Uniform-Price Auctions with a Lowest Accepted Bid Pricing Rule

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1 Unform-Prce Auctons wth a Lowest Accepted Bd Prcng Rule PRELIMINARY DRAFT Justn Burkett and Kyle Woodward August 9, 017 Astract We model mult-unt auctons n whch dders valuatons are multdmensonal prvate nformaton. We show that the last accepted d unform-prcng rule admts a unque equlrum wth a smple characterzaton, whle the frst rejected d unform-prcng rule admts many equlra, many of whch provde zero expected revenue. In a natural example, equlrum strateges n the last accepted d aucton are constructed from famlar strateges for sngle-unt frst-prce auctons. In contrast wth the nformaton poolng we prove to arse n the frst rejected d and pay-as-d auctons, the unque equlrum of the last accepted d aucton s fully revealng. 1 Introducton In a unform-prce aucton dders sumt demand functons to a seller who awards m unts of a homogeneous good to the hghest m ds at a sngle clearng prce that apples to all ds. 1 These rules govern well-known large-scale auctons, such as those run y the U.S. Treasury and the ndependent system operators n charge of electrcty dstruton, ut they have also een used to model decentralzed olgopoly markets under the guse of competton n supply functons [Klemperer and Meyer, 1989, Vves, 011]. Despte ther mportance and apparent smplcty, we have a lmted understandng of equlrum ehavor n these auctons when dders have prvate nformaton. Analyss of these auctons s complcated y the multdmensonal structure of ds, and the cross-dmensonal nformaton couplng that ths mples. One urketje@wfu.edu; Wake Forest. kyle.woodward@unc.edu; UNC at Chapel Hll. 1 These rules are easly modfed to accommodate dders sumttng supply curves to sell goods, sale of a dvsle good and/or a seller usng a non-constant supply or demand schedule. Wth ndvsle goods the clearng prce may n prncple e the last accepted d, the frst rejected d or any amount n etween. 1

2 such source of complcaton, dentfed y Vckrey [1961], s the fact that a d on one unt may nfluence the prce pad on pror unts. A straghtforward consequence s that ddng up to one s value on every margnal unt s weakly a domnated strategy dders engage n demand reducton [Ausuel et al., 014]. The degree of demand reducton for a sngle d may depend on the how many pror unts a dder has d on as well as on the dder s margnal values for those unts. A competng dder s nferences aout the proalty of wnnng aganst each d must take these factors nto account. To avod these dffcultes n models wth prvately nformed dders, authors have restrcted attenton to cases n whch dders demand exactly two goods [Ausuel et al., 014, Engelrecht-Wggans and Kahn, 1998], 3 or to dvsle-good models n whch dders have lnear demands determned y normally dstruted ntercepts [Kyle, 1989, Vves, 011]. We present a new approach towards modelng ths aucton n whch dders have mult-dmensonal prvate nformaton aout ther demand for any numer of unts of an ndvsle good. The model allows one to flexly specfy each dder s expected numer of unts demanded at each prce, referred to as a dder s mean demand curve, whle mposng restrctons on the dstruton of realzed demand curves aout ths mean demand curve. We show that ths model, appled to the last accepted d unform-prcng rule, s tractale and yelds equlrum ehavor wth several desrale propertes, especally when compared to exstng models. We frst show that wth two dders the equlrum ds for each margnal unt take the form of ds from an asymmetrc frst-prce aucton for a sngle unt. If the two dders demands are symmetrc, the d curves can e solved for n closed form as n the frst-prce aucton. Whereas the modern lterature on unform-prce auctons and mult-unt auctons more generally emphaszes dfferences etween equlrum ddng strateges n a mult-unt aucton and a sngle-unt aucton, we fnd a close connecton etween a partcular verson of the unform-prce aucton and the frst-prce aucton. 4 An mmedate mplcaton s that many of the results from the frst-prce aucton lterature translate drectly to our envronment. For example, we are ale to draw a connecton etween the relatonshp etween two dders mean demand curves and how aggressvely they d n the aucton. By extendng the work of Maskn and Rley [000] we can classfy the mean demand curves as strong or weak and te these demand curves to ddng aggressveness. Equlrum strateges depend crtcally on the clearng prce charged y the seller. Although t may e true that for a fxed set of strateges the dders Intutvely, a dder reduces hs d elow hs demand curve for the same reason a monopolst s margnal revenue curve falls elow ts demand curve. 3 Wth two goods and a prcng rule that specfes that the clearng prce s equal to the frst rejected d, dders have a weakly domnant strategy to d ther margnal value on the frst unt. Ths leaves one d per dder to e determned n a natural class of equlra. 4 Hstorcally, the lterature on mult-unt auctons assumed wthout analyss they were strategcally analogous to sngle-unt auctons; see, e.g., Fredman [1991]. Our results should not e mstaken for a return to ths vew. As stated aove, our results suggest a strategc connecton etween sngle-unt frst-prce auctons and mult-unt last accepted d auctons.

3 payoffs are not affected much y choosng the last accepted d or the frst rejected d as the clearng prce, our results show that the choce of clearng prce does have a sgnfcant effect on equlrum strateges. The underlyng reason s that the equlrum ds are determned y condtonng on the low proalty event that that partcular d s selected as the clearng prce, and focusng on these events, our analyss makes clear that whether the clearng prce s the last accepted d or the frst rejected d has sgnfcant mplcatons for how the d s chosen. 5 Snce our model accommodates any numer of unts, the equlrum n our model provdes a natural equlrum selecton n the dvsle goods case, where the equlrum need not e unque. One example of non-unqueness n a unform-prce aucton for a dvsle good comes from Back and Zender [1993]. 6 In the Back and Zender [1993] model, although dders receve prvate sgnals aout the common value of the good, there s a contnuum of equlra n whch all types of dders sumt the same d curves whch splt the quantty awarded to the dders for the lowest possle prce. Ths equlrum has at least three undesrale propertes, each of whch s not present n our model. Frst, there are multple equlra n Back and Zender [1993], where we show that the equlrum we descre s unque n a road class of strateges. Second, no prvate nformaton s revealed y the equlrum all types pool n equlrum; n our case, a dder s vector of prvate nformaton s fully revealed n equlrum. Fnally, the seller receves mplausly low revenue, where the equlrum n our model generates nonzero revenue wth proalty one. To understand the revenue n our model, the strong analogy we draw to the frst-prce aucton s useful. For example, revenue comparsons etween frst- and second-prce auctons can e translated nto revenue comparsons etween the last-accepted-d unform-prce aucton and the Vckrey aucton n our case. Pedagogcally, these results suggest the care e pad to the selecton of salent features of equlrum n aucton models. For example, dders report truthfully the canoncal equlrum n a second-prce sngle-unt aucton; t s known that the optmalty of truthful reportng does not extend to mult-unt frst-rejected d prcng see aove, and also Back and Zender [1993], Engelrecht-Wggans and Kahn [1998], Wang and Zender [00], and Ausuel et al. [014] among many others. 7 It s also known that the ntutve ehavor n a sngle-unt frstprce aucton does not translate to mult-unt dscrmnatory auctons, n spte of dders payng ther ds n oth settngs see, e.g., Woodward [016]. Our results show strategc equvalence etween sngle-unt frst-prce auctons and mult-unt last-accepted-d auctons, suggestng that the strategcally salent feature of these auctons s the selecton of the hghest market-clearng prce. 5 Ths dstncton s relevant untl the numer of unts ecomes so large that the envronment approaches a dvsle good model n whch the dstncton etween last accepted and frst rejected s typcally nconsequental. 6 Wth the frst-rejected-d rule, our model exhts multple equlra as well. For more on ths, we refer the reader to Back and Zender [1993]. 7 In partcular, the optmalty of truthful reportng n a second-prce sngle-unt aucton derves from ts equvalence to a Vckrey aucton. Wth multple unts ths equvalence evaporates. 3

