lm sup n W n t n ) > 0. Proof Appendx E.5. By Eq. 32), z k s the lmt of the supremum zk n of bdder k s types that bd below bn n the m n -equlbrum. Hence the condton t n n t such that t < max k z k mples that, for all approxmaton equlbra suffcently far along the sequence, bdder can proft from resellng the good to hs rvals f he wns wth the bd b n. By Eq. 43), the mass of rval-types surpassed by the bd ncrease does not vansh along the sequence. Hence the bd ncrease brngs about a nonvanshng ncrease of resale probablty and expected revenue at resale, gven the possble gan of resale hypotheszed n ths lemma Step 3: Pvotal Effect Case One: Bypassng the Mddleman Two cases need to be consdered on the pvotal effect Π n t n ). In the frst case, bdder s type s so hgh that, n the event of tyng at b and he loses to bdder j, he buys the good nearly for sure from bdder j. Essentally a mddleman, bdder j charges ths type of a prce markup n addton to the perod-one prce. In makng the bd ncrease thereby surpassng j, bdder avods payng the prce markup, whch consttutes the pvotal effect n ths case. More precsely, for any k I and any x T k, defne t k F k x) F k t k ))/f k t k ) f 0 t k x V k,x t k ) := x f t k x. By Lemma 25 Appendx E.4, due to Eq. 42)), when bdder j wns wth a bd b n the collapsng c n j, b + δ n ), every losng bdder k s posteror vrtual utlty functon converges to V k,zk as n. Hence the precse meanng of our frst case s that at the lmt bdder outranks everyone else n terms of V k,zk ) k j,.e., V,z t ) max k z k as n the next lemma. Lemma 10 If t n n t such that V,z t ) max k z k, then lm n Π n t n ) > 0. Proof Appendx E.6. Snce the types of j that bd n c n j, b + δ n ) would nearly for sure resell the good to bdder when s type happens to satsfy the hypothess of the lemma, the expected payment extracted from such a hgh type of bdder s larger than j s expected resale revenue by a nonvanshng margn, as could be of low types accordng to j s posteror belef Lemma 23, Appendx A.3). Wth j s expected resale revenue never below hs perod-one wnnng bd 23 46)

26 Lemma 6), ths nonvanshng margn mples a nonvanshng markup between the current prce for the good and the expected payment that the hgh-type bdder would need to delver to reseller j. Ths markup consttutes the pvotal effect of the bd ncrease Step 4: Pvotal Effect Case Two: Becomng the Mddleman Here comes the other case for the pvotal effect, where bdder s type s not hgh enough to nearly for sure buy the good at resale from bdder j. Dfferent than the prevous case, bdder j s perod-one bd, whch s approxmately the current prce n the event that bdder s devaton s pvotal, could be hgher than the prce that j wll charge at resale: Even f the revenue extracted from s less than what j pays at perod one, j can stll proft from the revenues extracted from the other potental buyers. 16 Then the devant bdder suffers a wnner s curse n the magntude of the perod-one prce mnus the lower prce at resale. The soluton stems from an dea of turnng the table: In the same way that j s loss from dealng wth s balanced by j s revenues extracted from other bdders, s wnner s curse s balanced by the revenues from the same clentele f becomes the reseller status nstead of j. Denote Ω n := t T : max β k n t k ) < b n ; c n < β j n t j ) < b n, 47) k /,j ψ n t n ) := W c n, t n, β n) c n E L t t n, β n ) Ω n. 48) Hence Ω n s the pvotal event of s bd ncrease, and ψ n t n ) hs expected payoff from wnnng mnus hs wnnng bd and mnus hs wnner s curse. Lemma 11 If β n t n ) = c n for each n and t n ) n=1 converges, then lm n ψn t n ) lm E 1 t j < V,z n n t n ) W j t n, t,j) β j n t j ), t j, β n) β j n t j ) ) Ω n. 49) 16 For example, suppose that n the contnuaton game where bdder j s the reseller, t j = 2, t s unformly dstrbuted on 0, 4, and t k unformly dstrbuted on 0, 10. In j s optmal resale mechansm, the maxmum of bdder s expected payment when t = 4) s equal to whle the reseller j s expected payoff equals t k 3)dt k /10 = 2.15, t +3 t + 3) dt k dt t k 3 t k 3) dt 4 Thus, at perod one, t s possble for bdder j to submt a bd strctly between 2.15 and dt k