ASYMMETRIC AUCTIONS* Eric Maskin and John Riley ** Revised November 6, 1998

Transcription

1 ASYMMETRIC AUCTIONS by Erc Makn and John Rley Forthcomng n the Ree of Economc Stude Reed Noember 6, 1998 The comment of Suan Athey, Km Border, Etelle Cantllon, Brt Grokopf, Robert Hanen, Paul Klemperer, Bernard LeBrun, Huagang L, Robert Marhall, Barry Nalebuff and a referee are gratefully acknoledged We thank the Natonal Scence Foundaton for reearch upport Th paper uperede UCLA Workng Paper #254 enttled "Aucton th Aymmetrc Belef" Department of Economc, Harard Unerty and UCLA

2 1 The reenue-equalence theorem 1 for aucton predct that expected eller reenue ndependent of the bddng rule, a long a equlbrum ha the properte that the buyer th the hghet reeraton prce n and any buyer th the loet poble reeraton prce ha zero expected urplu Thu, n partcular, the to mot common aucton nttuton - the open "Englh" aucton and the ealed hgh-bd aucton - are equalent depte ther rather dfferent trategc properte Th trong predcton of equalence eem at odd, hoeer, th the emprcal oberaton that rarely any gen knd of commodty old through more than one ort of aucton Thu, for example, art nearly alay auctoned off accordng to the Englh rule, herea job contract are normally aarded through ealed bd Admttedly, n the publc ector, there hae been a fe attempt to ue both method (lumber contract n the Pacfc Northet) or to tch from one to the other (Treaury Bll) But change hae typcally met great retance Th alo n conflct th theory, nce a corollary of the reenue equalence theorem that the expected urplu for any buyer the ame n the to aucton Thee dcrepance ugget that the hypothee of the reenue-equalence theorem may be too trong The prncpal aumpton are () rk neutralty, () ndependence of dfferent buyer' prate gnal about the tem' alue, () lack of colluon among buyer, and () ymmetry of buyer' belef Oer the lat ffteen year, a number of paper hae explored the mplcaton of relaxng thee aumpton Typcally a clear-cut predcton ha emerged n each of thee paper n faor of ether the hgh-bd or open aucton Thu under ncreangly general aumpton, Holt (1980), Rley and Samuelon (1981), and Makn and Rley (1984) ho that, hen buyer are rk aere, the hgh-bd aucton hould be faored by a eller een f he alo exhbt rk aeron 2 1 Th a ndependently dered by Myeron (1981) and Rley and Samuelon (1981) Tenty year earler Vckrey (1961) etablhed the reenue equalence of the to mot common aucton (open and ealed hgh-bd) under the aumpton that reeraton prce ere ndependent dra from a unform dtrbuton 2 Matthe (1987) extend th reult to ho that f all buyer exhbt contant abolute rk aeron, the aucton are equalent from ther perpecte The ealed hgh bd aucton thu Pareto-domnate the open aucton

3 2 In turn, the aumpton of ndependence of prate gnal of the tem alue relaxed by Mlgrom and Weber (1982) If reeraton prce are "afflated" (techncally, par-e potely correlated), they ho that the Englh aucton generate hgher expected reenue than the hgh-bd aucton Fnally Graham and Marhall (1989) and McAfee and McMllan (1992) allo for the poblty that buyer collude In partcular, Graham and Marhall argue that uch colluon facltated n an open aucton, here buyer can drectly npect one another' behaor Hence expected reenue ll tend to be hgher n hgh-bd aucton In all thee paper, buyer are ymmetrc ex ante n the ene that ther preference parameter (e, ther "type") are dran from a ymmetrc jont probablty dtrbuton Thu f to buyer are of the ame type they ll hae the ame belef about the remanng buyer' preference Gen th ymmetry, there ll ext a ymmetrc equlbrum (f an equlbrum ext at all 3 ) In uch an equlbrum, all buyer ue the ame equlbrum trategy a a functon of ther type (Mot author hae confned attenton to ymmetrc equlbra 4 ) In th paper e drop the ymmetry aumpton 5 Aymmetre are often mportant n contract bddng Each potental contractor ha eentally the ame nformaton about the nature of the project but a dfferent opportunty cot of completng t Wheneer ome apect of thee dfference common knoledge, belef are aymmetrc In major art aucton a ell, there are obou ex ante aymmetre aocated th dfferng budget contrant In the follong ecton e decrbe the bac model and then preent three llutrate example Thee ho that, th aymmetry, reenue equalence no longer 3 See Makn and Rley (1995) for condton under hch an equlbrum ext n the hgh-bd aucton 4 Makn and Rley (1996) prode condton under hch the hgh-bd aucton ha no aymmetrc equlbra 5 Although preou lterature on aymmetrc aucton not large, t doe date back many year Vckrey (1961) analyzed equlbrum under the extreme aumpton that one buyer' reeraton alue publc nformaton hle the other buyer' aluaton dran from a unform dtrbuton Gremer, Letan and Shubk (1967) extend Vckrey' analy to the cae of to unform dtrbuton

4 3 hold and that, under dfferent aumpton about the nature of the heterogenety, expected reenue n the hgh-bd aucton may be hgher or loer than n the open aucton 6 We lay tre on the nformal logc behnd thee reult In Secton 2 e begn the formal analy by characterzng buyer' equlbrum bddng tratege n the hgh-bd aucton (Propoton 22 and 24) We alo ho qute generally that "trong" buyer prefer the open aucton, herea "eak" buyer prefer the hgh-bd aucton (Propoton 25) Then, n ecton 3, e extend the example of ecton 1 to obtan general comparon of the reenue yelded by the hgh-bd and open aucton (Propoton 33-35) Fnally, n order to llutrate the economc gnfcance of our analytcal reult, e preent ome numercal oluton n ecton 4 1 THREE EXAMPLES Conder a to-buyer aucton for a ngle tem In th paper, e aume prate alue, e, that one buyer' nformaton doe not affect the other' preference and that buyer are rk neutral Let be buyer ' reeraton prce or "aluaton" Then e can expre buyer ' urplu f he n the tem and pay b a (11) u ( b, ) b We aume that buyer ' aluaton prate nformaton From the other buyer' perpecte, t a random arable ~ th cdf F ( ) We aume that the upport of an nteral [ β, α ] Heterogenety thu captured by the aumpton that the buyer' aluaton are dran from dfferent dtrbuton Throughout e ll decrbe one buyer a "trong" () and the other a "eak" () (Roughly peakng, thee term mean that F ( ) frt-order tochatcally domnate F ( ) ) Hence the ndex range n the et {,} Fnally e uppoe that the random arable for the to buyer' aluaton, denoted, ~ and ~, are ndependent 7 6 Graham and Marhall (1995), Bulo, Huang and Klemperer (1997) and Klemperer (1997) alo conder aymmetre Each paper pont to cae n hch expected reenue hgher n the hgh-bd aucton 7 A L and Rley (1998) ho, eentally equalent reult hold f the true aluaton a conex combnaton of the prate gnal Thu buyer ' payoff u ( b,, ) α + ( 1 α) b, j, j j