4 To our knowledge ths has gone unaddressed n the lterature. Fnally, we compare the last accepted d aucton to the commonly-analyzed frst rejected d unform-prce and the pay-as-d rules. We extend known neffcency results n oth the frst rejected d and pay-as-d auctons see, e.g., Ausuel et al. [014] to our model of dder values. Ths provdes a clear contrast wth the unque equlrum n the last accepted d aucton wth symmetrc dstrutons: ths equlrum s always effcent, whle nether the frst rejected d nor the pay-as-d auctons are effcent. We expose a connecton etween ths neffcency and the tractalty prolems whch are known to complcate the frst rejected d and pay-as-d auctons. In these two auctons, we show that nformaton s pooled n all well-ehaved equlra. It s therefore mpossle to acheve effcency, and solvng analytcally for equlrum nvolves the determnaton of pooled ntervals, whch mply regons over whch the frstorder condtons cannot e naïvely appled. 8 Ths provdes a clear contrast wth the tractale unque equlrum we fnd n the last accepted d aucton. Last, we show that the low-revenue collusve-seemng equlra whch are known to plague the frst rejected d aucton cannot e supported n the last accepted d aucton. These ponts together provde straghtforward justfcaton for employng the last accepted d prcng rule. Leadng example: dders, goods To llustrate the concepts we are uldng on, we egn wth a smple example of our model. There are two dders, each wth demand for up to two unts. An auctoneer s sellng two unts n a mult-unt aucton: he solcts weakly decreasng demands for each unt from each of the dders, and awards the two unts to the agents sumttng the two hghest ds. Bdders have ndependent prvate values: dder s value for her k th unt s v k. For each dder, v s determned y orderng two ndependent draws from a U0, 1; n partcular, v k s margnally dstruted accordng the kth order statstc of a unform dstruton on [0, 1], v k U k 0, 1. In ths analyss we elde some techncal detals and focus on well-ehaved strateges; n partcular, tereakng s a proalty-zero event, so we do not need to worry aout allocatons when dders sumt the same d. Denote the nverse d functons mappng ds to values y ϕ k. Denote the margnal dstruton of v k y F k. Because values are dstruted as order statstcs, F 1 x = x, F x = x x. 8 In the frst rejected d aucton, poolng arses for relatvely low valuatons for whom the margnal gan assocated wth an ncrease n wnnng proalty s outweghed y the ncreased proalty of settng the market prce. In the pay-as-d aucton, poolng arses due to the constrant that ds e weakly decreasng whle agents would sometmes prefer to sumt nonmonotone or even weakly ncreasng d functons. The mathematcs underlyng poolng ehavor are dstnct n these two auctons, ut they mply smlar ssues for tractalty. 4

5 We consder two payment rules. In last accepted d LAB, dders pay the second-hghest d sumtted for each unt they receve. In frst rejected d FRB, dders pay the thrd-hghest d sumtted for each unt they receve. 9 We defer the dscusson of pay-as-d auctons, n whch dders pay ther sumtted d for each unt they receve, untl later n the paper. Several statstcal events are of mportance n the prcng rules we nvestgate. Dervatons of the assocated proaltes may e found n the Appendx..1 Last accepted d In LAB, three statstcal events are salent. Frst, dder can wn unts; ths occurs when 1. Second, dder can wn 1 unt whle dder sets the prce; ths occurs when 1 1 >. Thrd, dder can wn 1 unt and set the prce; ths occurs when 1 > 1.10 Wth these events, nterm utlty n LAB can e expressed as u ; v = v1 + v Pr 1 + v1 E [ > ] Pr 1 1 > + v1 1 Pr 1 > 1. As we show n Appendx A, n a symmetrc equlrum the agents frst-order condtons are gven y ϕ 1 1 ϕ dϕ ϕ ϕ ϕ 1 = 0; unt 1 These equatons mply that n equlrum, ϕ ϕ 1 dϕ 1 ϕ 1 = 0. unt ϕ 1 = ϕ =, 1 v = v = 1 v. That s, equlrum n LAB s exactly equlrum n a standard frst-prce aucton for a sngle unt. It s mmedate to verfy that ths equlrum s effcent.. Frst rejected d In FRB, three statstcal events are salent. Frst, dder can wn unts; ths occurs when 1. Second, dder can wn 1 unt and set the prce; ths 9 As we wll dscuss later, LAB and FRB correspond to the hghest and lowest respectvely market-clearng prces n a Walrasan market wth nelastc supply and demands gven y the sumtted ds. 10 There s also a fourth relevant event, that dder wns zero unts. Because ths yelds 0 utlty, t s of no consequence to the formal analyss. 5

