UNIQUENESS IN SEALED HIGH BID AUCTIONS. Eric Maskin and John Riley. Last Revision. December 14, 1996**

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1 u9d.doc Uqueess u9a/d.doc UNIQUENESS IN SEALED HIGH BID AUCTIONS by Ec Mas ad Joh Rley Last Revso Decembe 4, 996 Depatmet of Ecoomcs, Havad Uvesty ad UCLA A much eale veso of ths pape focussed o the symmetc case ad the bdde case ude asymmety. Commets by Bead LeBu ae gatefully acowledged. 9

2 u9d.doc Uqueess Whle much has bee wtte o the theoy of auctos, almost all of ths wo focusses exclusvely o the symmetc equlbum of a aucto whch bddes ae symmetc. That s, two bddes wth the same pvate fomato have exactly the same belefs about all of the opposg bddes. I a compao pape (Mas ad Rley (994, we have examed the questo of exstece of equlbum a sealed hgh bd aucto the absece of the symmety assumpto. Thee we show that ude qute wea assumptos thee exsts a equlbum whch bds cease mootocally wth bddes' esevato pces fo the tem. I ths pape we tu to the questo of uqueess. Ude the symmety assumpto t s well ow that thee s a uque symmetc equlbum (Mlgom ad Webe, 98, Mas ad Rley, 984. Howeve, t s ot ueasoable to suppose that a patcula buye mght establsh a eputato as a aggessve bdde f t s hs teest to do so. Rley (980 povdes a example of the "wa of attto" whch ths s deed the case. I fact thee s a cotuum of asymmetc equlba whch oe buye bds "aggessvely" ad the othe "passvely". Futhemoe, the geate the degee of aggesso, the lage s the equlbum expected ga of the aggessve buye. A secod example of a cotuum of equlba occus the commo value aucto, f the tem s sold by ope ascedg bd. As fst oted by Mlgom (98 thee s always a cotuum of equlba the two buye case. Bchada ad Rley (99 also peset a example whch, wth bddes, thee s a cotuum of equlba. Fo the symmetc sealed hgh bd aucto, howeve, we show that thee ca be o asymmetc equlbum ude the assumpto of depedece. Thus equlbum s uque. 0

3 u9d.doc Uqueess Whe we dop the symmety assumpto we have a vey geeal uqueess esult f thee ae oly bddes. All we eque s a mld addtoal estcto o pefeeces whch esues mootocty. Fo the geeal case of bddes, ou esults ae moe lmted. If dffeeces ca be paametzed smply as vaatos the dstbuto of esevato pces, we have a futhe qute stog uqueess esult. But whe buyes also have dffeet pefeeces all we ae oly able to establsh uqueess whe these dffeeces ae small. We descbe the model secto. I secto we peset chaactezato esults. We use these secto 3 to deve ou ma theoems. Secto 4 cosdes the possblty that buyes mght sometmes "ovebd", that s, bd moe tha the esevato pces. Cocludg emas ae secto 5.. THE MODEL Thoughout the pape we shall mae the followg assumptos about the aucto ad those patcpatg t. A sgle tem s to be sold to the buye who maes the hghest oegatve bd. If two o moe te, the we s selected at adom fom amog the hgh bddes. Thee ae potetal buyes. Buye has a utlty of u ( b, s f he ws wth a bd of b ad s of type s. Buye 's type has suppot [, s ] ad s dstbuted wth c.d.f. F (. We assume that F s cotuously dffeetable ad that the desty s stctly postve o [, s ]. Wthout loss of geealty we omalze so that the utlty of buye s zeo f hs α bd s usuccessful. We futhe assume that u ( b, s s cotuously dffeetable, α wth u b < 0 ad u s > 0, =,...,.

