Robustly Optimal Auctions with Unknown Resale Opportunities

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1 Robustly Optmal Auctons wth Unknown Resale Opportuntes Gabrel Carroll Ilya Segal Department of Economcs Stanford Unversty, Stanford, CA 94305, USA February 13, 2018 Abstract The standard revenue-maxmzng aucton dscrmnates aganst a pror stronger bdders so as to reduce ther nformaton rents. We show that such dscrmnaton s no longer optmal when the aucton s wnner may resell to another bdder, and the auctoneer has non- Bayesan uncertanty about such resale opportuntes. We dentfy a worst-case resale scenaro, n whch bdders values become publcly known after the aucton and losng bdders compete Bertrand-style to buy the object from the wnner. Wth ths form of resale, msallocaton no longer reduces the nformaton rents of the hgh-value bdder, as he could stll secure the same rents by buyng the object n resale. Under regularty assumptons, we show that revenue s maxmzed by a verson of the Vckrey aucton wth bdder-spec c reserve prces, We are grateful to Adren Auclert, Lawrence M. Ausubel, Ben Brooks, Peter Cramton, Drew Fudenberg, Smon Grant, Paul Mlgrom, Isaac Sorkn, Glen Weyl, and partcpants of the 2016 Stony Brook Workshop on Complex Auctons and Practce, the NBER Market Desgn Workng Group, the NYUAD Advances n Mechansm Desgn conference, the NSF-CEME Decentralzaton conference, and semnars at ANU, UNSW, UQ, HU-Berln, Mannhem, Bonn, Northwestern, and Harvard-MIT, for helpful comments. We thank the Smons Insttute at UC Berkeley for ts hosptalty durng the Fall 2015 Economcs and Computaton Program. Segal acknowledges the support of the Natonal Scence Foundaton (grant SES ). Carroll s supported by a Sloan Research Fellowshp. Alex Bloedel and Seunghwan Lm provded valuable research assstance. 1

2 rst proposed by Ausubel and Cramton (2004). The proof of optmalty nvolves constructng Lagrange multplers on a double contnuum of bndng non-local ncentve constrants. 1 Introducton Standard aucton theory says that when bdders are a pror asymmetrc, revenue-maxmzng auctons dscrmnate aganst stronger bdders,.e. those who are more lkely to have hgher values. The optmal aucton requres them to pay a premum over weaker bdders values n order to wn (Myerson 1981, McAfee and McMllan 1989). Ths dscrmnaton enhances revenue because t ncreases competton between weaker and stronger bdders and so reduces the latter bdders nformaton rents. One mght suspect that the bene ts of such dscrmnaton would be vtated f bdders could resell to each other after the aucton, snce a strong bdder mght then prefer to st back and let a weaker bdder wn, n the hopes of buyng from hm later at a better prce. In the words of Ausubel and Cramton (1999, p. 19): The possblty of resale undermnes the seller s ablty to gan by msassgnng the good. The best that the seller can do s to conduct an e cent aucton [.e. sellng only to the hghest-value bdder], perhaps wthholdng... the good. 1 However, ths s not supported by exstng results on optmal auctons wth resale (Zheng 2002, Calzolar and Pavan 2006), n whch revenue s typcally maxmzed by a based aucton, whch nduces resale n equlbrum wth a postve probablty. For a smple example due to Zheng (2002), suppose one bdder s known to have a zero value for the good and also to have full barganng power n resale. Then the auctoneer would want to smply sell to ths bdder, who would then resell the good usng Myerson s (1981) revenue-maxmzng mechansm. The auctoneer could charge the zero-value bdder a prce equal to the Myerson expected revenue. (Ths s the best she can do as long as bdders values reman prvate nformaton, snce the combned outcome of the aucton and resale would have to satsfy the same ncentve compatblty constrants as 1 The noton that prce dscrmnaton s ne ectve under resale s also famlar n the context of prcng n markets. For example, Trole (1988, p. 134) wrtes: It s clear that f the transacton (arbtrage) costs between two consumers are low, any attempt to sell a gven good to two consumers at d erent prces runs nto the problem that the low-prce consumer buys the good to resell t to the hgh-prce one. 2

3 Myerson s mechansm.) Ausubel and Cramton (1999, 2004) clam that t s optmal for the auctoneer not to msallocate when resale s perfect. Spec cally, they propose a way of addng bdder-spec c reserve prces to a Vckrey (second-prce) aucton, whch we call the Ausubel-Cramton-Vckrey (ACV) aucton and descrbe n detal below. The ACV ether allocates the object e cently among bdders, or wthholds t (when no bdder beats hs assgned reserve). 2 Ausubel and Cramton (1999) assert that the ACV aucton s optmal. They do not formalze the assumpton of perfect resale, and as s well known, ef- cent resale would generally be mpossble under prvate nformaton; 3 but one mght justfy the assumpton by assumng that the partes values exogenously become publcly known before resale. However, even f we operatonalze perfect resale n ths way, t could stll be optmal for the auctoneer to msallocate. Indeed, f we modfy the zero-value bdder example above by lettng the zero bdder observe all other bdders values before resellng, the optmal mechansm would extract full rst-best surplus by agan sellng the object to the zero bdder, now at a prce equal to the expectaton of the hghest value, whch s exactly what the zero bdder expects to receve n resale. Smlarly, n the symmetrc settng wth perfect resale studed by Bulow and Klemperer (2002) and Bergemann et al. (2017a), where the aucton s wnner s assumed to have full nformaton and full barganng power n resale, the auctoneer can command a hgher prce by allocatng to an ne cent bdder than to the e cent one, because the ne cent bdder can later extract the e cent bdder s surplus n resale; thus the optmal aucton msallocates the good. 4 2 At least two other ways to ntroduce asymmetrc reserves nto Vckrey have been studed: lazy and eager Vckrey (Dhangwatnota et al., 2010). Eager Vckrey allocates to the hghest bdder among those who met ther reserve prces as long as one such bdder exsts, and so t msallocates n equlbrum. Lazy Vckrey allocates to the hghest bdder provded that he met hs reserve prce and does not allocate otherwse. It wll emerge as the soluton to our relaxed problem n Subsecton For example, when the aucton smply gves the object to bdder 1 for zero payment, so that no nformaton s revealed, resale must be ne cent accordng to the theorem of Myerson and Satterthwate (1983). The same deas apply to resale followng more general aucton mechansms. 4 Bulow and Klemperer (2002, Example 3) nd that n ths settng, sellng to a randomly chosen bdder at a xed prce low enough to guarantee a sale yelds a hgher revenue than the Vckrey aucton. Bergemann et al. (2017a) show that the xed-prce mechansm s n fact the optmal mechansm among those that always sell the good. They also derve the 3