5 4 In the ealed hgh-bd aucton, buyer ubmt bd multaneouly The nner the hgh bdder (n our formal analy e ork th contnuou dtrbuton and o the probablty of a te zero), and he pay h bd Suppoe that the trong buyer' equlbrum bddng trategy to bd b b ( ) a a functon of h aluaton Under eak aumpton, one can ho that both b ( ) and b ( ) (the equlbrum bddng tratege of the trong and eak buyer) are trctly ncreang functon (ee Makn and Rley (1996)) Hence for each, b ( ) ole 1 (12) Max F ( b ( b))( b), b here b 1 ( b ) the nere of b ( ) Smlarly, b 1 (13) Max F ( b ( b))( b) b ( ) ole In the open (or "Englh") aucton, buyer call out ucceely hgher bd The lat buyer to bd the nner, and he pay h bd Under our aumpton, a buyer ll clearly be llng to top h opponent' current bd a long a that bd le than h on aluaton Hence, bddng ll proceed untl the loer of the to buyer' aluaton reached Thu, n equlbrum the nner ll be the hgh-aluaton buyer, but he ll pay a bd equal to the other buyer' aluaton The Englh aucton, therefore, equalent to a ealed-bd aucton n hch the hgh bdder n but pay only the econd-hghet bd (a econd-prce or "Vckrey" aucton), nce, a Vckrey (1961) hoed, t a domnant trategy n uch an aucton to bd one' aluaton Thu although e are ntereted n comparng the hgh-bd and open aucton, e ll analyze the latter a a econd-prce aucton We begn by examnng eeral leadng example of deaton from ymmetry Example 1: The trong buyer ' dtrbuton "hfted" to the rght Suppoe that the eak buyer aluaton dtrbuted unformly on the nteral [0,1] and that the trong buyer' aluaton dtrbuted on the nteral [2,3] That, the trong buyer' dtrbuton hfted to the rght here 0 < α 1 The crtcal aumpton that the gnal be ndependent

6 5 Let u frt conder the hgh-bd aucton Aume, for the moment, that the eak buyer bd h aluaton, e, he et b ( ) What the trong buyer' bet 1 repone? If he bd b [0,1], he n th probablty F ( b ( b)) b Hence, her maxmzaton problem Maxb( b) But for 2, the oluton b [ 0, 1] b 1 That, the trong buyer' bet-repone b ( ) 1 for n the nteral [2,3] Notce, moreoer, that f he behae n th ay, t ndeed optmal for the eak buyer to et b ( ), nce he cannot proftably n the aucton anyay We hae therefore exhbted an equlbrum for the hgh-bd aucton Moreoer, a hon n Makn-Rley (1995), th equlbrum (eentally) unque The alent feature that the trong buyer tand to gan o much from nnng that t pay her to be ure that he out-bd the eak buyer Clearly, the expected reenue from the aucton 1 No let u turn to the open aucton Notce that (a n the hgh-bd aucton) the trong buyer alay the nner, nce he alay ha the hgher aluaton Hoeer, becaue he pay only the econd-hghet aluaton, her expected payment E{ ~ } 1 2 We hae thu exhbted an example n hch an aymmetry beteen the buyer faor the hgh-bd oer the open aucton More generally, heneer the trong buyer' dtrbuton uch that, th hgh probablty, her aluaton a great deal hgher than that of the eak buyer, the hgh-bd aucton ll tend to generate more reenue; to guarantee nnng, the trong buyer ll be nclned to enter a bd equal to the maxmum aluaton n the eak buyer' upport, herea, under the open aucton, he ll pay only the expected alue of the eak buyer' aluaton Th prncple mght termed the "Getty effect," after the ealthy art mueum knon for out-bddng t competton And clearly, a the art orld llutrate, t an mportant prncple n practce But a our generalzaton of Example 1 (Propoton 33) demontrate, the hgh-bd aucton alo emerge a uperor een for horzontal hft of the trong buyer' dtrbuton that are not o extreme a to nduce her to preempt her opponent

7 6 Example 2: The trong buyer 2' dtrbuton "tretched" For our econd example, uppoe that the eak buyer' aluaton dtrbuted 1 unformly on [ 0, ] and the trong buyer' aluaton dtrbuted unformly on 1+ z 1 [ 0, ], for z > 0 That, the trong buyer ha the ame dtrbuton a the eak 1 z buyer, only "tretched out" oer a der nteral that When z 0 (e, the dtrbuton are both unform on [0,1]), t eay to erfy 1 (14) b ( b) 2b buyer ' equlbrum nere bd functon n the hgh-bd aucton 8 Becaue a buyer th aluaton 2b n th probablty Fj ( 2b) 2b, j, h expected payment Fj ( 2b) b 2b 2 In the open aucton, a 2b-buyer alo n th probablty 2b, and h payment (condtonal upon nnng) the mean of the other buyer' aluaton, condtonal upon t beng le than 2b Becaue of our unformty aumpton, th condtonal mean jut b Thu the uncondtonal expected payment 2b 2, the ame a n the hgh-bd aucton (th an llutraton of the reenue equalence theorem) No let u ee hat happen a z become pote In the open aucton, the eak buyer th aluaton 2b n th probablty 2b( 1 z), and h uncondtonal 2 expected payment 2b ( 1 z) In the hgh-bd aucton, f the trong buyer ere to contnue to ue the trategy gen by (14), the eak buyer' bet repone ould alo be gen by (14) Thu the eak buyer' expected payment ould be Fj ( 2b) b ( 1 z) 2b 2, exactly a n the open aucton Smlarly, the payment by the trong buyer ould be ( 1+ z) 2b 2 n ether aucton Thu reenue ould be the ame n the to aucton 8 When z 0 and one buyer bd accordng to (14), the probablty that the other buyer, th aluaton, n th a bd of b 2b Then the trong buyer ha an expected proft of 2b( b) Th take on t maxmum at b 1 2

8 7 a 1 1 In the hgh-bd aucton, hoeer, buyer do not contnue to ue (14) If they dd, aluaton trong buyer ould outbd a z 1 1+ aluaton eak buyer by z 1 1 to, and o can reduce her bd to the latter hle tll nnng th 2( 1 z) 2( 1+ z) probablty 1 Thu, for equlbrum, the trong buyer mut reduce her bd a a functon of her aluaton, relate to (14) But uch a reducton ll nduce the eak buyer to bd more aggreely than he ould f the trong buyer ued (14) Th becaue the trong buyer' bd are no dtrbuted more denely than before Hence the margnal beneft to the eak buyer of bddng lghtly hgher re: a mall ncreae n h bd lead to a greater ncreae n h probablty of nnng than under (14) In equlbrum, 9 the eak and trong bdder' nere bd functon are b 1 2b ( b) 1 + z( 2b) 2 and b 1 2b ( b) 2 1 z( 2b) What effect doe th change n bddng tratege hae on reenue? For the hghbd aucton, the cdf G H ( b) for the nnng bd atfe 1 1 H G ( b) Pr( b ( ~ ) b) Pr( b ( ~ ) b) F ( b ( b)) F ( b ( b)) 1 1 ( 1 z )( 2b) ( 1 z) b ( b)( 1+ z) b ( b) z ( 2b) 2 2 It readly confrmed that th a decreang functon of z Thu expected reenue n the hgh-bd aucton re th z For the open aucton the econd aluaton le than b f and only f t not the cae that both aluaton are hgher Thu G ( b) 1 ( 1 F ( b))( 1 F ( b)) F + F F F O ( 1 z) b + ( 1+ z) b ( 1 z ) b 2b ( 1 z) b 9 It readly confrmed that thee nere bd functon atfy the equlbrum dfferental equaton and endpont condton that e ge belo (ee (212) and (213))