4 u9d.doc Uqueess Let b ( s be the esevato pce of buye f hs type s s, that s u ( b ( s, s = 0. We assume that fo all, buye 's hghest esevato pce, b ( s, s stctly postve (othewse buye eve has a cetve to bd. Clealy buyes caot ga by bddg moe tha the esevato pces. I fact we wll beg by assumg that they do ot. Assumpto : No buye bds eve bds moe tha hs esevato pce. Next, defe s to be the lowest type wth a o-egatve esevato pce. If s > α so that F ( s > 0, all those types s < s have a egatve esevato pce ad ae theefoe stctly wose off submttg a wg bd tha stayg out of the aucto. Thoughout we assume that such types eve bd ad theefoe focus o types daw fom the teval S [ s, s ]. We ext toduce the usual "sgle cossg popety" whch udeles so much of cetve theoy. Let M ( b, s be the ate at whch a buye s wllg to cease hs bd etu fo a geate pobablty of wg. As s easly cofmed, Assumpto s the equemet that M ceases wth s. Assumpto : Sgle Cossg Popety Fo all u ( b, s > 0, l u b s stctly ceasg wth s. We have agued elsewhee (Mas ad Rley, 984 that ths s a wea assumpto. Ideed f u ( b, s = V ( s b so that s s buye 's esevato pce, Assumpto holds f buye s s eutal o s avese. Fo some of ou esults we wll also explctly toduce the assumpto of s aveso. Assumpto 3: Buyes ae (wealy s avese. u ( b, s > 0 u ( b, s s cocave b.

5 u9d.doc Uqueess. CHARACTERIZING THE EQUILIBRIUM BID FUNCTIONS Above we defed b ( s to be the esevato pce of buye f hs type s s, that s, u ( b ( s, s = 0. It wll be useful below to defe the vese fucto (. φ ( b b ( b. That s, φ ( b s the smallest type buye wllg to pay b fo the tem. Fom Mas ad Rley (994 we have the followg esults. Lemma : If Assumptos ad hold, the dstbuto of wg bds has suppot [ b, b ] ad c.d.f. whch s cotuous o ( b, b ]. Lemma : Mootocty Suppose that a ealzato of 's equlbum stategy s b' f hs type s s' ad b" f hs type s s" > s'. Suppose, moeove that the expected etu to bddg b' s stctly postve. The b' b" As ou fst pelmay hee, we chaacteze b, the lowe suppot of the dstbuto of wg bds. Lemma 3: Chaactezato of the mmum bd Let b ( s be the lowest oegatve esevato pce of buye, =,...,. Wthout loss of geealty we suppose that b ( s... b ( s b ( s If Assumpto holds, the mmum bd satsfes (- b ( s b b ( s Moeove, ethe both of these ae equaltes o both ae stct equaltes. If the latte, 3

6 u9d.doc Uqueess (-3 b = Max ag Max F ( φ ( b u ( b, s b = whee φ ( b b ( s, =,..., Poof: Suppose that b < b ( s. By Lemma thee ae o mass pots o ( b, b ]. The both buye ad buye, egadless of type, have a stctly postve expected payoff fom bddg b λ, whee bλ λb + ( λ b ( s, 0 < λ <. Let p, =, be the pobablty that buye bds b. If both p ad p ae stctly postve, bddg b esults a te wth postve pobablty. The buye, egadless of hs type, s stctly bette off bddg slghtly above b sce ths beas the te ad ceases hs w pobablty by a fte amout. Hece p ad p caot both be stctly postve. Suppose the that p s zeo. Cosde a bd b by buye the eghbohood of b. Sce p = 0, buye 's pobablty of wg ad hece hs expected utlty decles towads zeo as λ. But we have aleady agued that buye 's equlbum expected utlty s stctly postve so aga we have a cotadcto. Next suppose that b > b ( s. Ay buye type who submts a bd must have a esevato pce of at least b. The ay such type must have a stctly postve expected utlty sce a bd the teval ( b ( s, b ws wth postve pobablty. It follows that ay buye type who submts a bd has a esevato pce exceedg b. But the at most oe buye bds b wth postve pobablty. (Fo othewse t would pay to bea the te by bddg slghtly moe. Aga we have a cotadcto, hece b satsfes (.. 4