4 Our paper shows how the ntuton expressed by Ausubel and Cramton can nevertheless be valdated when the aucton desgner s uncertan about the resale procedure (ncludng possble exogenous revelaton of prvate nformaton) and desres the revenue to be robust to ths uncertanty. Indeed, notce that some of the optmal based auctons descrbed n the above examples are not robust to even a small amount of uncertanty about resale. 5 Thus, we formulate the problem of maxmzng worst-case expected revenue, where the expectaton s taken over buyers prvate values drawn from known dstrbutons, and the worst case s over the possble resale procedures. We refer to ths problem as the robust revenue maxmzaton problem. To solve ths problem, we begn wth a smpl ed model n whch the auctoneer s requred to always sell the object (Secton 3). Wth no resale, the optmal aucton n ths model would be based. In contrast, robust revenue maxmzaton under resale uncertanty s acheved by the smple second-prce aucton (wth no reserve), whch allocates e cently. To prove ths, we guess a worst-case resale procedure, n whch bdders values exogenously become publc knowledge after the aucton, and then the losng bdders compete à la Bertrand to buy the object from the wnner n resale. Wth ths resale procedure, n any aucton that always sells the object, each bdder can guarantee hmself a payo equal to hs margnal contrbuton to socal surplus, by sttng out to let another bdder wn and then buyng from the wnner. Gven ths lower bound on the nformaton rents captured by the bdders, the desgner cannot do better for herself than the Vckrey aucton. Snce ths aucton also sustans truthful bddng as an ex post equlbrum under any other resale procedure, t solves the robust revenue maxmzaton problem. As a sde bene t, the optmal aucton turns out to be ndependent of the bdders value dstrbutons,.e., completely pror-free. Whle the must-sell model cleanly llustrates the robust optmalty of e cent auctons, n most real-lfe settngs (and n the classcal theory) the auctoneer has the ablty to ncrease revenue by sometmes wthholdng the object, e.g. by usng reserve prces. In Secton 4, we turn to study such settngs, and show that under approprate regularty assumptons on the optmal aucton that can wthhold the good, and show that t msallocates wth a hgh probablty. Whle wrtten ndependently of our paper, ther paper shares some techncal smlartes wth our analyss, whch we pont out below. 5 In ether of the zero-bdder examples, suppose the auctoneer s guess about resale s mstaken, and there s actually an " probablty that no resale opportunty wll arse. Then the zero bdder refuses to buy at the proposed prce, and revenue drops to zero. 4

5 dstrbutons, an ACV aucton wth approprately chosen reserve prces s optmal. Thus, the auctoneer agan never msallocates the good, but takes advantage of the known asymmetres by settng d erent reserve prces to d erent bdders. We show ths result by provng that the ACV aucton s optmal under the partcular worst-case resale procedure descrbed above. Snce, by an argument of Ausubel and Cramton (2004) (whch we nclude n the appendx for completeness), ths aucton sustans truthful bddng as an equlbrum under any resale procedure, t solves the robust revenue maxmzaton problem. Characterzng an optmal aucton under our worst-case resale procedure requres new technques that may be of separate nterest. Frst, by foldng the outcome of resale nto reduced-form payo functons, we model the auctoneer s problem as a sngle-stage aucton desgn problem, albet one wth externaltes and nterdependent values (snce a bdder who does not wn the object cares whether another bdder wns, and what the wnner s value s.) 6 We can then apply the standard rst-order approach to such problems, whch consders only the local ncentve constrants, and rewrtes the objectve as an approprately-de ned vrtual surplus. The soluton found by ths standard method renforces our basc ntuton: t never msallocates the good; t ether allocates e cently or wthholds the good. However, ths soluton s not the rght answer, because t s not ncentve-compatble. It s vulnerable to non-local devatons, where a bdder underbds to lose and then buys the good n resale. To nd the true optmal aucton, we need to account for such non-local ncentve constrants. We guess that the optmum s the ACV aucton. To verfy optmalty, we explctly construct supportng Lagrange multplers on the double contnuum of bndng non-local ncentve constrants. 7 Our approach also yelds an teratve constructon of the optmal bdderspec c reserve prces n ACV auctons. Appendx F llustrates by workng 6 Other examples of externaltes and nterdependent values caused by post-aucton nteractons among buyers can be found n Jehel and Moldovanu (2000) and Bulow and Klemperer (2002). 7 Earler work that constructed Lagrange multplers as a measure over a double contnuum of bndng ncentve constrants ncludes the transport theory approach to multdmensonal screenng (e.g., Daskalaks et al. (2013)). Also, Bergemann, Brooks, and Morrs (2017a) ndependently develop a treatment of non-local ncentve constrants that s the closest techncally to our approach: our analyss mplctly shares wth thers the feature of consderng randomzed msreports that are drawn from the same dstrbuton as the true type, but truncated. 5

6 through an example n whch bdders values are dstrbuted unformly wth d erent upper lmts. In ths case, wth bdders ordered from stronger to weaker, the optmal reserve prce for the kth bdder s obtaned by solvng a kth-degree polynomal equaton. Some readers may nd a tenson between the auctoneer s use of Bayesan prors to construct optmal reserve prces n the ACV aucton and her complete gnorance of the resale procedure. We do beleve that there are reallfe stuatons n whch the auctoneer has knowledge that some bdders are stronger than others, whch s tradtonally captured by means of Bayesan prors, whle beng gnorant about the resale procedure. 8 Furthermore, as we detal n Secton 5, the tenson can be drectly amelorated n two ways. Frst, we o er a model wth uncertanty both about value dstrbutons and about resale, n whch our concluson about optmalty of ACV auctons carres over. Second, we o er addtonal results suggestng that settng the rght reserve prces s quanttatvely less mportant than gnorance of resale, so that smply usng Vckrey wth no reserves s a good pror-free aucton choce. Our paper jons the growng lterature usng the maxmn crteron to model robust mechansm desgn (ncludng, n partcular, Frankel (2014), Carroll (2015), and Bergemann et al. (2017b)). Conceptually, t s the closest to the work showng that a strategy-proof mechansm may emerge as optmal when the desgner s uncertan about bdders belefs about each other s values (Chung and Ely 2007; Yamashta and Zhu 2017) or about each other s strateges (Yamashta 2015). Lkewse, n our paper, a resale-proof mechansm may emerge as optmal when the desgner s gnorant about the resale procedure. Also, observe that ACV auctons are also robust to bdders belefs and nformaton about each other s values and ther belefs about the resale procedure, hence we obtan these addtonal robustness bene ts for free. 8 To gve one example, n spectrum auctons run by the U.S. Federal Communcatons Commsson, some bdders (such as AT&T, Verzon, and T-Moble) are beleved to have strong busness cases for the use of addtonal spectrum, whle other bdders (such as Dsh and Comcast) are beleved to be bddng speculatvely, wth ther values determned to a large extent by expectaton of resale, even though the structure and outcomes of the resale market are hard to antcpate. 6