9 8 The cdf for the open aucton therefore ncreang n z Snce the to dtrbuton yeld the ame expected reenue n the ymmetrc cae (z0), expected reenue trctly greater for the hgh-bd aucton than for the open aucton hen z>0 We hae been dcung the cae of the unform dtrbuton But, a Propoton 34 belo make clear, the ame concluon apple to a large cla of other dtrbuton Let u turn to our lat example Example 3: Probablty reallocated to the loer end pont of buyer 1' dtrbuton The dea of the example eaet to preent for to-pont dtrbuton - one n hch all probablty ma confned to the pont 0 and 2 Suppoe frt that both buyer hae degenerate dtrbuton n hch all probablty concentrated on 2 Suppoe that e no hft half the probablty ma for the eak buyer to the pont 0 Thu the probablte that the eak buyer ha aluaton 0 or aluaton 2 are 1 2 each In the open aucton, expected reenue jut the probablty that the eak buyer' aluaton hgh (e, 1 2 ) tme the payment made n that cae (e, 2) Thu expected reenue 1 In the hgh-bd aucton, the trong buyer can n th probablty (at leat) 1 2 f he bd (jut aboe) zero Her expected payoff from dong o (at leat) 1 Th mple that he ll neer bd more than 1 n equlbrum (nce her payoff ould then be trctly le than 1) Hence the eak buyer can n for certan by bddng 1+ ε, and o h ex ante expected payoff (e, h payoff before h type realzed) exceed 1 2 ( 2 ( 1+ ε)) ( 1 ε ) for all ε > That, h expected payoff at leat 1 2 We conclude that the um of the equlbrum expected payoff to the to buyer at leat Becaue the nner alay ha a aluaton of 2, the ocal urplu from the aucton 2 It follo that expected reenue - the dfference beteen urplu and buyer' expected payoff - at mot 1 2 Hence, the open aucton uperor The mportant feature n th example that n the hgh-bd aucton the trong buyer doe not get a pote payoff from bddng o hgh that he aured of nnng

10 9 (gen that the eak buyer ha aluaton 2 th pote probablty, the trong buyer ould hae to bd 2 to be aured of nnng) Hoeer, he doe obtan a pote expected payoff from bddng ery lo It th ncente to "lo ball" that ork agant the hgh-bd aucton here Our general reult along thee lne can be found n Propoton 35 2 EQUILIBRIUM BIDDING WITH ASYMMETRIC BELIEFS We no turn to the detaled analy For upport [ β, α ], 0 β < α, th cdf F {, }, buyer ' aluaton ~ ha ( ) on [ β, α ] that tce contnuouly dfferentable on ( β, α ] We aume alo that the denty F ( ) trctly pote on [ β, α ] Remark 1: We allo for the poblty of a ma pont at the loer end of the dtrbuton (e F ( β ) > 0 ) One ay that th could come about (een f ntrncally buyer' aluaton are dtrbuted contnuouly, f the eller et a reere prce β that hgher than the mnmum aluaton of at leat one buyer In equlbrum, anyone th a aluaton greater than the eller' reere bd trctly greater than β Thu by bddng β, the trong buyer n th probablty F ( β ) and the eak buyer n th probablty F ( β ) We hall aume that the dtrbuton of the trong buyer' aluaton frt order tochatcally domnate that of the eak buyer: (21) F > F, for all ( β, α ) Notce that (21) mple that (22) β β and α α Actually, e ll requre a condton omehat tronger that (21), z, condtonal tochatc domnance Specfcally, uppoe that, for all x < y n ( β, α ) (23) Pr{ ~ ~ F ( x) F ( x) x y} Pr{ ~ ~ < < < < x < y} F ( y) F ( y) Rearrangng (23) e obtan:

11 10 F ( x) F ( y) < for all x < y n ( β, α ) F ( x) F ( y) Thu for (23) to hold, F ( ) F mut be trctly ncreang on the nteral [ β, α ] The defnton of Condtonal Stochatc Domnance that e hall ue a lghtly eaker condton than th Condtonal Stochatc Domnance (CSD) Suppoe that (22) hold There ext λ ( 0, 1 ) and γ [ β, α ] (th γ β f β > β ) uch that, and () F λ F for all [ β, γ ] () d d F F > 0 10 for all [ γ, α ], Note that CSD mple that (24) > F F, for all ( γ, α ] In addton, e hae the follong reult Lemma 21: CSD mple frt order tochatc domnance, that, (21) In addton, t mple that ether F ( β ) > F ( β ) or F ( β ) F ( β ) 0 Proof: If β < β, then for ( β, β ), F > 0 F and for [ β, α ), CSD () mple (25) If β F F ( α ) < 1 F F ( α ) β, then, for ( β, γ ) 10 If γ > β, th hould be nterpreted a the rght derate of F F

12 11 F > λf F and for ( γ, α ), () mple that (25) hold Hence CSD mple (21) But f (26) β β β and F F > 0, then (24) mple that F > F, for all near β, a contradcton of (21) We conclude that f (26) hold, and F F, then (27) F F 0 QED Before preentng the man reult, e frt conder hat nght can be ganed from a mechanm-degn perpecte From the reenue-equalence theorem, reenue can be computed mmedately for any mechanm, once the probablty of nnng for each buyer-type acertaned Wth to bdder hang aluaton and, let π (, ), {, }, be the equlbrum probablty that buyer n n a gen mechanm We conder the cae n hch the mnmum aluaton are zero ( β β 0 ) Then a Myeron (1981) ho, expected reenue can be expreed a z z 0 0 α α (28) R [ J ( ) π (, ) + J ( ) π (, )] df df here (29) J 1 F F We are ntereted n undertandng the dfference n expected reenue generated by the hgh-bd and open aucton Thee aucton belong to the cla of ellng mechanm for hch the tem alay old n equlbrum That, for all poble par of aluaton (, ), π (, ) + π (, ) 1 Thu (28) can be rertten a z z z z α α α α (210) R [ J ( ) J ( )] π (, ) df df + J ( ) df df

13 12 A Bulo and Robert (1989) pont out, J ( ) the expected margnal reenue generated f the tem agned to buyer of type Thu, under the aumpton that J and J are trctly ncreang, and gen the contrant that the tem mut be old, expected total reenue maxmzed by ellng to the trong buyer (ettng π 1) f and only f J exceed J Suppoe, to begn th, that, for each aluaton, the eak buyer' margnal reenue no greater than that of the trong buyer ( J J ) Fgure 1 depct the mplct mappng T J 1 ( ) ( J ( )) That, the par of aluaton (, ) here J ( ) J ( ) If J J, th cure le on or belo the 45 lne In the open aucton the hgh-aluaton buyer the nner Thu to maxmze reenue, α 45 lne Q( ) J J T( ) O α Fgure 1: Open aucton uperor

14 the playng feld need to be tlted n faor of the trong bdder In th paper, hoeer, e are concerned th comparng the reenue from the to common aucton hch both employ ymmetrc rule (a leel playng feld) In the open aucton, the hgh-aluaton bdder n Thu, from the reenue perpecte, the tem ncorrectly agned n the open aucton heneer the aluaton le n the healy haded regon bounded by the 45 lne and the cure T( ) A e hall ho, t typcally the cae n the hgh-bd aucton that the trong buyer hade h bd more than the eak buyer, that (211) b < b, ( 0, α ) Then, for any aluaton of the eak buyer, he n f < b 1 ( b ( ) Q( ) Gen (211) the mplct mappng Q( ) le aboe the 45 lne Thu mallocaton occur n the hgh-bd aucton heneer the aluaton le ether n the healy or lghtly haded regon It follo mmedately that, under thee condton, expected reenue hgher n the open aucton The aumpton that J J, hoeer, a trong one Suppoe, for example, that the eak buyer' aluaton dtrbuted accordng to F, [ 0, 1 ], hle F a, [ a, 1 + a] That, the trong bdder' dtrbuton hfted to the rght by a It readly confrmed that margnal reenue repectely J 2 1, and J 2 1 a Thu the eak bdder ha a hgher margnal reenue 11 In the pecal cae F ων+ ( 1 ω ) F, [ 0, α], 1 F ( 1 ω )( 1 F ) and o J J Thu the crtcal cure T( ) the 45 lne In th cae the open aucton optmal from the reenue perpecte 12 One ay to do o to ue a modfed Vckrey aucton n hch the eak buyer gen an "effecte bd" of T( ) f he ubmt a bd of See alo McAfee and McMllan (1989)