6 occurs when 1 >. Thrd, dder can wn 1 unt whle dder sets the prce; ths occurs when 1. Wth these events, nterm utlty n FRB can e expressed as u ; v = v1 + v E [ 1 ] 1 Pr 1 + v1 Pr 1 > + v1 E [ 1 > ] Pr 1 >. As we show n Appendx A, n a symmetrc equlrum the agents frst-order condtons are gven y ϕ 1 ϕ dϕ = 0; unt 1 ϕ ϕ 1 dϕ 1 ϕ ϕ ϕ 1 = 0. unt The frst-order condton wth respect to the d for the frst unt, 1, confrms the ntuton that truthful reportng s a weakly domnant strategy. Ths follows from standard second-prce aucton logc: the d for the frst unt s never the clearng prce when the agent wns so t s effectvely costless to ncrease the d. 11 In an equlrum n whch agents d truthfully for ther ntal unts the frst-order condton wth respect to the second-unt d s no longer meanngfully a dfferental equaton: ϕ 1 =, and hence dϕ 1 = 1. Suttutng through, symmetrc equlrum ds for the second unt must solve v v v = 0. Ths s a negatve quadratc n, wth a soluton of = v ± v v. Snce v [0, 1], t must e that v v 0 and ths nequalty s strct when v / {0, 1}; then the negatve quadratc has no real zeroes, and the frstorder condton wth respect to s negatve everywhere. It follows that = 0 ndentcally, ndependent of v..3 Comparson Because ds n the LAB aucton are ndependent of the unt for whch they are sumtted that s, ecause k v = v k/ for oth unts outcomes are effcent. Ths contrasts strongly wth the results of the FRB aucton. Because second-unt ds are always zero, neffcent outcomes wll arse whenever one 11 Our techncal results show that nformaton confoundng n equlrum s ndependent of whether frst-unt ds are truthful; t follows mmedately that FRB equlra are never effcent. We conjecture that revenue-domnance of LAB over FRB holds across all FRB equlra, ut we do not formally demonstrate ths result. 6

7 agent s value for her frst unt s elow the other agent s value for her second unt n our model, ths proalty s 1/3. Due to the lnearty of ds n the LAB aucton, expected revenues may e smply computed as half of the expected second-hghest draw from four draws of a unform dstruton. Aggregate expected revenue s then 3/5, and perunt expected revenue s 3/10. Ths agan contrasts strongly wth the expected revenue of the FRB aucton. Because second-unt ds are always zero, the clearng prce s always zero. Then aggregate and per-unt expected revenues are zero. In the remander of ths paper we demonstrate that certan of these propertes generalze. When market demand satsfes a smple algerac condton market alance there s an equlrum of the LAB aucton n whch ds are ndependent of the unt for whch they are sumtted, mplyng effcency. We prove generally that there s a unque well-ehaved 1 equlrum n the LAB aucton, estalshng effcency of a natural outcome of the aucton. Contrarwse, we demonstrate that there s always nformaton confoundng and some degree of poolng n the FRB aucton, thus all outcomes of the FRB aucton are neffcent. We also demonstrate that the FRB aucton always admts an equlrum wth expected revenue gven y the reserve prce tmes the quantty sold, whle the LAB aucton admts no such equlrum. 3 Model An auctoneer sells m unts of a homogeneous good to n rsk-neutral dders who, wth proalty 1, have strctly postve aggregate demand for at least m unts. Bdder values unts accordng to the ordered realzatons of m ndependent draws from the asolutely contnuous dstruton F : [0, 1] [0, 1] wth densty f. The orderng ensures that margnal values are weakly declnng for every realzaton. For example, the dder s margnal value for the frst unt s the frst order statstc of m ndependent draws from F. We denote the ordered vector of dder s valuatons y v, so that vk s her value for her k th unt. By defnton, v1 vm. For smplcty we frequently reference the n m model, n whch n symmetrc agents each have demand for m/n 1 N unts, and m unts are avalale n the aucton. Generally, when m = j m j for any dder, we say that the market s alanced. We consder sealed-d auctons, where dders sumt weakly decreasng demand vectors to the auctoneer. Bdder sumts a weakly-postve demand vector, so that k s her d for her kth unt. Where helpful, we wll take a mechansm desgn approach and consder ds as functons of dders prvate values, k k v. Wthout a reserve prce, the auctoneer allocates the aval- 1 The proper noton of well-ehavedness s defned later. 7

8 p p q Fgure 1: Maxmum and mnmum market-clearng prces dsplayed on an aggregate demand curve, Dq = nf{p : #{, k : k p} < q, when m = 7 unts are avalale. ale unts to the m hghest ds. 13,14,15 Denote the maxmum and mnmum market-clearng prces y p and p, respectvely, where p = mn { p : # {, k : k p } m }, p = max { p : # {, k : k p } m }. Each dder s rsk-neutral and her utlty s quaslnear n payments. Condtonal on allocaton q and payment t, dder s ex post utlty s u [ q, t ; v q ] = t We focus most of our attenton on the last accepted d LAB unform-prcng rule, n whch each dder pays the same prce for each unt she otans, and ths prce s equal to the m th hghest d; ths s equvalent to clearng the market at p, the hghest market-clearng prce, t q = max{p, r}q. For mechansm comparsons, we also dscuss the frst rejected d FRB unform-prcng rule, where the per-unt prce s the m + 1 th hghest d equvalent to the lowest market-clearng prce, t q = max{p, r}q, and the pay-as-d PAB prcng rule, n whch for each unt a dder receves she pays her d for ths specfc unt, t q = q k=1 k. 13 Bd monotoncty s a constrant typcally oserved n practce. However, under the assumpton that the auctoneer accepts ds n decreasng order d monotoncty s also a smplfyng assumpton that can e made wthout loss of generalty. 14 Where the m th hghest d s not well-defned some form of tereakng or ratonng s necessary. Because the tereakng rule s not of mportance to our analyss, we leave t unspecfed. Ths pont has een noted n the mult-unt and dvsle-good aucton lterature; see, e.g., Häfner [015]. 15 When a nontrval reserve prce s present, ds are accepted n decreasng order untl ether all m unts are allocated or there are no remanng ds weakly aove the reserve prce. For the most part our results are not meanngfully affected y the presence or asence of a reserve prce n lght of what s known of ehavor n sngle-unt auctons wth reserve prces. Therefore t s natural to gnore reserve prce to avod unnecessary techncaltes. k=1 v k 8