7 2 Setup There are n 2 bdders. Bdder s prvate type s hs value for the object, whch s dstrbuted on [0; 1] accordng to a c.d.f. F wth a contnuous strctly postve densty f. 9 Values are ndependent across bdders. We wrte = ( 1 ; : : : ; n ) for the pro le of values. The space of (possbly randomzed) allocatons s X = fx 2 [0; 1] n : P x 1g, where x 2 [0; 1] s the probablty of allocatng the object to bdder. We use tldes to denote random varables: ~ for the random varable representng s value; for a spec c realzaton. A general aucton mechansm s a trple = hm; ;, where M = (M 1 ; : : : ; M n ) s a collecton of measurable message spaces for the bdders, such that ; 2 M for each, where ; denotes the specal non-partcpaton message; : Q M! X s a measurable allocaton rule and : Q M! R n s a measurable payment rule, wth (m) = (m) = 0 whenever m = ;. Followng allocaton spec ed by the aucton, resale may take place. We model resale n reduced form by an n-tuple of measurable functons v = (v 1 ; : : : ; v n ), where v (x; ) gves bdder s post-resale payo (net of payments n the aucton) followng allocaton x 2 X spec ed by the aucton when the bdders value pro le s. Ths formalsm captures a settng n whch all bdders values exogenously become publc after the aucton, so that the outcome of resale depends only on the ntal allocaton and the values. (In Appendx C, we descrbe a more general class of resale procedures that does not assume values are revealed.) We requre that the total reduced-form payo s not exceed the maxmal total surplus avalable n resale: P v (x;) max ( P x ) : (1) We also requre the resale procedure to be ndvdually ratonal: v (x; ) x for each : (2) 9 The assumptons of common support and contnuous postve denstes are made for expostonal smplcty: n Appendx F we explan how our results can be extended to some cases n whch these assumptons do not hold. 7

8 A gven mechansm and resale procedure v together nduce a Bayesan game: the acton space of player s M, and hs payo s v ( (m) ; ) (m), wth correspondng revenue P (m) for the auctoneer. Let Rev( ; v) denote the supremum of expected revenue over all Bayes- Nash equlbra of ths game. We state the robust revenue maxmzaton problem as 10 max nf Rev( ; v) : (3) v We wll establsh that a spec c aucton solves the robust revenue maxmzaton problem, by constructng a resale procedure v and a revenue target R such that the followng two condtons are sats ed: Rev( ; v) R for all auctons, (4) Rev( ; v) R for all resale procedures v: (5) (4) means that gven resale procedure v, the desgner could not desgn an aucton yeldng expected revenue above R, whle (5) means that gven aucton, an adversary could not construct a resale procedure to reduce the desgner s expected revenue below R. The logc of the Mnmax Theorem then mples (the formal proof of ths and all other results are n the appendx) Lemma 1. If (4)-(5) hold then aucton solves the robust revenue maxmzaton problem max (nf v Rev( ; v)), whle resale procedure v solves the worst-case resale problem mn v (sup Rev( ; v)), and the value of both problems s Rev( ; v) = R. To establsh (4) for a partcular resale procedure v, we can apply the Revelaton Prncple and restrct attenton to drect mechansms, n whch each bdder s message space s M = [0; 1] [ f;g, and to the Bayes-Nash equlbrum n whch all bdders partcpate and report truthfully,.e., sats es 10 Note, n partcular, that Rev( ; v) = 1 f the nduced game has no equlbrum, so the desgner must guarantee equlbrum exstence for all v. We vew ths requrement as not too onerous; for example, any mechansm wth nte message spaces guarantees equlbrum exstence (Mlgrom and Weber, 1985, Theorem 1). It s n lne wth the standard assumpton n mechansm desgn that agents must play an equlbrum. 8

9 the followng ncentve compatblty and ndvdual ratonalty constrants: 11 E ~ [v (( ; ~ ); ; ~ ) ( ; ~ )] E ~ [v ((^ ; ~ ); ; ~ ) (^ ; ~ )] for all ; ^ ; (6) E ~ [v (( ; ~ ); ; ~ ) ( ; ~ )] E ~ hv ((;; ~ ); ; ~ ) : for all : Then, we show (4) by showng that the expected revenue n any drect aucton satsfyng (6)-(7) cannot exceed the target R, whch we take to be the expected revenue of when bdders behave truthfully. In Secton 3, we do ths for the specal case n whch the seller must sell the object wth probablty 1, and s the standard Vckrey aucton. In Secton 4, we do ths for the general case n whch the seller can wthhold the object, and s an ACV aucton (formally de ned n De nton 1 ahead) wth approprately constructed bdder-spec c reserve prces. These results consttute the heart of our contrbuton. Then, by an argument of Ausubel and Cramton (2004), truthtellng s an equlbrum of for any resale procedure v, not just for v. Thus, Rev( ; v), whch s the hghest expected revenue of across all equlbra, s bounded below by R, establshng (5). Lemma 1 then mples our man result. 3 The Must-Sell Case We begn by consderng a smpler model n whch the seller must sell the object. (For example, ths could be mcrofounded by assumng the seller has a prohbtvely hgh cost of keepng the object.) To adapt the robust problem (3) to ths settng, just rede ne the objectve Rev( ; v) as the supremum of expected revenue over those Bayes-Nash equlbra n whch the object s sold wth probablty 1, and Rev( ; v) = 1 f no such equlbrum exsts. In the absence of resale, f bdders values are drawn from d erent dstrbutons, the optmal must-sell aucton msallocates the object towards weaker bdders, wth the goal of reducng the nformaton rents of stronger bdders 11 Note that the Revelaton Prncple would not generally apply to the robust maxmzaton problem, snce n a gven aucton, truth-tellng may be an equlbrum for some resale procedures but not for others. Ths s why the robust problem s formulated usng ndrect mechansms. (7) 9