15 14 Fgure 2 depct th cae In the open aucton the mallocaton agan the healy haded regon For the hgh-bd aucton the mallocaton occur hen the aluaton le beteen the 45 lne and the cure Q Note that for hgh aluaton the α Q( ) T( ) 45 lne J J O α Fgure 2: Shft n the mean hgh bd aucton allocate the tem to the eak buyer too often hle the reere true for lo aluaton It follo that, from geometry alone, e cannot mmedately rank the to aucton (although, n fact the hgh-bd aucton turn out to be better, a e hall ee n the next ecton) Thu n the cae n hch J J need not hold, mechanm degn conderaton do not ettle the matter of hch aucton generate more reenue We no characterze equlbrum n the hgh-bd aucton Rather than attemptng to ole drectly for the equlbrum bd functon, t conenent to ork th nere bd functon From Makn and Rley (1995 and 1996), e kno that, proded that

16 15 β not too much bgger than α 13, there are unque mnmum and maxmum nnng bd b and b for hch there ext a oluton to the follong par of dfferental equaton (212) R S T ( φ ) ( b) F ( φ ) φ 1 φ b ( φ ) ( b) F ( φ ) φ 1 φ b atfyng the boundary condton (all functon φ and φ are ealuated at b for all b [ b, b ] ) F ( φ ( b )) 1, {, }, (213) β β b φ ( β ) φ ( β ) β β < β b Max arg Max{( β b) F ( b)}, φ ( b ) b b Moreoer, th oluton unque and conttute the (unque) equlbrum par of nere bd functon That, (214) φ ( b) b ( b), {, }, 1 here b ( ) buyer ' equlbrum bd a a functon of h aluaton To ee that (214) hold, note that f the trong buyer bd accordng to φ and the eak buyer ubmt a bd of b, then he n f and only f < φ ( b) (It can be hon that the oluton functon to (212) and (213) are trctly ncreang and tce dfferentable) The eak buyer' expected urplu therefore (215) ( b) Pr{ < φ ( b)} F ( φ ( b))( b) Takng logarthm and then dfferentatng by b, e obtan the frt-order condton at ( φ ) F ( φ ) φ 1, th boundary condton y ( β ) β b φ ( b), hch the econd equaton n (212) 13 A e hae een, n Example 1, the trong bdder ll bd α regardle of h aluaton, f β uffcently bgger than α

17 16 We characterze the equlbrum bd functon by comparng bddng n the aymmetrc aucton th that hen buyer are ymmetrc, e ether both trong or both eak Let y ( b) be the ymmetrc equlbrum nere bd functon From (212) (216) Rearrangng e obtan F ( y ) y F y y b 1, {, }, th boundary condton y ( β ) β ( ) bf y dy F y y F y dy ( ) + ( ) ( ) db db Let b be the correpondng equlbrum bd functon (e, the nere of y ( b ) ) Integratng the lat equaton, e hae z β (217) b F yf ( y) dy It follo mmedately that n the equlbrum here both buyer are of type ( or ), a buyer' maxmum poble bd equal to the mean aluaton z α b ( α ) yf ( y) dy E{ ~ } µ β By Lemma 21 the CSD aumpton mple (21), n hch cae (218) µ < µ Note that e can rerte (217) a F ( x) (219) b dx zβ F From (219) and CSD, e hae, (220) b b for all ( β, α ) Our frt general reult concern the buyer' equlbrum bd dtrbuton Defne (221) p ( b) F ( φ ( b)), {, } Alo defne (222) H ( ) F 1 ( ), {, } Gen tochatc domnance, H ( p) > H ( p) for all p ( 0, 1 ) Subttutng (221) and (222) nto (212) e obtan

18 (223) R S T p 1 p H ( p ) b p 1 p H ( p ) b 17 Smlarly, for the ymmetrc equlbra, defne (224) π ( b) F ( y ( b)), {, } Then from (216) (225) R S T π π 1 H ( π ) b π 1 π H ( π ) b The reult that follo are proed n the appendx Propoton 23 part () tell u that n the hgh-bd aucton the equlbrum bd dtrbuton of the trong buyer tochatcally domnate that of the eak buyer Part () ndcate that f a eak buyer face a trong buyer rather than another eak buyer, he repond th a more aggree bd dtrbuton (n the ene of tochatc domnance) And ymmetrcally, Propoton 23 part () etablhe that f a trong buyer face a eak buyer rather than another trong buyer, he ll repond th a le aggree bd dtrbuton A for Propoton 25, part () ndcate that, n the aymmetrc equlbrum, the trong bdder hade h bd further belo h aluaton than the eak bdder Part () tell u that f a eak bdder face a trong bdder rather than a eak bdder he ll bd more aggreely (cloer to h aluaton 14 ) Argung ymmetrcally, part () ndcate 14 It may eem a though frt-order tochatc domnance ould be enough to mply th reult Conder, hoeer, the example n hch F ( ) 3 2 and F ( ) It follo from equaton (A4)-(A7) n the appendx that φ ( b) > y ( b) for b uffcently mall

19 18 that f a trong bdder face a eak bdder rather than a trong bdder he ll bd le aggreely A e hall ee n Propoton 26, thee reult allo u to rank the hgh-bd and open aucton from perpecte of each buyer Lemma 22: If (21) hold, β β β, F F 0 and there ext δ > β uch that for all b [ β, δ ] () π ( b) > π ( b) () p ( b) > p ( b) () π ( b) > p ( b) () p ( b) > π ( b) d d F > F β 0, then Propoton 23: Comparon of equlbrum bd dtrbuton If CSD hold, 15 then () π ( b) > π ( b), for all b ( β, µ ) () p( b) > p( b), for all b ( b, b ) () π ( b) > p ( b), for all b ( b, b ) () p( b) > π ( b), for all b ( b, µ ) Corollary 24: Gen CSD, e hae µ b µ th at leat one trct nequalty Proof: If b < µ, then p ( b ) > π ( b ) and o p ( b) > π ( b) for b near b, a contradcton of part () of Propoton 23 A mlar contradcton follo from µ b The reult then follo from (214) QED 15 Actually, Propoton 23 goe through proded that (21) hold and that, f β β β, the hypothee of Lemma 22 hold