9 Market clearng mples that dder receves unt k f and only f her opponents receve n aggregate less than m k + 1 unts. 16 It s helpful to consder dder competng for her k th unt aganst the aggregate demand of her opponents for m k + 1 unts. Let H e the margnal dstruton of her opponents m k + 1 th hghest d, and let h e the assocated densty where well-defned. 3.1 Matchng demand curves There s a natural nterpretaton of our order statstcs model n terms of dders mean demand curves n the followng sense. Fxng a unform prce p, the expected numer of unts demanded y dder s 1 F pm. 17 The specfcaton of the mean demand curve s therefore flexle ecause F s artrary, whle the dstruton of demand curves aout the mean s determned y the propertes of the order statstc model Equlrum of LAB We frst derve equlrum ddng strateges n the last-accepted-d aucton wth symmetrc demands when the alanced market condton holds, showng that they have closed form representatons. 4.1 Symmetrc Demand Curves n a Balanced Market Recall that a alanced market s one where m = j m j. Ths mples that each dder faces exactly m ds from opponents n equlrum. The case where two dders each demand m unts s one such example. By symmetrc demand curves, we mean that each dder s k th unt s dstruted accordng to the k th order statstc from the same dstruton F.e., vk F k for each. We use Y k to denote a random varale that s the k th order statstc from m ndependent draws from F. Buldng on the example n Secton, suppose that each opposng dder shades her d consstently for every margnal unt n the sense that j k = vj k for some ncreasng and each j and k. Let ϕ e the nverse of v. The dstruton of the opponent s k th d s therefore F k ϕ. If dder places the d k = v k on her kth unt, under the last accepted d rule, dder wns exactly k unts and pays Y m k f and only f vk Y m k vk+1. On the other hand, dder wns k unts and pays v k f and only f Y m k vk Y. If k > 1, dder s d on the k th unt 16 Wth a reserve prce the only f s stll vald, ut the f may fal. Nonetheless the competton faced for unt k s aganst opponents aggregate demand for m k + 1 unts. 17 For a fxed prce, the numer of unts demanded out of a maxmum of m s a random varale wth a nomal dstruton wth proalty of success gven y 1 F p. 18 For example, the oservatons n Footnote 17 mply that the varance of the numer of unts demanded at prce p must e F p1 F pm, or large for ntermedate prces and small for prces near 0 or 1. 9

10 also affects the proalty of wnnng exactly k 1 unts when the realzaton of values s such that vk 1 Y m k 1 vk. These three events account for all the ways n whch the d on the k th unt affects dder s payoff. To derve a necessary condton for to e an equlrum consder a d v vk 1, v k+1 and the assocated contruton to dder s payoff. v k vl kx f m k x dx v k+1 l=1 k + vl kv F v F m k v + l=1 v k 1 k 1 v vl k 1x f m k 1 x dx l=1 The dervatve of ths expresson wth respect to v evaluated at vk s v k vk f vk k vkf vk F m k vk m = k1 F v m k k m k F vk k 1 { fv k vk vk vkf vk } 1 One can check that the famlar soluton to the symmetrc frst-prce aucton for a sngle unt, v = 1 v F v xfx dx makes ths expresson zero, and ths s n 0 fact the equlrum mappng from margnal values to ds. Proposton 1. If margnal values for each dder are the order statstcs from ndependent draws from F the equlrum ds for dder n the last-acceptedd aucton are v = 1 F v k v k 0 xfx dx k {1,..,m } Proof. From the dscusson precedng the proof, t s clear that the canddate equlrum d, vk, satsfes the kth frst-order condton. Standard arguments estalsh that the partal dervatve s negatve postve for v when v > vk v < vk, whch also means that the ojectve must e lower at the end ponts, vk 1 and v k+1, gven y the monotoncty constrant. Furthermore oserve that all cross-partal dervatves are zero, from whch t follows that the second-order condtons are satsfed. For k = 1, m, the relevant components of the ojectve are respectvely v v 1 x f m 1 x dx + v 1 v F m v F m 1 v. v and m vj mv F 1 v + j=1 v m 1 v m 1 vj m 1x f 1 x dx. j=1 10

11 Smlar arguments estalsh the optmalty of settng 1 = v1 and m = vm. Ths equlrum s effcent, and hence standard arguments mply that the expected payment should e equal to the Vckrey payment. To see ths, consder the event that vk s the last accepted d.e., Y m k vk Y. In ths event the dder pays kvk, whch s the expected payment made for k unts n a Vckrey aucton condtonal on ths event ecause k E [ Y m k+j Y m k vk ] Y j=1 = k E [ ] Y j:k v k Y 1:k j=1 = ke [ Y v k Y ] = kv k, where the notaton Y j:k denotes the j th hghest value out of k ndependent draws from F. To understand the frst equalty, oserve that condtonal on the event Y m k > v k > Y the frst m k random varales provde no addtonal nformaton aout the last k random varales, so the expectaton reduces to one nvolvng just the last k. Recall that n the Vckrey aucton a dder who wns k unts n ths envronment would e requred to pay the sum of the k rejected ds made y the opponents. In other words, the ds n ths equlrum are set so that the expected payment equals the expected Vckrey aucton payment condtonal on the event that the d determnes the payment. 4. Asymmetrc Demand Curves wth Two Bdders When margnal values for two dders, who demand all m unts, are drawn from dfferent dstrutons, the equlrum wll n general no longer e effcent or have closed-form expressons, as s the case n asymmetrc frst-prce auctons for a sngle good. However, gven that frst-order condtons n ths aucton take forms very smlar to that of the frst-prce aucton for a sngle good, many of the results wll carry over. Most of the lterature on asymmetrc frst-prce auctons focuses on the twodder case. An analogous case n ths model s set up as follows. Suppose that there are two dders, = 1,, who each value margnal unts accordng to m draws each from F, where F 1 F. Suppose that s ds for each margnal unt are determned y the ncreasng, dfferentale functon vk wth nverse ϕ. Then the event that s d on the k th unt, k, s selected as the last accepted d occurs f and only f Y m k ϕ k Y. Accountng for the other two events nfluenced y the choce of k see Secton 4.1, the 11