10 (Myerson (1981), McAfee and McMllan (1989)). As we shall see, ths bene t of msallocaton can be hampered by resale. We guess a worst-case resale procedure to be the Bertrand game n whch the aucton s losers make competng prce o ers to acqure the object from the wnner, who then accepts one of the o ers or rejects all of them and keeps the object to hmself. As a result, f the aucton s wnner s not the hghest-value bdder, the former sells the object to the latter at prce (2), the second-hghest value of all bdders. 12 Thus, bdder s post-resale payo from aucton allocaton x n state (exclusve of payments made n the aucton) can be wrtten as v (x; ) = maxf ; (2) g x + maxf0; (2) g P j6= x j: (8) Note that ths resale procedure v s e cent (.e., sats es (1) wth equalty) and ndvdually ratonal (sats es (2)). Resale procedure (8) s a natural guess for the worst case because t makes the hghest-value bdder a resdual clamant for surplus, thus allowng hm to capture nformaton rents even f the object s allocated to another bdder. More spec cally, n any must-sell aucton, any bdder can, by lettng another bdder wn and then buyng from hm f possble, assure hmself an expected payo E ~ hmaxf ~ ~ (2) ; 0g. Ths payo s bdder s expected margnal contrbuton (that s, the total surplus avalable, mnus the surplus that would be achevable f were absent), and concdes wth hs expected payo n the Vckrey aucton (.e. second-prce aucton) wth no reserve prce. If the seller cannot avod concedng at least ths much surplus to the bdders, she cannot do any better than usng the Vckrey aucton wth no reserve prce. The above argument s not complete because t s not clear that bdder can ensure that another bdder wns whle avodng payng anythng to the auctoneer. For example, f bdder bds exactly 0, the auctoneer mght wthhold the good (snce she only needs to sell wth probablty 1), or mght 12 Spec cally, ths s the unque outcome arsng n a subgame-perfect equlbrum n undomnated strateges n the full-nformaton Bertrand game. Ths s not the only worstcase resale procedure: for example, the argument below would also work f the hghestvalue bdder, when he fals to wn the object, were to buy t back by makng a take-tor-leave-t o er to the aucton s wnner. On the other hand, the results below do rely on the extreme dvson of barganng power n these games. For example, f there are two bdders and bdder 1 captures a xed share 2 (0; 1) of resale surplus, we can have an example where the seller can do better than the Vckrey aucton. 10

11 sell t to bdder but charge hm a postve prce, lettng hm recoup t n resale. We can address these ssues by notng that snce the aucton must sell wth probablty 1, bdder could ensure that the good s sold wth probablty 1 (to ether another bdder or hmself) by makng a bd ^ that s arbtrarly small. Then, we can show that the ndvdual ratonalty constrant of type ^ and resale procedure (8) guarantee bdder an expected payo close to hs expected Vckrey payo, whch mples the result. Theorem 1. Under resale procedure (8), equlbrum expected revenue n any aucton that sells wth probablty 1 does not exceed the expected truthtellng revenue n the Vckrey aucton. Ths establshes condton (4) above for resale procedure (8). As for condton (5), t follows from the observaton that truthful bddng s an ex post equlbrum of the Vckrey aucton for any resale procedure satsfyng (1)-(2) (and then no resale occurs, snce the hghest-value bdder has already won). Whle ths wll follow from Theorem 2 below, the nformal argument s as follows. Snce the usual arguments show that a bdder can never bene t by devatng from truthful bddng n a way that does not nvolve resale, we just need to check that there are no pro table devatons that nvolve resale ether. If bdder makes a downward devaton that causes hm to lose, he cannot buy the object n resale unless he pays at least the wnner s value, (2), whch s what he would have pad to wn the object n the aucton anyway. And f he makes an upward devaton that causes hm to wn, he cannot resell the object for more than the hghest value, (1), whch s the prce he would have to pay to wn the aucton. In both cases he does no better than bddng hs true value. Wth (4) and (5) thus establshed, Lemma 1 mples Corollary 1. The Vckrey aucton solves the robust revenue maxmzaton problem (3) for a seller that must sell the object wth probablty 1. 4 The Can-Keep Case We now turn to the optmal aucton when the seller can wthhold the object, whch s the verson of the model more wdely studed n the lterature. As n the prevous secton, we conjecture that a worst-case resale procedure s gven by (8), and show that gven ths procedure, an approprately-desgned ACV aucton s optmal. 11

12 4.1 Regularty Assumptons on Dstrbutons The man result of ths secton wll use the followng assumptons on bdders value dstrbutons. (However, many of the ntermedate steps wll not rely on all of the assumptons, and so we wll mpose them only when needed.) A1. For each bdder, the hazard rate f () = [1 F ()] s nondecreasng. A2. For each bdder, the reverse hazard rate f () =F () s nonncreasng. A3. The vrtual value functons () = (1 F ())=f () satsfy 1 () : : : n () for each 2 [0; 1]. (A1) mples, n partcular, that each dstrbuton F s regular,.e., each functon () s ncreasng. The value r = 1 (0) s then unquely de ned; t s the optmal prce for sellng to bdder alone, and s also the optmal reserve prce for bdder n Myerson s optmal aucton. Note also that both (A1) and (A2) are sats ed, n partcular, when the densty f s log-concave, whch s sats ed by many standard dstrbutons (Bagnol and Bergstrom 2005). Assumpton (A3) s equvalent to requrng that the dstrbutons F be ordered from stronger to weaker under the hazard rate orderng. In the absence of resale, t s a su cent condton (and, for two-bdder must-sell auctons, a necessary condton) for the optmal aucton to always dscrmnate aganst the stronger bdders (McAfee and McMllan 1989, Theorem 3). In partcular, (A3) ensures that the optmal bdder-spec c reserve prces would satsfy r 1 : : : r n. Whle assumptons (A1)-(A3) are not unreasonable and are famlar from the lterature on optmal auctons, they are not the weakest possble to ensure the optmalty of ACV auctons. In Secton 5 we dscuss the possblty of relaxng the assumptons and the challenges that arse. 4.2 Vrtual Surplus and the Relaxed Problem We begn wth the standard approach of usng rst-order ncentve compatblty constrants to substtute out the payment functons. To ths end, for any proposed (drect) aucton (; ), let U ( ) = E ~ [v (( ; ~ ); ; ~ ) ( ; ~ )] (9) 12