20 19 Propoton 25: Characterzaton of equlbrum nere bd functon If CSD hold, then () y( b) y( b), for all b ( β, µ ) and () φ ( b) > φ ( b), for all b ( b, b ) () y ( b) φ ( b), for all b ( b, b ) and () φ ( b) > y ( b), for all b ( β, b ) Ung Propoton 25 e can dere the follong comparate reult Propoton 26: Rankng of the to aucton by the buyer If CSD hold, the trong buyer trctly prefer the open aucton hle the eak buyer prefer the hgh-bd aucton (here the preference trct for all aluaton exceedng the mnmum bd b n the hgh-bd aucton 16 H Proof: For {, }, let U (, F, F ) be buyer ' expected equlbrum urplu from the hgh-bd aucton hen h reeraton prce and the to buyer' reeraton prce are dtrbuted accordng to F and F repectely Smlarly, let U ' expected urplu from the open aucton From part () of Propoton 25 O (, F, F ) be buyer F ( φ ( b)) p ( b) > π ( b) F ( y ( b)) for all b ( b, b ) Thu for all ( b, α ] H U (, F, F ) Max p ( b)( b) b p ( b )( b ) > ( b )( b ) π H U (, F, F ) O U (, F, F ) from ymmetry and reenue equalence ( x) df ( x) zβ O U (, F, F ) 16 Under the eaker aumpton of frt order tochatc domnance, t can be hon that the rankng by buyer contnue to hold for all thoe buyer th uffcently hgh aluaton

21 20 O A for [ β, b ], U (, F, F ) 0 becaue < β and o the eak buyer eakly prefer the hgh-bd aucton From part () of Propoton 25 π ( b) > p ( b) for all b ( β, b ) Hence for all ( β, α ], O O U (, F, F ) U (, F, F ) H U (, F, F ) by the Reenue Equalence Theorem Maxπ ( b)( b) b π ( b )( b ) > p ( b )( b ) H U (, F, F ), here b ( ) the trong buyer' equlbrum bd functon n the hgh-bd aucton hen the dtrbuton are ( F, F ) QED Th lat reult eem to hae been undertood by buyer ho perceed themele to be "trong" before the recent pectrum aucton held by the FCC There a a clear preference for ome form of open aucton rather than a ealed hgh-bd aucton Smlarly, n the lumber tract aucton n the Pacfc Northet, the local "nder" th neghborng tract hae forcefully (and uccefully) lobbed for open aucton and the elmnaton of ealed hgh-bd aucton Our reult alo prode ome nght nto the logc behnd Propoton 33 belo If the trong buyer a much tronger bdder, then the eak buyer n th only a mall probablty, o that h expected payoff mall n ether aucton It follo that the dfference n expected payoff from the to aucton mall for the eak buyer Total urplu loer n the hgh-bd aucton nce the hgh-aluaton buyer n th probablty le than 1 But agan, f the trong buyer almot alay n, th lo n

22 21 urplu mall Then, roughly peakng, the loer expected payoff for the trong buyer n the hgh-bd aucton offet by an ncreae n payoff to the thrd party -- the eller That, expected reenue hgher n the hgh-bd aucton 3 REVENUE COMPARISONS -- GENERAL RESULTS We no dere our general reenue comparon Throughout th ecton e hall noke Propoton 25 () Hence, the functon Q(), mplctly defned by the equaton (51) φ ( b) Q( φ ( b)), a mappng from [ b, α ] onto [ β, α ] th Q > for all ( b, α ) Let u adopt the conenton that Q for all [ β, b ) For each aluaton, the trong buyer bd loer than the eak buyer (e,φ ( b) > φ ( b), b ( b, b ) ) n equlbrum Hoeer, from Propoton 23 (), the dtrbuton of h bd frt-order tochatcally domnate that of h opponent, that, for all b ( b, b ), Pr{ b ~ b} p ( b) F ( φ ( b)) < F ( φ ( b)) p ( b) Pr{ b ~ b} Thu, from the defnton of Q, t follo that (52) F > F ( Q), for all ( b, α ) We hae the follong general expreon for expected eller reenue n the to aucton Proof of thee and later propoton can be found n the Appendx Lemma 31: Expected eller reenue from bdder (, ) n the ealed hgh bd aucton R H, here α H d (53) R F Q d F d b F b ( 1 ( ( ))) ( ( 1 ( ))) + ( 1 ( )) and bz H (54) R ( 1 F ( Q)) Q d + b F ( b )( 1 F ( β )) α bz

23 22 Lemma 32: Expected eller reenue from buyer n the open aucton can be expreed a R O, here α O d (55) R F d F d F ( 1 ( )) ( ( 1 ( )) + β ( 1 ( β )) and zβ O (56) R ( 1 F ) df + β F ( β )( 1 F ( β )) α zβ We no turn to the reenue comparon If β β β, the mnmum bd, b β Then, from the aboe Lemma, the dfference n expected reenue from the to aucton H H O D R + R R R O z z α ( F ( Q))[( F ) ( Q ) F ] d ( F )( F ) d b b Rearrangng e obtan z z α (57) D [ 1 F ( Q)( Q ) F ( 1 F )( F ( Q) F )] d here b ( Q )( 1 F ( Q))( 1 F ) C(, Q) d, zb α F ( Q) F (58) C(, Q) 1 F ( 1 F ( Q))( Q ) If β < β, then, from (28), α b α R O α d < ( 1 F d F d b F b ( )) ( ( 1 ( ))) + ( 1 ( )), bz proded that ( ) + 1 F ( ) 0 for all ( b, β ) Thu e hae an upper bound for the expected reenue from the eak buyer n the open aucton And, therefore, a loer bound for the dfference beteen the hgh-bd and open aucton' reenue zα b b zβ ( Q )( 1 F ( Q))( 1 F ) C(, Q) d ( 1 F ) df + b F ( b )( 1 F ( β )) β F ( β )( 1 F ( β )),

24 23 hch no le than the rght-hand de of (14) The follong reenue rankng are obtaned by fndng condton uffcent to gn the functon C( Q, ) Propoton 33: Hgh-bd aucton uperor for dtrbuton hft Suppoe that (59) (a) d ( ) 0, and (b) d Gen a < (510) F < 0 on [ β, α ] F α β, uppoe that, for all [ β, α + a] R S, < a + T 0 β F ( a), a + β and that ( ) + F ( ) 0 for all [ β, β + a] Then the hgh-bd aucton generate hgher expected reenue than the open aucton Notce that F n Propoton 33 jut a hft to the rght (by a) of the dtrbuton F Thu the Propoton a generalzaton of Example 1 n ecton 1 (In that example a2) We next turn to a generalzaton of Example 2 Imagne "tretchng out" dtrbuton F by multplyng t by a calar λ < 1 Snce λ F ( α ) < 1, e hae to ay hat happen for alue > α, n order to obtan a ne cdf F Let G( ) be the "extenon" of F to th range of alue (In Example 2, G ) We hae 1+ x Propoton 34: Hgh-bd aucton uperor for dtrbuton "tretche" Suppoe that F ( ) atfe F ( β ) 0 and (511) d d < F 0 on [ β, α ]

25 24 For λ ( 0, 1 ), let the trong buyer hae dtrbuton F ( ), here [ β, α ] ( α < α ), uch that (512) F here G( α ) λ, G( α ) 1, and R S T λf, [ β, α ] G, ( α, α ] (313) G ( ) > 0, for all [ β, α ] and [ α, α ] Then the hgh-bd aucton generate more expected reenue than the open aucton Recall that Example 3 a obtaned by takng a one-pont dtrbuton and hftng probablty ma to the zero pont We conclude the ecton by generalzng th example o that, at each pont of a dtrbuton F, a fracton 1 θ( ) of the denty hfted to the loer end- pont of the dtrbuton, Propoton 35: Open aucton uperor for hft of probablty ma to the loer end pont Suppoe that the trong buyer' aluaton ~ dtrbuted accordng to F, [ β, α ] here F ( β ) 0 and (314) ncreang 1 F Buyer 1' aluaton ~ dtrbuted o that, for all [ β, α ], t denty at a fracton θ ( 0, 1 ) (th θ 0 ) of here the remanng denty reagned to β That, (315) F ( t) df ( t) + θ zβ γ here z α (316) γ ( 1 θ( t )) df ( t ) β Then, the open aucton generate hgher reenue than the hgh-bd aucton