12 frst-order condton assocated wth k f k k+1, k 1 s m k1 F ϕ m k k m k F ϕ k k 1 { ϕ j kf ϕ kvk k F ϕ k } = 0 3 whch reduces to the expresson studed n Maskn and Rley [000]. Suppose 1 v and v are the equlrum d functons from the frst-prce aucton nvolvng two dders wth correspondng value dstrutons F 1 and F, then t s mmedate that settng k = vk wll satsfy dder s frst-order condton for good k when v = vk k {1,...,m}. Proposton. Wth two dders whose m margnal values are the order statstcs from F 1 and F, f 1 v and v are the equlrum d functons n the frst-prce aucton for a sngle unt wth two dders whose values are dstruted accordng to F 1 and F, then the strateges v = v k k {1,..,m} consttute an equlrum of the last-accepted-d aucton. Proof. Analogous to the proof of Proposton General Case If the market s ether alanced and dders have symmetrc demands or there are two dders wth asymmetrc demand for all unts, we can dentfy equlrum strateges wth a correspondng frst-prce aucton. Ths s no longer true n the general case where the market s ether unalanced or there are more than two asymmetrc dders. A common property of the equlra n oth of the prevous sectons s that there exsts a unvarate functon whch dder uses to determne the ds on all of hs margnal unts from ther margnal values. In general ths property, whch allows for the reducton to a frst-prce aucton, does not hold n equlrum, and dders may shade ther ds on margnal unts dfferently dependng on the unt to whch the d s on. Despte not eng ale to pn down equlrum strateges n the general case, we show n ths secton that we can utlze technques from the frstprce auctons lterature to estalsh that some key propertes stll hold. For example, we present an unalanced, symmetrc demand case n whch we can prove a unqueness result for equlrum strateges usng an argument that closely resemles the unqueness argument typcally gven for the equlrum of a frst-prce aucton. Frst, we provde an exstence result for the general model. Proposton 3. Wth n dders {1,..., n}, where dder s m margnal values are the order statstcs from the dstruton F, the last-accepted d unform prce aucton admts a pure-strategy Bayesan Nash equlrum. Proof. Ths follows from Corollary 5. n Reny [011] Reny [011] nvestgates the FRB aucton. Wth regard to exstence although not, as argued aove, the structure of equlrum the arguments do not change n a sustantve way. 1

13 Next, we descre and analyze a case of the model that departs from the symmetrc, alanced case n a mnmal way. There are two goods, m =, and n > dders wth symmetrc demands for two unts, F = F and m = for all {1,..., n}. In a symmetrc, separatng equlrum, there are two d functons, 1 v 1 and v, that determne the ds on the frst and second unts for each dder as functons of the margnal value of the respectve unt. For the equlrum to e separatng, we requre that 1 v 1 > v for all v 1 > v. If we denote the nverse d functons when 1 and are ncreasng y ϕ 1 and ϕ, t follows that ϕ 1 < ϕ. 4.4 Implcatons Gven the relaton etween equlrum ddng n the last-accepted-d model and ddng n asymmetrc frst-prce auctons, a numer of results follow mmedately. Instead of exhaustvely lstng them here, we emphasze ther nterpretaton n ths model. Recall that we may nterpret the functon 1 F pm as dder s mean demand curve. It follows from the prevous secton, that 1 F ϕ m represents the mean numer of ds placed y dder that exceed n equlrum, referred to as the mean quantty demanded n equlrum. Bdder s mean resdual supply curve s therefore F ϕ m, whch s proportonal to the equlrum d dstruton of a dder wth type dstruton F n a frst-prce aucton. The stochastc domnance propertes used n the asymmetrc frst-prce aucton lterature have mmedate analogues to propertes of the mean demand curves n ths model. For example, dder havng weakly hgher mean demand than dder at each prce s equvalent to F frst-order stochastcally domnatng F. An mplcaton from the frst-prce aucton lterature s that dder s mean quantty demanded weakly exceeds dder s n equlrum [Krkegaard, 009, Corollary 1]. The stronger dstrutonal orderng property of reverse hazard rate domnance can e stated as follows. Defnton 1 Reverse Hazard Rate Domnance. F rh G d F x 0, x. dx Gx When F and G admt denstes at x, ths mples fx/f x gx/gx. If F xm s the mean resdual supply curve that dder would present to dder f she were to d her value for each unt, then xf x/f x s the elastcty of that supply curve. The reverse hazard rate condton can then e nterpreted as requrng that these elastctes are ordered. From Proposton 3.5 of Maskn and Rley [000] we can therefore conclude that ths orderng of elastctes s suffcent to order the d curves of the dders, meanng F F mples v < v or v < v. Ths s the well-known weakness leads to aggresson result. Fnally, we make one more connecton to work on nvestment ncentves n sngle unt auctons. In ther Proposton 3 Arozamena and Cantllon [004] 13

14 show that f one dder s gven the opportunty to upgrade ther type dstruton ex ante y makng t stronger wth respect to hazard-rate domnance, the nvestment ncentves are stronger n the second-prce aucton than n the frst-prce aucton. Furthermore, ther Proposton 4 shows that nvestment ncentves are optmal n the second-prce aucton. Upgradng the dstruton has a natural nterpretaton n our model. It s equvalent to a dder n our model nvestng to ncrease her mean demand curve n such a way as to weakly ncrease the elastcty of the mean resdual supply curve at every pont. From the Arozamena and Cantllon [004] results we get mmedate comparsons of the nvestment ncentves n the last-accepted-d unform-prce aucton to those n the Vckrey aucton, whch s the extenson of the second-prce aucton to ths envronment. 4.5 Equlrum unqueness The work n the prevous sectons shows that there s a close connecton etween the equlrum of the frst-prce aucton and that of the LAB aucton. Several authors have nvestgated the unqueness of equlrum ddng strateges n the frst-prce aucton, notaly Maskn and Rley [003], Bajar [001], and Lerun [006]. Ther arguments for unqueness are ased on analyses of the system of dfferental equatons n the nverse d functons derved from frst-order condtons. In our dervaton of equlrum for the LAB aucton, we show that under the assumpton that the opponent uses the same unvarate d functon for each margnal unt we recover the same system of dfferental equatons e.g., see 3. It follows that f the dders are restrcted to usng the same d functon for each margnal unt the exstng unqueness results for the frst-prce aucton apply n our settng. In ths secton, we extend ths result to show unqueness over a larger set of strateges. We show that the equlrum dentfed n the prevous secton s unque among all separatng strateges, where separatng means that each dder s monotoncty constrant nds wth proalty zero. More precsely, we show that whenever the correspondng frst-prce aucton admts a unque equlrum, the equlrum we have dentfed s unque among separatng strateges. The arguments for unqueness n the frst-prce aucton gven n the lterature follow the same asc steps. 0 Frst, one shows that the largest equlrum d or smallest n the case of procurement s the same for every dder. Second, one defnes a system of ordnary dfferental equatons nvolvng nverse d functons. The equatons n the system are shown to e necessary and suffcent for optmalty and also to satsfy the Lpschtz condton at every d excludng the lowest d. The ntal value prolem startng from a partcular hghest d therefore has a unque soluton. Thrd, one shows that f and are two ntal 0 We refer to Lerun [006] for a dscusson of unqueness results n the frst-prce aucton lterature and the assumptons requred to prove unqueness. There s a unque equlrum n the asymmetrc frst-prce aucton under farly general condtons, ut as argued n Lerun [006] some pror proofs have reled on unjustfed uses of L Hôptal s rule. 14