13 be the nterm expected payo enjoyed by when hs type s, and note that by the standard envelope-theorem argument, ncentve compatblty (6) mples that U s absolutely contnuous and ts dervatve s gven almost everywhere by 13 U 0 ( ) = E ~ [v 0 (( ; ~ ); ; ~ )]: (10) Here v 0 denotes the dervatve of v (x; ) wth respect to, whch s de ned when does not te wth another bdder, and then takes the form v 0 (x; ) = 1 = (2) x + 1 = (1). As for partcpaton constrants, as usual we consder (7) only for type 0, for whch t takes the form U (0) 0. If only ths partcpaton constrant s mposed, t wll be optmal to satsfy t wth equalty, and so ntegratng (10) and usng (9) yelds the followng expresson for the nterm expected payment of bdder of type : E ~ [ ( ; ~ )] = E ~ [v (( ; ~ ); ; ~ )] Z 0 E ~ hv 0 ((^ ; ~ ); ^ ; ~ ) d^ : (11) The usual ntegraton by parts then allows us to rewrte the aucton s expected revenue n terms of the allocaton rule as " # " X X E ~ ( ~ ) = E ~ v (( ~ ); ~ X 1 F ( ) ~ # ) f ( ~ v 0 (( ~ ); ~ ) : (12) ) The standard relaxed problem maxmzes (12) over all allocaton rules, gnorng (6), as well as (7) for types other than 0. Ths can be done by maxmzng the vrtual surplus (the expresson nsde brackets) for each pro le separately. At any pro le wthout tes, f we allocate to the hghest-value bdder, the vrtual surplus s 1 F ( ) f ( ) = ( ): 13 Spec cally, for any gven report ^, v ((^ ; ); ; ) s Lpschtz contnuous n and s d erentable n except when there are tes n values. Ths mples that E ~ [v ((^ ; ~ ); ; ~ )] s Lpschtz contnuous and d erentable n, and allows the applcaton of Mlgrom and Segal s (2002) Corollary 1 to establsh absolute contnuty of U and (10). 13

14 If we allocate to the second-hghest bdder j, then the v 0 terms are 1 for both = and = j, so the vrtual surplus s 1 F ( ) f ( ) 1 F j ( j ) : f j ( j ) Fnally, f we allocate to a bdder other than or j, the vrtual surplus s the same as from allocatng to. Thus, msallocaton could only ncrease nformaton rents: allocatng to a bdder other than stll concedes nformaton rents to hgher types of (who can buy the good cheaply n resale), and may leave nformaton rents to the ne cent wnner as well. So the seller cannot do better than allocatng to the hgh-value bdder, whch leaves rents to hm only. Furthermore, we should allocate to the hghest-value bdder f and only f hs vrtual value s postve, whch s equvalent to > r = 1 (0) provded that hs vrtual value functon crosses zero just once (whch s ensured under assumpton (A1)). In partcular, f there are two bdders and each crosses zero once, the allocaton rule solvng the relaxed problem s as shown n Fgure 1, and the soluton s unque up to measure-zero sets. Gven the allocaton rule, transfers consstent wth (11) can be obtaned, n partcular, by chargng aucton wnner the threshold prce maxfr ; (2) g, whch s hs lowest value for whch he would have won the good gven the values of others; and chargng losers nothng. The resultng mechansm s known as the Vckrey aucton wth lazy reserves (Dhangwatnota et al., 2010), and t s domnant-strategy ncentve compatble n the absence of resale. However, wth our resale procedure v, the soluton to the relaxed problem volates global ncentve compatblty constrants (6) unless all bdders have the same optmal reserve r. For example, consder the case of two bdders wth r 1 > r 2, and consder bdder 1 of type 1 2 (r 2 ; r 1 ) devatng to report ^1 < 1, as llustrated wth the horzontal arrow n Fgure 1. Ths devaton a ects the aucton s outcome when 2 2 (maxf^ 1 ; r 2 g; 1 ), and n these cases t changes the outcome from leavng the object unsold to gvng t to bdder 2, allowng 1 to pro tably buy t back n resale. Thus, n order to nd the correct soluton, we need to consder non-local ncentve constrants Note that we could deter all local msreports wthn dstance " > 0 by perturbng the relaxed soluton to wthhold the good whenever j 2 1 j < ", but the perturbed mechansm would stll be vulnerable to global devatons of the form descrbed above. Ths s n contrast to ronng n the standard screenng settng, where global ncentve 14

15 Fgure 1: Allocaton rule from relaxed problem. 1 means allocate to bdder 1; 2 means allocate to bdder 2. In the remanng regons, the good s not sold. 4.3 Ausubel-Cramton-Vckrey Auctons Intutvely, to avod the ncentve to underbd to cede the object and then buy t back, we mght use an allocaton rule n whch a lower bd never causes the object to be sold. (Ausubel and Cramton (1999) call ths property monotoncty n aggregate.) In the two-bdder example, we mght try to x the allocaton rule by llng n the trangular regon r 2 < 2 < 1 < r 1 n Fgure 1, allocatng to bdder 1 n ths regon (based on the above ntuton that we prefer to allocate to the hgh-value bdder or to nobody). Note, however, that ths soluton can be mproved: snce bdder 1 s vrtual value n the lled-n trangle s negatve, the seller would rather shrnk the sze of ths trangle by rasng the reserve prce for bdder 2 above r 2, even though dong so also means mssng out on pro table sales to bdder 2. The optmal reserve prce for bdder 2 trades o these two e ects. The resultng allocaton rule s shown n Fgure 2. compatblty s mpled by local ( rst- and second-order) ncentve constrants (Archer and Klenberg, 2014; Carroll, 2012). 15

16 Fgure 2: Allocaton rule for ACV aucton. 1 means allocate to bdder 1; 2 means allocate to bdder 2. In the remanng regons, the good s not sold. Ths aucton belongs to the followng class of auctons, ntroduced by Ausubel and Cramton (1999, 2004): De nton 1. An Ausubel-Cramton-Vckrey (ACV) aucton wth reserve prces p 1 ; : : : ; p n 2 [0; 1] s the drect revelaton mechansm descrbed as follows: Allocaton rule: 1 f = () = () and j > p j for some j; 0 otherwse, where () 2 arg max (wth arbtrary te-breakng.) Payments: 8 < max fp ; max j6= j g f () = 1 and j p j for all j 6= ; () = max j6= j f : () = 1 and j > p j for some j 6= ; 0 otherwse, and non-partcpaton message ; s treated the same as a report of 0. 16