26 25 Fnally, e note that, n combnaton, our three reenue comparon, Propoton embrace all frt-order deaton from ymmetry Specfcally, gen F ( ), uch deaton can be obtaned by takng: F λf ( ω + α) + δ and conderng λ 0, ω 1, α 0, and δ 0 Decreae n α from 0 correpond to Propoton 33; araton n λ(or mlarlyω ), here δ 0, to Propoton 34; and decreae n λ, here δ ncreae correpondngly, to Propoton 35 Hence, by combnng the effect decrbed by the three propoton, one can examne any frtorder aymmetry 4 Numercal Reult Becaue the reult n ecton 3 are purely qualtate, t eem orthhle to complement them th ome example 17 to ee ho large the quanttate effect can be The unquene reult of Makn and Rley (1995) prode a natural ay to ole numercally for the unque equlbrum Frt e conder the cae of unform dtrbuton and conder the effect of a mple hft of one dtrbuton a n Propoton 33 TABLE 41: Percentage gan n reenue under the hgh-bd aucton oer the open aucton under mple hft n the unform dtrbuton F (, a) a, [ a, 1+ a ] cdf for eak buyer a 0 a 0 a 0 a 0 17 See Rley and L (1996) for a much more complete numercal analy, and alo Marhall et al (1994) The program "BIDCOMP²" hch compute nere equlbrum bd functon and compare expected reenue aalable for ue by any ntereted reader The FORTRAN ource fle are alo aalable to reearcher hng to comple modfed eron of the program

27 26 cdf for trong buyer a 0 a 1 4 a 1 2 a 3 4 Reenue n hgh-bd aucton Reenue n open aucton Percentage dfference We next conder the effect of a dtrbutonal tretch, a n Propoton 34 Suppoe that F unform on [0,1] and that F unform on [ 0, α ] TABLE 42: Percentage ncreae n expected reenue under the hgh-bd aucton F / α, [ 0, α ] cdf for eak buyer α 1 α 1 α 1 α 1 cdf for trong buyer α 1 α 2 α 3 α 4 Reenue n hgh-bd aucton Reenue n open aucton Percentage dfference Fnally e turn to Propoton 36 Suppoe that (41) F + η, [ 0, 1] 1+ η Then, th probablty η 1+ η 1 buyer ha a aluaton of zero and th probablty 1+η h aluaton a dra from a unform dtrbuton th upport [0,1] Table 43 ummarze dfference n expected reenue for dfferent parameter alue Note that the hgh-bd aucton doe ore a the eak buyer' probablty of not bddng ncreae Th becaue the trong buyer' ncente to "loball" alo re TABLE 43: Percentage ncreae n expected reenue under the hgh-bd aucton F + η, [ 0, 1] 1+ η

28 27 cdf for eak buyer η 0 η 0 η 0 η 0 cdf for trong buyer η 0 η 1 η 2 η 3 Reenue n hgh-bd aucton Reenue n open aucton Percentage dfference Concludng Remark We noted n the ntroducton that art aucton are nearly alay conducted openly herea job-contract bddng normally ealed It temptng to try to explan thee regularte ung our reult on aymmetre (Of coure focung on other olaton of the reenue equalence theorem' hypothee mght ge re to alternate explanaton) One of the ell-knon pecularte about people' tate for art that thee are doyncratc Idoyncratc tate mean that the market for any gen tem may be extremely thn Suppoe, for example, that a gen buyer happen to be enthuatc about a partcular pantng He mght reaonably conjecture that he alone n h enthuam But, f o, lo-ballng n a ealed hgh-bd aucton become a good trategy A e hae een, an open format help afeguard the eller n uch a tuaton, e, Propoton 35 apple If e take the cae of defene contractng, by contrat, e fnd at leat to bdder n erou competton on almot eery occaon (ee Alexander (1992)) Thu loballng tend not to be a able trategy Rather, Propoton 33 and 34 are the releant fndng We therefore expect the ealed hgh-bd aucton to ork better than open aucton for defene procurement Indeed, the edence appear to bear th out The US defene Department ha almot alay ued ether "prototype" or "paper" competton for aardng contract In the prototype mode, a compettor mut produce an actual orkng arplane, or hateer, a t "bd" Snce uch prototype are normally contructed n ecret and are dffcult to modfy ex pot, th competton reemble a hgh-bd aucton By contrat, n a paper aucton, a ould-be contractor need only

29 28 produce the blueprnt for the arplane Thee are comparately eay to modfy after the contractor learn hat t compettor hae done, and o the competton more cloely reemble an Englh aucton Alexander ha found that the Defene Department ha fared conderably better th the prototype than th paper competton

30 29 Reference Alexander, B, (1992), "Prme Contract Competton Regme and Subcontractng Among Mltary Arcraft Manufacturer," mmeo, Brande Unerty Bulo, Jeremy, Huang Mng and Paul Klemperer, (1997) "Toe-hold and Takeoer," forthcomng n Journal of Poltcal Economy Bulo, Jeremy and John D Robert "The Smple Economc of Optmal Aucton" (1989) Journal of Poltcal Economy, 97, Graham, Danel A and Robert C Marhall (1987), "Collue Bdder Behaor at a Sngle Object Second Prce and Englh Aucton," Journal of Poltcal Economy, 95, Gremer, JH, RE Letan and M Shubk, (1967) "Toard a Study of Bddng Procee, Part Four: Game Wth Unknon Cot," Naal Reearch Logtc Quarterly, 14, Holt, C, (1980) "Compette Bddng for Contract Under Alternate Aucton Procedure," Journal of Poltcal Economy, 88, Klemperer, Paul (1997), "Aucton th almot Common Value," forthcomng n European Economc Ree L, Huagang and John G Rley (1998), "Aucton th Related Value" mmeo, UCLA Marhall, RC, MJ Meurer, J-F Rchard and W Stromqut (1994), "Numercal Analy of Aymmetrc Frt Prce Aucton" Game and Economc Behaor, 7, Makn, ES, and JG Rley, (1984) "Optmal Aucton Wth Rk Aere Buyer," Econometrca, 52, and, (1995) "Extence of Equlbrum n Sealed Hgh Bd Aucton," UCLA Dcuon Paper #407, reed and, (1996) "Unquene of Equlbrum n Sealed Hgh Bd Aucton," UCLA Dcuon Paper, reed

31 30 Matthe, S, (1983) "Sellng to Rk Aere Buyer Wth Unoberable Tate," Journal of Economc Theory, 30, McAfee P and J McMllan (1989) "Goernment Procurement and Internatonal Trade" Journal of Internatonal Economc, 26, (1992) "Bddng Rng," Amercan Economc Ree, 82, Mlgrom, PR, and RJ Weber, (1982) "A Theory of Aucton and Compette Bddng," Econometrca, 50, Myeron, RB, (1981) "Optmal Aucton Degn," Mathematc of Operaton Reearch Rley, JG, and WF Samuelon, (1981) "Optmal Aucton," Amercan Economc Ree, 71, June Rley, JG, (1989) "Expected Reenue from Open and Sealed Bd Aucton," Journal of Economc Perpecte, 3, and H L, (1994) BIDCOMP² -- A Freeare Program to COMPute equlbrum bd and COMPare expected reenue, (1996) Aucton Choce: A Numercal Analy Mmeo Vckrey, W, (1961) "Counterpeculaton, Aucton, and Compette Sealed Tender," Journal of Fnance, 16, 8-37