15 values wth < then the solutons to the ntal value prolem usng are greater than those to the prolem usng at every nteror. In other words, the solutons are monotone n the hghest d. Fnally, one shows that the second and thrd results mply that there can only e one hghest d yeldng a soluton that s also an equlrum. To estalsh unqueness of the LAB equlrum among separatng strateges n our model, we follow the frst two steps n the pror paragraph ut then appeal to the unqueness of the correspondng frst-prce aucton soluton to complete the proof. We restrct attenton to separatng strateges, ecause our argument reles on the analyss of a system of dfferental equatons that s only vald for separatng strateges. Allowng the monotoncty constrant to nd for artrary ds leads to a system of equatons that s sustantally more dffcult to analyze. The most mportant step n our argument s to estalsh that the hghest d sumtted for any unt y any dder n equlrum s the same. Ths does not follow drectly from the analogous argument n the frst-prce aucton, although there are smlartes. The added dffculty here arses from the facts that there s a monotoncty constrant on ds and that the proalty that a d on unt k wns depends on the dstruton of two of the opponent s ds. We show that the hghest equlrum d s the same for all unts n Lemma, after provng an ntermedate lemma next. Lemma 1. Suppose that a type-v dder sumts a constant d {k,...,k+a} for unts k,..., k + a and let l v l and sv s wth l, s {k,..., k + a} e respectvely any of the dder s largest and smallest unconstraned ds for these unts. Then l v l > sv s mples l v l > {k,...,k+a} > sv s. Proof. The frst-order condton for the constraned d s k+a j=k j U {k,...,k+a} ; v = 0, 4 or the sum of the unconstraned d frst-order condtons. Note that the ojectve s quas-concave n each j. At the largest unconstraned d, l, the frstorder condtons for the other ds cannot e postve, due to quas-concavty, and gven l v l > svs at least one s negatve. Therefore, at l, 4 s negatve. A smlar argument mples that 4 s postve at s. Lemma. In equlrum, there s a such that for all and k, k =. Proof. Frst, t cannot e that k = for all and k ut ecause the type of dder who sumts the hgher maxmmum d could lower all of hs ds and reduce hs payment wthout reducng the proalty of wnnng any unts. Therefore f the lemma s false k > k+1 for some k and. Let ˆk to e the lowest k for whch k > k+1. 15

16 We clam that m ˆk+1 = m ˆk = ˆk. It must e that m l+1 l= ˆk for all l ˆk ecause otherwse the type of dder placng these ds could weakly reduce all of these ds wthout reducng the proalty of wnnng any of the tems. If m l+1 m ˆk+1 < ˆk for all l ˆk, then dder should respond y reducng hs maxmum ds on the frst ˆk unts for the same reason. Hence, m ˆk+1 = ˆk. Now m ˆk+1 = m ˆk follows ecause otherwse m ˆk > max{ m ˆk+1, ˆk+1 } and m ˆk can e reduced wthout lowerng the proalty of wnnng or volatng monotoncty. Fnally, for v m ˆk close to 1, v m ˆk m ˆk ˆk+1 < ˆk = m ˆk+1. Lemma 1 then mples that the optmal choce of constraned d for dder s strctly elow ˆk, whch s a contradcton. Havng estalshed that there s a common maxmum d for all unts and dders, we next descre the system of dfferental equatons we evaluate. As wth frst-prce auctons, the arguments are made smpler y wrtng the dfferental equatons n terms of an unknown dervatve wth respect to a d dstruton cf. Lerun [006]. Recall that n ths secton we are assumng that the dders use separatng strateges. Ths mples that ϕ k ϕ k+1 for all k and. Consequently, the dstruton of the k th d of dder s Fk ϕ k. Furthermore, dder s frst-order condton wth respect to hs m k + 1 th d ecomes [ϕ k] f k ϕ kv m k + 1 F kϕ k F k 1ϕ k 1 = 0. We create a system of m dfferental equatons out of the frst-order condtons for each d y each dder. Instead of wrtng the system n terms of unknown nverse d functons, we wrte t n terms of unknown d dstrutons as follows. Defnton. Let H k F k ϕ k, H 0 0, and 0, 1 e gven. Fnd H 1 k, H k k=1,...,m such that for all 1 k m, {1, }, and 0, ] d d H k = m k + 1 H ϕ k Hk 1 5 H k = 1 Ths ntal value prolem nvolves a system of m equatons n m unknown functons, Hk. Note that ϕ = [F ] 1 H. The next lemma estalshes that an equlrum of the LAB aucton s necessarly a soluton to ths ntal value prolem. Lemma 3. Any equlrum d profle n separatng strateges must satsfy 5. Proof. Ths s mpled y contnuty and dfferentalty of the d dstruton functons. These results are smlar to the arguments famlar from the frstprce aucton, ut we reproduce them here due to the changes n the agents 16