17 In words, f at least one bdder beats hs reserve prce p, then the good s allocated to the hghest-value bdder, otherwse, the good s left unsold. Importantly, snce the reserve prces are asymmetrc, a bdder can wn the good wthout meetng hs reserve prce p f another bdder j wth a lower reserve has met hs reserve p j. A wnner s payment n the ACV aucton s hs threshold prce the mnmal bd that would have allowed hm to wn. By standard arguments, ths ensures that the aucton s strategy-proof wthout resale. 15 More mportant for us s that (n contrast to the soluton to our relaxed problem) n an ACV aucton, bdders are ncentvzed to bd truthfully even wth resale. Spec cally, truthful bddng s an ex post equlbrum, for any resale procedure and for any pro le of values: Theorem 2. Consder an ACV aucton wth reserve prces p 1 ; : : : ; p n. For any resale payo functons v 1 ; : : : ; v n satsfyng (1) (2), t s an ex post equlbrum for all bdders to partcpate and report ther true values (after whch no resale occurs). Formally, for any values 1 ; : : : ; n, and any possble devaton ^ 2 [0; 1] [ f;g for any bdder, () () v (( ^ ; ); ) ( ^ ; ): The proof essentally follows the nformal argument gven n Secton 3 for the Vckrey aucton wthout reserves, but also usng the fact that n an ACV aucton, a downward devaton would never cause the object to be sold. Whle ths result was proved by Ausubel and Cramton (2004), for completeness we gve a proof n Appendx C. Also, n that appendx we develop addtonal formalsm for a broader class of resale procedures that drops the assumpton that values are revealed; thus, barganng may take place under asymmetrc nformaton. The proof shows that the result holds for ths broader class of resale procedures. We note n passng that ths theorem only gves us ex post equlbrum, not domnant strateges as we would have n Vckrey auctons wthout re- 15 An ACV aucton can also be ndrectly mplemented as a deferred acceptance clock aucton of the knd descrbed by Mlgrom and Segal (2017). In ths mplementaton, the aucton o ers the same ascendng prce to all bdders, lettng them ether accept the prce or ext at any pont, and stoppng when both () there s a sngle bdder who s stll acceptng the current prce, and () at least one bdder (not necessarly the one who s stll bddng) has ever accepted a prce above hs reserve prce. (Note, however, that one advantage of the clock aucton format ts obvous strategy-proofness (L, 2017) does not hold when resale s possble.) 17

18 sale. 16 Ths s to be expected snce our settng s one of nterdependent values (compare Perry and Reny (2002) or Chung and Ely (2006)). 4.4 Constructon of Optmal Reserve Prces In the two-bdder ACV aucton llustrated n Fgure 2, bdder 1 s prce p 1 can be optmzed wthout regard to bdder 2 (snce t only matters when bdder 2 bds below p 2 < p 1, and therefore does not wn), hence t s optmal to set p 1 = r 1. On the other hand, the optmal prce for bdder 2 s rased above r 2 to ncrease the expected revenue on bdder 1. Extendng ths dea to n bdders, we construct a weakly decreasng sequence p 1 ; : : : ; p n of reserve prces and a correspondng sequence R 1 ; : : : R n of revenue levels on the rst k bdders recursvely, ntalzng p 0 = 1 and R 0 = 0 and lettng for each k 1, R k = R k (p k ) + F k (p k ) Rk 1 R k 1 (p k ) (13) = max Rk (p) + F k (p) Rk 1 R k 1 (p) ; (14) p2[0;p k 1 ] where R k (p) s the expected revenue from the (symmetrc) Vckrey aucton on the rst k bdders facng the same reserve prce p, wth R 0 (p) 0. Formula (13) gves an nductve constructon of the expected revenue R k from the ACV aucton on bdders 1; : : : ; k wth reserves p 1 ; : : : ; p k, assumng the bdders bd ther true values. To see ths, compare ths ACV aucton to the symmetrc Vckrey aucton wth reserve p k on the same bdders. Notce that the only states n whch the two auctons could yeld d erent revenues are those n whch bdder k s value k s below p k, whch happens wth probablty F k (p k ). Condtonal on any such value of k, the former aucton reduces to the ACV aucton on the rst k 1 bdders wth reserves p 1 ; : : : ; p k 1, yeldng expected revenue R k 1, whle the latter aucton reduces to the Vckrey aucton wth reserve p k on the rst k 1 bdders, yeldng expected revenue R k 1 (p k ). Condton (14) requres that bdder k s reserve prce p k maxmze revenue on the strongest k bdders takng as gven the stronger bdders reserves 16 To see ths concretely, magne that there are two bdders, no reserve prces, and bdder 1 expects 2 to bd hgher than hs true value. Then, under our resale procedure (8), 1 has an ncentve to underbd and make 2 wn, snce f 1 wns the object n the aucton he has to pay 2 s (exaggerated) bd, whereas to buy t n resale he only has to pay 2 s true value. Thus, truthful bddng s not a domnant strategy. 18

19 p 1 ; : : : ; p k 1 and the constrant p k p k 1. Ths condton does not mmedately mply full optmalty of the reserve prces, snce we do not know whether the constrant bnds, or how the choce of p k n turn constrans the revenues on the weaker bdders. However, we wll show a stronger result that the resultng aucton s ndeed optmal, and not just among ACV auctons but among all possble auctons Optmalty of ACV Aucton We now come to our man theorem. Theorem 3. Under assumptons (A1)-(A3), the formulas (13)-(14) unquely de ne reserve prces p 1 ; : : : ; p n. The ACV aucton wth these reserve prces s an optmal aucton gven resale procedure (8), and the resultng revenue s R n. Ths mples a soluton to our orgnal problem: Corollary 2. Under assumptons (A1)-(A3), the ACV aucton wth reserve prces p 1 ; : : : ; p n unquely de ned by (13)-(14) solves the robust revenue maxmzaton problem (3). We sketch the man steps of the proof of Theorem 3. Snce Theorem 2 showed that the ACV aucton s feasble (.e. satsfes the constrants (6) (7)), and ts revenue s ndeed R n by (13), we focus on showng that R n s an upper bound for revenue n any aucton. As dscussed n Subsecton 4.2, we need to make actve use of the nonlocal ncentve constrants, whch, usng the formula (11) for transfers, can 17 The workng paper of Ausubel and Cramton (1999) formulated the problem of optmal aucton desgn, assumng no msallocaton and monotoncty n the aggregate, motvatng both propertes nformally by ncentve compatblty under perfect resale. They also clamed (wthout proof) that, n the two-bdder case, the problem s solved by an ACV aucton. In contrast, we derve no msallocaton and monotoncty n aggregate as propertes of a soluton to the seller s maxmn problem under regularty assumptons (A1)- (A3). As argued n the Introducton, the optmal aucton need not have these propertes when resale s perfect but not of the worst-case form. Also, n Appendx G we show that the robust-revenue-maxmzng aucton need not have these propertes wthout regularty assumptons. 19