32 31 APPENDIX Suppoe that β β β Defne ( β) F e, {, } F We can therefore rerte (212) a follo: (F1) e( φ ) φ φ β ( b) φ b e( φ ) φ φ β ( b) φ b Applyng l'hôptal' Rule e nfer from the defnton of e ( ) that (F2) F 0 R S T e 1 e 2F Dfferentatng F / F, e obtan d d F F F F F F, for F 2 > 0 and F > 0 F Agan applyng l'hôptal' Rule, e obtan d F (F3) F F 0 d F β 1 F F [ ] 2 If d d F > F β 0, the bracketed expreon n (14) trctly pote Lemma 22: If (21) hold, β β β, F F 0 and there ext δ > β uch that for all b [ β, δ ] () π ( b) > π ( b) () p ( b) > p ( b) () π ( b) > p ( b) () p( b) > π ( b) d d F > F β 0, then

33 32 Proof: Snce F F 0, p π 0, {, } Applyng l'hôptal' Rule to (23), e obtan φ ( ) φ β φ 1 e It follo from (22) that φ ( ) and e φ β φ 1 (F4) φ ( β ) 2, {, }, and, n ymmetrc equlbrum, (F5) y 2, {, } Next, takng the logarthm of (23) and dfferentatng, e fnd e + φ e φ φ φ φ β φ 1 φ b Subttutng for φ on the rght hand de, ung (23) yeld 1 1+ φ e + φ e φ e φ φ b Applyng l'hôptal' Rule to th lat expreon hen b β and makng ue of (28) and (217) ge u F + 1 φ φ 2 2 Rearrangng, e obtan 1 3F φ β β + φ ( ) ( ) 2 2F A ymmetrc argument mple that 1 3F φ β β + φ ( ) ( ) 2 2F Solng thee equaton yeld:

34 33 β F β (F6) φ ( ) j β j F ( ) ( ) [ 2 ], and, n ymmetrc equlbrum, (F7) y F By defnton of p ( b ) and π ( b) e hae (F8) p ( b) F ( φ ) φ ( b) and π ( b) F ( y ) y ( b) It follo mmedately from (221) and (222) that (F9) p 2 F π From (212) j (F10) and p 2 ( φ ) + F φ y 2 π ( ) + F y Subttutng ung (216) - (F7), e obtan F (F11) p j + F ( ) ( ) ( )[ F β 4 β β 2 ], j j and (F12) π β F β β β + F ( ) β F ( ) ( ) 4 ( ) ( )[ 2 ] F By (F9) p π ( β ) From (F3) and the hypothee of the Propoton, F >, and o, from (F3), (F11), and (F12), π > p Hence () hold A ymmetrc argument etablhe () hold alo From (21), ( β ) If the nequalty trct, then (F8) and (F9) mply that () and () hold If

35 34 then, nce d d F > F β 0, t follo from (A3) that F > ( β ), but th contradct (21) QED Propoton 23: Comparon of equlbrum bd dtrbuton Gen CSD, () π ( b) > π ( b), for all b ( β, µ ) () p( b) > p( b), for all b ( b, b ) () π ( b) > p ( b), for all b ( b, b ) () p( b) > π ( b), for all b ( b, µ ) Proof: To etablh (), note frt that, from (218), 1 π ( b) > π ( b) for all b [ µ, µ ) Contrary to (), uppoe that there ome b $ ( β, µ ) uch that π ( b $ ) π ( b $ 1 We hall ) argue that π ( b) π ( b) ncreang at b $ Snce H ( p ) > H ( p ) for p (0,1), π π mple that H ( π ) H ( π ) > H ( π ) Then from (223) Hence π 1 1 π > at b b $ π H ( π ) b H ( π ) b π d db π π π π It follo that, for ome δ> 0, π π > π π [ ] 0, at b b $ (F13) π ( b) > π ( b) for all b ( b $ δ, b $ ) Let δ be the bgget alue for hch (F13) hold If b $ δ > β, then (F14) π ( b $ δ ) π ( b $ δ ), and from the aboe argument, π ( b) > π ( b) for b ( > b$ δ ) near b $ δ, a contradcton of (F13) Aume, therefore, b $ δ β In the ymmetrc aucton th to trong

36 35 buyer, both buyer bd aboe β f and only f they hae aluaton exceedng β Hence π ( β ) F ( β ) F ( β ) π ( β ) and o, from (26), (214) hold We conclude that F F ( β ) F ( β ) and o β β β Thu, from Lemma 21, e mut hae F 0 If γ > β, then F λf, [ β, γ ], and o, from (216) y ( b) y ( b) for b n ome neghborhood of β, and o π ( b) > π ( b) n that neghborhood, a contradcton of (211) Hence γ β Then, from part () of Lemma 22, π ( b) > π ( b) for all b n a neghborhood of β, a contradcton of (F13) We conclude that b$ ( β, µ ) atfyng π ( b $ ) π ( b $ 1 doe not ext, and o () etablhed ) To proe (), uppoe that there ext $ b ( b, b ) uch that p b ( $ ) p ( b $ ) 1 Snce H ( p) > H ( p), for p ( 0, 1 ), t follo from (223) that p 1 1 p < at b b $ p H ( p ) b H ( p ) b p Hence p b ( ) p ( b) ncreang at b $ Becaue the ame argument apple to any b o > b $ for hch o p( b ) o p ( b ) 1, p b p b ( ) < ( ) for all b ( b $, b ) But from (213) p( b ) p( b ), and o b $ cannot ext Hence () hold unle, for all b ( b, b ), p ( b) < p ( b), hch ould conflct th Lemma 22 () To proe (), uppoe that for ome $ b ( b, b ), π ( b $ ) p ( b $ ) 1 If µ b $, then 1 π ( b $ ) p ( b $ ) and 1> p ($ b), a contradcton Hence, aume that b $ < µ Snce () hold, π ( b $ ) p ( b $ ) < p ( b $ ) Thu (F15) π π 1 1 p > at b H ( π ( b)) b H ( p ( b)) b p b $ The ret of the proof parallel that of () but ue part () of Lemma 22 ntead of part () A ymmetrcal argument etablhe ()

37 36 QED Propoton 25: Characterzaton of equlbrum nere bd functon If CSD hold, then () y ( b) y ( b), for all b ( β, µ ) and () φ ( b) > φ ( b), for all b ( b, b ) () y ( b) φ ( b), for all b ( b, b ) and () φ ( b) > y ( b), for all b ( β, b ) Proof: For b [ µ, µ ], 1 y ( b) > y( b) For, b ( β, µ ), CSD mple that () follo mmedately from (220) To demontrate (), e frt argue that () hold n a punctured neghborhood of b If α α φ ( b ) > φ ( b ) α < α, th mmedate becaue then If α α, then φ ( b ) φ ( b ), and o, from (212), (F16) ( φ ) F ( φ ) φ 1 1 φ b φ b ( φ ) F ( φ ) φ, at b b Gen CSD t follo that φ ( b) < φ ( b) and o () hold n a punctured neghborhood of b, a clamed Suppoe that there ext $ b ( b, b ), uch that φ ( b $ ) φ ( b $ 1 Then (F16) hold at ) b b $ Hence, Aumpton CSD mple that φ ( b) 1 for all b b b φ ( b) ($, ), a contradcton of our fndng aboe Thu φ ( b) < 1 for all b b b φ ( b) (, ) To proe (), note frt that, by Corollary 24, b µ Hence φ ( b ) y ( b ) From (213), (216) and part () of th Propoton, for any b ( β, b ) uch that φ ( b) y ( b) Hence, ( φ ) > F ( φ ) φ φ b φ b y b ( y) y F ( y )