17 utlty functons nduced y shftng to a mult-unt model. We say that unt k s opposed to unt m k + 1, n the sense that agent wns unt k f and only f agent j wns unt m k + 1. Recall the separale utlty representaton for the LAB aucton, u ; v = m k=1 v k H k H k H m k k k k k k 0 xdh m k x + k 1 k 0 xdh x.1 Frst, there are no gaps n equlrum d dstruton functons. If there s a gap n H, then a d for agent s unt k strctly nsde ths gap nduces no addtonal wnnng proalty ut ncurs addtonal expected costs vs a vs ddng the lower ound. It follows that any gaps n H are shared y the opposng dstruton Hk. Snce there s no proalty gan wthn the gap, for a d to e placed at the upper end of the gap there must e a mass pont; there are therefore dentcal mass ponts for the opposng unts k and m k + 1. Identcal mass ponts cannot arse for standard tereakng reasons, therefore ths s not supportale n equlrum. Second, aove the reserve prce there are no mass ponts n equlrum d dstrutons.e., equlrum d dstrutons are contnuous. Suppose that there s a mass pont n H at d, ut no mass pont n H m k. Snce ds are n general strctly elow values 3 and there are no gaps n the d dstrutons, there s a value v such that k v = ε for any ε > 0. For ε small enough, a slght ncrease to k v = + ε yelds a dscrete jump n expected utlty; ths mples that gaps exst n response to mass ponts, and we have already estalshed that gaps cannot exst. Otherwse, suppose that there are mass ponts n oth H and H m k at, so that the aove logc does not apply. However, f ths s the case, then there s a mass pont n H m k, the unt opposed to dder s unt k + 1. Then the prevous argument holds unless there s also a mass pont n H m k 1, and so on. Snce there are no mass ponts n the degenerate dstruton H0 the H correspondng to m k k = m the orgnal argument must hold for some unt, volatng the no-gaps property estalshed aove. Lastly, equlrum d dstrutons are dfferentale aove the reserve prce. Suppose that H s not dfferentale at whle H m k s. As s 1 Ths expresson appears to presuppose the dfferentalty of dh k for all k, however t s a re-expresson of one n terms of well-defned condtonal expectatons; snce we estalsh that n equlrum the d dstrutons are contnuously dfferentale ths expresson s ultmately correct. We do not presuppose the correctness of ths expresson, and avod ths potental crcularty n our formal arguments. Ths analyss gnores the posslty that the support of the d dstruton aove the gap s left-open. For a d suffcently close to ths upper endpont, the arguments are the same. 3 Ths somewhat ovous pont s proved explctly n a Lemma n an earler verson of ths paper. 17

18 famlar, ths mples the exstence of a gap n the case of an upward knk or a mass pont n the case of a downward knk n the opposng d dstruton for agent s unt k. Snce oth of these have een ruled out aove, ths nondfferentalty s not possle. The case n whch oth H and H m k are nondfferentale at can e handled smlar to the analyss of mass ponts aove. Then equlrum d dstrutons must e dfferentale. Snce equlrum d dstrutons are contnuous and dfferentale aove the reserve prce, the frst-order condtons must e satsfed n any equlrum n separatng strateges. Compare the prolem n Defnton to the followng correspondng one for the frst-prce aucton. Defnton 3. Let H F ϕ and 0, 1 e gven. Fnd H 1, H such that for all 0, ] and {1, }, d d H = H ϕ H = 1. 6 For an artrary, ecause 6 satsfes the Lpschtz condton for all 0, ], the Fundamental Theorem of Ordnary Dfferental Equatons FTODE mples there s a unque soluton to the ntal value prolem n Defnton 3. Furthermore, when there s a unque equlrum n the frst-prce aucton, a sngle such yelds a soluton that also satsfes the oundary condton H = F ϕ = 0, where s the lowest equlrum d. Snce the system n 5 also satsfes the Lpschtz condton for all 0, ], the FTODE mples that there s a unque soluton to the prolem n Defnton gven a. But these two solutons must concde n the sense that f ϕ 1, ϕ s a soluton to the frst-prce aucton prolem y settng ϕ k = ϕ for all k and and examnng Equaton 3 we fnd the unque soluton to the ntal value prolem correspondng to the LAB aucton as well. The fnal step s to oserve that whle Proposton gves us that the equlrum value of generates equlrum solutons to oth prolems a dfferent would generate a soluton that s not an equlrum of the frst-prce aucton y unqueness and cannot e an equlrum of the LAB aucton. Proposton 4. Consder the last-accepted-d aucton etween two dders whose m margnal values are the order statstcs from F 1 and F and the correspondng frst-prce aucton nvolvng two dders wth value dstrutons F 1 and F. If the equlrum n the frst-prce aucton s unque, then there s one equlrum of the last-accepted-d aucton n whch the dders use separatng strateges.e., ones n whch the monotoncty constrant nds wth proalty zero. 18

19 5 Poolng and nformaton revelaton Each of the auctons we analyze has a structure n whch ds for dfferent unts are co-determned only when the monotoncty constrant that ds must e weakly decreasng n quantty s ndng. 4 We capture ths structure wth the noton of separale ncentves. Defnton 4 Separale ncentves. An aucton model has separale ncentves f there are functons u k m k=1 n =1 such that for each agent, each unt k, and all d profles, and value profles v, u, ; v m = u k k, ; v k. k=1 When quantty-monotoncty constrants are not ndng, a model wth separale ncentves can e analyzed dmenson-y-dmenson as a set of m ndependent optmzaton prolems. In lght of Lemma 4 ths features promnently n our analyss of the revelaton propertes of the FRB and PAB aucton formats. Lemma 4 Separalty of mult-unt auctons. The FRB, LAB, and PAB auctons each have separale ncentves. In each of these auctons, each dmensonal utlty functon u k satsfes ncreasng dfferences n k, v k. Lemma 4 s proved n Appendx B. If the quantty-monotoncty constrant does not nd, n models wth separale ncentves the d for any unt s determned solely y the value for ths unt and the opponent s ddng strategy. Thus n a separatng equlrum, separale ncentves mply that the agent s optmzaton prolem s well-ehaved and ndependent of any constrants. Defnton 5 Strctly separatng ds. A d functon s strctly separatng f the nverse d correspondence s at most sngle-valued; that s, for all type profles v, # { v : v = v } = 1. Defnton 6 Partal poolng. A d functon exhts partal poolng f the nverse d correspondence s mult-valued wth postve proalty; that s, Pr v { v : #ϕ v > 1 } > 0. There s a wedge etween strct separaton and partal poolng: nverse ds mght e mult-valued wth zero proalty. We are concerned wth ssues of nformaton confoundng, and n partcular n stuatons n whch nformaton s ofuscated n equlrum. If equlrum ds are non-separatng wth proalty zero, equlrum s essentally separatng, and ths dstncton s not meanngful. 4 Separately, these models exht the standard IPV mechansm desgn monotoncty-nvalue. Because ths s a result and not a constrant, when we refer to ndng monotoncty constrants we are referrng to monotoncty n quantty. 19