20 be rewrtten entrely n terms of the allocaton rule: Z E ~ [v 0 (( ; ~ ); ; ~ )] d E ~ [v ((^ ; ~ ); ; ~ ) v ((^ ; ~ ); ^ ; ~ )] 0: ^ (15) We account for these constrants by ntroducng Lagrange multplers on them. Snce there s a double contnuum of such constrants for each bdder, ndexed by ( ; ^ ), the Lagrange multplers (weghts) on them are descrbed wth some approprately constructed nonnegatve measure M on [0; 1] [0; 1]. The Lagrangan adds terms to the objectve functon (12) to penalze volatons of the constrants: " X E ~ v (( ~ ); ~ X 1 F ( ) ~ # ) f ( ~ v 0 (( ~ ); ~ ) + X ZZ S ( ; ^ ; ) dm ( ; ^ ); ) (16) where S ( ; ^ ; ) s the left sde of (15) the slack n bdder s ncentve constrant from type to type ^. We show Theorem 3 by constructng measures M = (M 1 ; : : : ; M n ) such that h; M sats es the followng condtons, where s the allocaton rule from the ACV aucton that s our canddate optmum. (a) Optmzaton: Allocaton rule maxmzes Lagrangan (16) gven measures M; (b) Complementary slackness: the support of M s con ned to constrants (15) that hold wth equalty at,.e., S ( ; ^ ; ) = 0. To see that these condtons mply the result, note that snce M s a nonnegatve measure, the expected revenue from any ncentve-compatble and ndvdually ratonal drect aucton wth allocaton rule cannot exceed the value of the Lagrangan (16) at h; M, whch by (a) does not exceed the value of the Lagrangan at h; M, whch by (b) equals the expected revenue from. We construct Lagrange multpler measures M to penalze devatons that would be pro table n the Vckrey aucton wth lazy reserves p 1 : : : p n, of the knd that emerged as a soluton to the relaxed problem. In ths aucton, as we argued, bdder < n wth value between p n and p cannot wn the object outrght, and so would want to underbd to ncrease the probablty that the object s sold to another bdder, from whom he can 20

21 then buy n resale. Thus, we let the support of M ( ; ^ ) for bdder < n be the trapezod descrbed by p n p and ^, and set M n 0 for the weakest bdder. Ths ensures complementary slackness n the canddatesoluton ACV aucton: Bdder s devaton n the support can only a ect the aucton s outcome f wns under truthtellng but hs devaton cedes the good to the next-hghest bdder. Then, after the devaton, buys n resale at prce (2), whch s the same prce at whch he would have won the aucton by bddng truthfully. So, s nd erent between truthful bddng and the devaton. We spec cally am to construct measures M of the followng form: for approprately chosen one-dmensonal measures and ^ wth supports [p n ; p ] and [0; p ] respectvely, take ther product (a two-dmensonal measure wth support [p n ; p ] [0; p ]), and then restrct to the lower half-plane ^ ; ths restrcton s M. We denote the two component measures dstrbuton functons by ( ) and ^ (^ ), respectvely. We then construct dstrbuton functons and ^ to satsfy the Lagrangan maxmzaton condton (a). Once these are constructed, we verfy (a) by expressng Lagrangan (16), whch s a lnear functonal of the allocaton rule, n the form # E ~ " X ( ~ ) ( ~ ) : (17) The coe cent (), whch we refer to as the mod ed vrtual value of bdder, combnes the ordnary vrtual value and the terms comng from ncentve compatblty constrants (15) weghted by the measures M. For the measures M we construct, we derve an explct formula for () n the appendx (see (31)). Condton (a) then amounts to requrng that wth probablty 1, the canddate allocaton rule allocate the object to a bdder wth the hghest mod ed vrtual value provded t s postve, and not allocate at all f all bdders have negatve mod ed vrtual values. To llustrate how condton (a) serves to pn down measures and ^, focus for smplcty on the two-bdder case, where we only need to construct a measure for bdder 1. In order for the proposed ACV aucton to be optmal, we need two propertes to be sats ed: 1. When 1 < p 2 and 2 = p 2, bdder 2 s mod ed vrtual value should be 0. 21

22 2. When 1 2 (p 2 ; p 1 ) and 2 = p 2, bdder 1 s mod ed vrtual value should be 0. Each property s needed for condton (a) because a slght ncrease n 2 should lead the auctoneer to sell the good to the hghest-value bdder (whose mod ed vrtual value must then be postve), whle a slght decrease n 2 should lead the auctoneer to wthhold the good (makng the mod ed vrtual value negatve). Property #1 pns down the densty of ^ 1 on [0; p 2 ]. It turns out to be proportonal to f 1 ( 1 ) (we normalze the proportonalty factor to 1). Ths occurs because sellng to bdder 2 n any such state ( 1 ; p 2 ) has two e ects on the Lagrangan: the drect e ect on revenue 2 ( 2 ), and the e ect of tghtenng the bndng non-local ncentve constrants of types of bdder 1 above p 2, who could devate to 1 and then buy the good n resale. The rst e ect appears wth a weght proportonal to the probablty of state ( 1 ; p 2 ), whle the second has weght proportonal to ^ 1 ( 1 ); these two weghts must be proportonal n order for these terms to cancel. Note that Property #1 does not pn down the densty of ^ 1 on the rest of ts doman, (p 2 ; p 1 ], but a natural guess s to take the densty as f 1 ( 1 ) here as well. Property #2 requres that the weghted slack n non-local ncentve constrants of bdder 1 s types above 1 nduced by sellng to 1 2 (p 2 ; p 1 ) exactly o set bdder 1 s negatve vrtual value 1 ( 1 ). Once ^ 1 s xed, ths condton pns down measure 1 on ts doman [p 2 ; p 1 ]. Wth more than two bdders, we need to smlarly construct measures for each bdder < n. We agan set ^ to have the same densty as s type dstrbuton, f, restrcted to the support [0; p ]. As for measures, they are pnned down by requrng the mod ed vrtual value of the hghest bdder to be exactly zero so that sellng to hm could be made contngent on whether some other bdder j has beaten hs reserve prce, smlarly to the argument above. The explct formula ((30) n the appendx) depends on an auxlary functon H, constructed below. We show that under assumptons (A1) (A3), the constructed functons are ncreasng, so that each M s n fact a nonnegatve measure. Fnally, we verfy that wth these measures, condton (a) holds,.e., the ACV aucton maxmzes the mod ed vrtual surplus (expressed as (17)) at every type pro le wthout tes, and not just on the regons used n the nd erence condtons nalng down the measures. That s, we show that the hghest-value bdder always has the hghest mod ed vrtual value, and 22