38 37 (F17) d F ( φ ( b)) φ ( b) y( b) > 0 db F ( y ( b)) For ome θ $ 1, uppoe that there ext $ b ( β, b ) atfyng (F18) F ( φ ( b $ )) F ( y ( b $ θ$ )) By (F17), Hence F ( φ ( b)) F ( y ( b)) trctly ncreang at b b $ (F19) d F ( φ ( b)) φ ( b) < y( b) and > 0, for all b [ β, b $ ) db F ( y ( b)) But y ( b ) b and o φ ( b ) y ( b ), a contradcton of (F19) We conclude that b $ cannot ext, and o () hold after all A ymmetrc argument etablhe that () hold alo QED Lemma 31: Expected eller reenue from bdder aucton R H, here α H d (33) R F Q d F d b F b ( 1 ( ( ))) ( ( 1 ( ))) + ( 1 ( )) and bz {, }n the ealed hgh bd H (34) R ( 1 F ( Q)) Q d + b F ( b )( 1 F ( β )) α bz Proof: The eak buyer' expected payment f he bd b b bf ( φ ( b)) Snce h equlbrum bd dtrbuton ha cdf F ( φ ( )), the expectaton oer all bd H R bf ( φ ( b)) df ( φ ( b)) b z b

39 38 hch, after ntegraton by part, can be rertten a b z H d R b F ( b )( 1 F ( b )) + ( 1 F ( φ ( b))) bf ( b) db db From (212) d db bf b F d ( ) φ ( φ ) φ db b Subttutng th expreon nto the ntegral e then obtan b z (F20) R b F F b F b b F b d H db db + φ ( β )( 1 ( )) ( 1 ( φ ( ))) φ ( ) ( φ ( )) Snce φ ( b) Q( φ ( b)), e can rerte th expreon a α z H R b F F b + F ( β )( 1 ( )) ( 1 ( )) ( Q) Q d Integratng agan by part, e obtan α z H d R b F b F Q d F d ( 1 ( )) + ( 1 ( ( ))) ( ( 1 ( ))) Appealng to ymmetry, e nfer from (F20) that b b b b z R b F b F F b b F b d H db db + φ ( )( 1 ( β )) ( 1 ( φ ( )) φ ( ) ( φ ( )) Agan ung the fact that φ ( b) Q( φ ( b)), e then obtan α z H R b F b F + F Q Q ( )( 1 ( β )) ( 1 ( ( )) ( ) d b b QED Lemma 32: Expected eller reenue from buyer expreed a R O, here α O d (35) R F d F d F ( 1 ( )) ( ( 1 ( )) + β ( 1 ( β )) and zβ O (36) R ( 1 F ) df + β F ( β )( 1 F ( β )) α zβ {, }n the open aucton can be

40 39 Proof: If the eak buyer ha a aluaton > β h expected payment +z β F ( β ) bdf ( b) Takng the expectaton oer, the expected reenue from the β eak buyer O R ( β F ( β ) + bdf ( b)) df ( ) β F ( β )( 1 F ( β )) + ( 1 F ) d α z z z β β Integratng by part once more, e obtan O R F d( ( 1 F ), α zβ hch can be rertten a (35) Appealng to ymmetry, e alo fnd α z O R β F ( β )( 1 F ( β )) + ( 1 F ) d β α β QED Propoton 33: Hgh-bd aucton uperor for dtrbuton hft Suppoe that CSD hold and, n addton (39) (a) d ( ) 0, and (b) d < 0, on [ β, α ] F Gen a < β α, uppoe that, for all [ β, α + a], < a + (310) F T 0 β F ( a), a + β R S and that ( ) + 1 F ( ) 0 for all [ β, β + a] Then the hgh-bd aucton generate hgher expected reenue than doe the open aucton Proof: From (39b) and (310), Propoton 25 apple and o, from part (), Q > for all [ b, α ] From (37)-(39), e need only ho that C (, Q ) pote Snce F conex (from (39)), o F Therefore Thu F ( Q) F ( Q) ( Q a) Q

41 40 But ( Q a) C(, Q) 1 F 1 F ( Q a) d F F F [ d F ] [ ( ) ( ) F F ] ( ) + > 0, nce F conex F Moreoer, from (32), F > F ( Q) F ( Q a) Thu Q a < and o C(, Q) ndeed pote QED Propoton 34: Hgh-bd aucton uperor for dtrbuton "tretche" (311) Suppoe that F ( ) atfe F ( β ) 0 and d d < F 0 on [ β, α ] For λ ( 0, 1 ), let the trong buyer hae dtrbuton F ( ), here [ β, α ] ( α < α ), uch that F, [, ] (312) F T λ β α G, ( α, α ] here G( α ) λ, G( α ) 1, and R S (313) G ( ) > 0, for all [ β, α ] and [ α, α ] Then the hgh-bd aucton generate more expected reenue than the open aucton Proof: From (312) and (313), CSD hold Hence, from Propoton 25 (), Q >, for all ( β, α ) No, (37) and the fact (from (32)) that F ( Q) < F mple that the dfference n reenue, D, atfe α F Q F D > Q F ( ( )) ( ) ( ( ) )( 1 ( ))[ ] d zβ Q Snce Q >, there ext ome $ [, Q] uch that F ( Q F ($) Q

42 41 Then D pote f ($) For the cae n hch $ > α, th follo mmedately from (313) Thu uppoe that $ (F21) ($) ($) F ($) F ($) F < α By (311) and (312) Alo nce $ Q, F ($) F ( Q) < F (here the lat nequalty follo from (32)) Then, from (F21) ($) QED Propoton 35: Open aucton uperor for hft to of probablty ma to the loer end pont Suppoe that the trong buyer' aluaton ~ dtrbuted accordng to F, [ β, α ] here F ( β ) 0 and (314) ncreang 1 F Buyer 1' aluaton ~ dtrbuted o that, for all [ β, α ], t denty at a fracton θ ( 0, 1 ) (th θ 0 ) of here the remanng denty reagned to β That, (315) F ( t) df + θ zβ γ here z α (316) γ ( 1 θ( t )) df ( t ) β Then, the open aucton generate hgher reenue than the hgh-bd aucton Proof: We frt etablh that CSD hold, e, (F22) > F F for all ( β, α )

43 42 But from (315), (F22) can be rertten a e, θ > F θ( t) df ( t) + zβ γ z (F23) θ( t) df ( t) + γ > θ F β And from (316), the left-hand de of (F23) can be rertten a zα F + ( 1 θ ( t)) df ( t), and o (F23) ndeed hold Thu, from (31) Q >, ( β, α ), e Q appear a depcted n Fgure 1 We next ho that, under our aumpton the eak buyer ha a loer margnal reenue Gen the dcuon at the begnnng of ecton 2, th ll mply that e hae the cae depcted n Fgure 1 and o reenue hgher n the open aucton From (316) and (317) 1 F θ z α θ d Then 1 F 1 F α z θ θ d α z d z α [ θ θ d z z α θ( ) d α d ]

44 43 0, nce θ( ) non-decreang Snce J 1 F, t follo mmedately that J J F