20 Our defnton of partal poolng s structured to capture two separate poolng effects. In the FRB aucton, truthful ddng for the frst unt s a weakly domnant strategy. However, we show that there s a range of last-unt valuatons such that a d of zero strctly domnates all others; ths occurs ecause resdual competton comes from opponents low unts, for whch dstrutons are relatvely strong. Increasng the d for the fnal unt has lttle margnal effect on the proalty of wnnng the unt ut a comparatvely strong margnal effect on the expected cost pad for all m 1 unts, condtonal on ther eng won. In an equlrum wth truthful ds for the frst unt, the proalty of wtnessng any partcular d profle s zero even though the proalty of wtnessng a zero d for an agent s fnal unt s strctly postve; partal poolng captures ths postve-proalty nonnvertlty. Partal poolng also captures the nformaton confoundng we oserve n the PAB aucton. In PAB, the dder s facng ncreasngly aggressve competton as she consders her d for hgher unts: her d for hgher unts s aganst her opponents ds for lower unts. We show that there s generally an ncentve for the dealzed d for the frst unt to e elow the dealzed d for the second unt, volatng the d monotoncty constrant. Ths mples that, for certan value profles, ds wll e flat for small quanttes. Contnuty of utlty n value mples that ths same flat wll e realzed for neary value profles f, for example, the value for the frst unt falls whle the value for the second unt rses or vce versa and thus upon wtnessng a partcular flat d the dder s value profle cannot e perfectly nverted. Agan, ths happens n spte of no d eng sumtted wth postve proalty. Asde from mplcatons for tractalty, nformaton revelaton s drectly related to effcency. An effcent mechansm must allocate unts to the agents wth the hghest values. When nformaton s confounded, ths s not possle: effcency entals knowng whch agents have the hghest values for the m avalale unts, and standard dentfcaton arguments mply that f ths s possle, ds must e separatng. We thus contrast the FRB and PAB auctons, n whch all equlra exht partal poolng and are thus neffcent, wth the LAB aucton, whch we have shown to admt a separale and effcent equlrum wthout poolng. Remark 1. Any pure-strategy equlrum can e transformed nto a monotone pure-strategy equlrum wthout affectng agents ncentves or payoffs. We therefore restrct attenton to equlra n monotone pure strateges. Lemma 5 Separale ds n separatng equlrum. In a monotone strctly separatng equlrum the LAB, FRB, and PAB aucton models, dder s equlrum d functon can e wrtten as v = 1 v 1,..., m v m. Corollary 1 No mass ponts n separatng equlrum. In a monotone strctly separatng equlrum, there s no dder, unt k, and nondegenerate nterval v, v such that k v,v s constant. 0

21 Taken together, the aove results mply that ether equlrum ds can e anlyzed ndependently, unt-y-unt, or equlrum exhts partal poolng. Followng the defnton of partal poolng, ths mples that when ds cannot e analyzed ndependently equlrum outcomes must e neffcent, and nformaton s not fully revealed. Helpfully these results allow us to analyze the revelaton queston dmenson-y-dmenson and, from these dmensonal analyses, to uld contradctons whch expose the relevance of partal poolng. 5.1 FRB aucton Lemma 6 Partal poolng n FRB. All well-ehaved equlra of the FRB aucton wth m unts exht partal poolng. Proof. The untwse utlty functon n the FRB aucton can e expressed as k, ; v k = vk k H k u k k 1 k 0 H m k+ x H x dx. Snce for any unt the agent has the opton of ddng 0 and otanng at worst zero utlty, u k k v, ; v k 0 whenever k s a est response ddng functon. As estalshed aove, f equlrum does not exht partal poolng, k v k v k. It follows that n an equlrum wthout partal poolng, v k k v k k 1 k v k 0 H m k+ H x H x dx k v k. Strct separaton, well-ehavedness, and est-responsveness requre that k v k > 0 whenever v k > 0, that ds are dense near 0, and that k 0 = 0.5 Then n the lmt, lm 0 0 H m k+ x H x dx H = 0. 6 When equlrum s well-ehaved and artrarly dfferentale, for a set of relevant t {0, 1,..., t} l Hôptal s rule mples d t H m k+ lm dt H 0 d t+1 H = In a workng verson of ths paper we provde arguments that these statements contnue to hold n the presence of a reserve prce r > 0. 6 Techncally only a weak nequalty, 0, s requred. Gven the relatonshp etween H m k+ and H t s straghtforward to show that strcty nequalty cannot e satsfed. 1

22 Quantty-monotoncty requres that j k v > j k +1 v, and hence y the nature of the order-statstc model there s some t such that lm d t H d m k+ > lm t H = At ths t, well-ehavedness requres that lm 0 d t+1 H 0 s fnte, hence the lmt s strctly postve, contradctng strct separaton. It s worth clarfyng the role of well-ehavedness n Lemma 6. If the necessary lmt n the proof does not exst, the Lemma s automatcally satsfed: the lmt wll fal to exst only when the rato can e dscretely postve for artrarly close to 0. Ths alone s suffcent to ndcate that poolng at 0 s advantageous, or ds are not dense. Thus well-ehavedness supports d densty, and provdes that d t+1 H 0 s fnte at the smallest t for whch t s nonzero. We do not have a clean economc nterpretaton for what t would mean for ths dervatve to e nfnte whle all lower dervatves are zero, ut nor can we rule t out out of hand. Remark. Wth n = dders and m unts, all equlra of the FRB aucton n weakly-domnant strateges exht partal poolng. Because truthful reportng for the frst unt s weakly-domnant, d t H1 0 s fnte for the lowest t at whch t s nonzero, mplyng that Lemma 6 can e appled drectly. Remark makes use of a further wrnkle n well-ehavedness. It s not necessary that all H e well-ehaved, only that there exsts an agent and a unt k such that H s well-ehaved. Ths allows for the followng Theorem. Theorem 1 Ineffcent equlrum n FRB. All equlra of the FRB aucton satsfy one or more of the followng two propertes:. Equlrum s neffcent.. For all agents and all unts k, d t H at whch t s nonzero. 5. PAB aucton 0 s nfnte at the lowest t That the PAB aucton exhts partal poolng follows from shft n the margnal dstrutons of values as quanttes demanded ecome larger. In the case of two dders and no reserve prce, a dder wll wn unt 1 f and only f her opponent does not wn unt m; smlarly the dder wll wn unt f and only f her opponent does not wn unt m 1. Snce her opponent s margnal dstruton of values for unt m 1 domnates the dstruton of values for unt m, the 7 Ths lmt makes clear the hdden role of the assumpton that m unts are avalale. When m = 1, H m k+ = 0, nvaldatng ths proof approach. Ths s to e expected, snce wth m = 1 unt avalale the FRB aucton s equvalent to a second-prce aucton, whch admts a well-ehaved, separatng, truthful equlrum.