23 t s nonnegatve when some bdder s value s above hs reserve prce and nonpostve otherwse. Now, for the Corollary, lettng be the descrbed ACV aucton, Theorem 3 establshes bound (4) wth R = R n, whle bound (5) follows from Theorem 2. Thus, by Lemma 1, solves the robust revenue maxmzaton problem (and Bertrand resale procedure v solves the worst-case resale problem). 4.6 Calculaton of Optmal Reserves We now ndcate bre y how one mght calculate the reserve prces p k n an applcaton. An nteror soluton p k to maxmzaton problem (14) must satsfy the followng rst-order condton: k (p k ) Y j<k F j (p k ) = R k 1 R k 1 (p k ). (18) Intutvely, ncreasng p k by " only matters when j < p j for all j > k, and n that case t has two rst-order e ects: () condtonal on k 2 (p k ; p k +"), the change a ects the expected revenue from the k 1 strongest bdders, changng t from R k 1 (p k ) (obtaned when bdder k beats hs reserve prce) to R k 1 (obtaned when bdder k does not beat hs reserve prce); and () the change also a ects the expected revenue from bdder k when all stronger bdders values are below p k, changng t from p k (1 F k (p k )) to (p k +")(1 F k (p k +")). The rst-order condton balances those two e ects. To avod havng to drectly calculate R k 1 and R k 1 (p k ), we de ne functon H on p 2 [p n ; 1) by H (p) = R k 1 R k 1 (p) j<k F j (p) for p 2 [p k ; p k 1 ), k = 1; : : : ; n, (19) and thereby rewrte rst-order condtons n the form k (p k ) = H (p k ). (20) Intutvely, H (p) re ects the shadow prce of sellng to a bdder wth value p on all bdders non-local ncentve constrants, and t s used n constructng the Lagrange weghts on these constrants n the proof of Theorem 3. 23

24 In Lemma 2 n Appendx D we show that functon H s contnuous, d erentable, and sats es the followng d erental equaton: d dp (H (p) j<kf j (p)) = ( j<k F j (p)) X j<k f j (p) j (p), for p 2 [p k ; p k 1 ], k 2: F j (p) (21) Furthermore, we establsh that the rst-order condtons (20) must be sats- ed regardless of whether the solutons p k to (14) are nteror. D erental equaton (21) together wth condtons (20) and the boundary condton H(p 1 ) = 0 yeld the followng teratve constructon of functon H and the reserve prces: Frst, p 1 = r 1 by the boundary condton and (20) for k = 1. Then for each k 2, gven p k 1, ntegratng (21) yelds an explct expresson for H(p) Q j<k F j(p) and therefore for H(p) on the nterval p 2 [p k ; p k 1 ], whose unknown left endpont p k s then dent ed as the unque soluton to equaton (20). In Appendx F, we apply ths method to an example where each bdder s value s dstrbuted unformly on [0; a ], wth the upper lmts satsfyng a 1 a n. We show that (20) then takes the form of a kth-degree polynomal equaton to compute the reserve prce p k. For example, p 1 = r 1 = a 1 =2, and then p 2 s gven by a quadratc equaton, whose postve root s p 2 = 1 2 q a (a 1 a 2 ) 2 (a 1 a 2 ) : (22) Wth bdders havng non-dentcal supports, the reserve p k may exceed a k, n whch case bdder k, as well as all the weaker bdders, are completely excluded from the aucton. In the unform example, bdder 2 s excluded f a 2 a 1 =4. Note that as long as all bdders supports are overlappng, such excluson cannot occur n the Myerson optmal aucton, nor even n the relaxed soluton from Subsecton 4.2. It s the non-local ncentve constrants nvolvng stronger bdders buyng from weaker bdders n resale that sometmes make t optmal to exclude weak bdders (and more generally, make the optmal aucton take the ACV form). 5 Dscusson We consder here n more detal the role of some of our modelng assumptons, and some possble alternatve models. 24

25 5.1 Robustness to Informaton Revelaton One mght be concerned wth an asymmetry n our model: we have allowed nformaton to be revealed exogenously after the aucton, but assumed t s not revealed before the aucton. A common perspectve on optmal aucton desgn s that deally t should not make assumptons on bdders nformaton before the aucton ether. Indeed, wthout resale, Myerson s (1981) optmal aucton, though derved under the assumpton that bdders share the auctoneer s pror, can actually be mplemented n domnant strateges, makng t robust to bdders belefs about each other. Smlarly, n our settng wth resale, we could magne a seller who s concerned about bdders learnng about each other both before and after the aucton, and desres robustness to nformaton arsng at ether stage. We have shown that an ACV aucton s optmal when bdders share the desgner s pror before the aucton and learn each other s values after t, followng the worst-case resale procedure. But snce truthful bddng s an ex post equlbrum n the ACV aucton, t sats es ths stronger form of robustness. Thus, we get robustness to pre-aucton nformaton revelaton for free, wthout needng to requre t explctly n the seller s problem. One could alternatvely try to restore symmetry to the model by assumng that exogenous nformaton revelaton cannot happen ether before or after the aucton. In ths case, our basc concluson that robustness to resale mples the auctoneer should not msallocate no longer holds. For an example, t su ces to consder the must-sell case, where we need not worry about (A1) (A3). Suppose that there are two bdders, wth bdder 1 s value equal to 1 for sure, and bdder 2 s value beng 1=3 or 2=3 wth probablty 1=2 each. Consder the mechansm that o ers the object to bdder 1 at a xed prce p = 2=3 ", and f he rejects, gves t to bdder 2 for free. Note that f bdder 1 rejects, he does not learn anythng about bdder 2 s value, so the hghest expected payo he could hope to get n resale s 1=3 (whch he could obtan f he has all the barganng power, by o erng ether a prce of 1=3 or 2=3 to bdder 2). Thus, bdder 1 would prefer to buy n the mechansm. Ths gves a revenue of 2=3 ", whch exceeds the Vckrey aucton s expected revenue of 1=2. Ths example nvolves dscrete dstrbutons, but t can easly be perturbed to a contnuous dstrbuton wth full support on [0; 1], and then any e cent must-sell aucton s revenue-equvalent to Vckrey; and stll, the seller would prefer our alternatve aucton over Vckrey, regardless of the resale procedure. So the optmal must-sell aucton must msallocate wth 25