7 u9d.doc Uqueess Suppose ext that b ( s < b ( s. Ay buye > s bette off bddg above b f hs esevato pce exceeds b. Gve Assumpto, buye bds less tha b f hs esevato pce s less tha b. If buye bds b wth postve pobablty ad buye 's esevato pce exceeds b he s stctly bette off espodg wth a bd ust above b sce hs pobablty of wg ses dscotuously at b. If buye bds b wth zeo pobablty, buye 's expected payoff s zeo f he bds b. Thus aga he s stctly bette off espodg wth a bd geate tha b. Combg these esults t follows that fo all >, buye outbds buye wth pobablty F ( φ ( b whe he bds b. The expected payoff to buye of type s f he maes hs equlbum bd of b s theefoe F ( φ ( b u ( b, s. = Sce we have assumed that o buye eve bds moe tha hs esevato pce, f buye bds b b hs expected payoff s at least F ( φ ( b u ( b, s. It follows that fo b to be a best espose fo type s, = F ( φ ( b u ( b, s F ( φ ( b u ( b, s = = Thus b ag Max F ( φ ( b u ( b, s b = Fally, suppose that both b' ad b" solve ths maxmzato poblem ad that b' < b". Buye of type s s at least dffeet betwee bddg b" ad ay lowe bd. Gve Assumpto, all othe buye types stctly pefe b" ove ay lowe bd. Thus the mmum bd fo all types s > s s at least b". But the b' s ot the lowe suppot of the equlbum dstbuto of wg bds. Q.E.D. 5

8 u9d.doc Uqueess Lemma 4: Stct Mootocty of the pobablty of wg: Let G w ( b be the c.d.f. of the dstbuto of wg bds. Suppose b ' < b" ad 0 < G ( b' < G ( b" <. The at least two bddes bd the teval ( b', b " wth postve w pobablty. w Poof: Suppose fst that thee exsts some such teval ove whch o oe bds wth postve pobablty. Defe b $ f{ b G ( b < G ( b'}. Wth o oe bddg ( b', b " wth w w postve pobablty t follows that b$ > b". By Lemma, ay buye bddg tes wth pobablty zeo. The such a buye ca lowe hs bd towads b' ad so ase hs ga to wg wthout loweg hs pobablty of wg. But the bddg $ b caot be a best espose. Suppose the that oly buye bds the teval ( b', b " wth postve pobablty. I ths case buye wll eve bd the teval ( b' + b", b" wth postve pobablty sce he ca lowe hs bd to ust above b' wthout loweg hs w pobablty. Thus o buye bds the teval ( b' + b", b" wth postve pobablty. But ths cotadcts ou eale cocluso. Q.E.D. Let ( b ~ (,..., ~ s b ( s be equlbum bddg stateges (possbly mxed stateges. Ay detemstc selecto b ( s fom b ~ ( s s stctly ceasg fo all s S that. It follows 6

9 u9d.doc Uqueess ~ y ( = b ( s a ceasg fucto that s well defed at all b fo whch thee exsts s wth b supp ~ b ( s. The, fo all bds exceedg the mmum bd b we ca defe (-4 φ ( b = sup{ y ( b $ b $ b, y ( b $ defed} Because y ( s ceasg, φ ( s odeceasg ad cotuous fo all b futhemoe, that the pobablty of wg ca be wtte as G ( b F ( φ ( b Sce φ ( b s cotuous fo all, so s G ( b. > b. Note, As a pelmay to povg uqueess we ow deve popetes of φ ( ad G b (. Poofs ca be foud the Appedx. Lemma 5: Stct mootocty popety of bd dstbutos. Let G ( b be the c.d.f. of the maxmum bd of all 's oppoets. The fo ay b $ = b ($ s such that 0 < G ($ b <, ad fo ay ε > 0, G ( b $ ε < G ( b $. Lemma 6: If φ ( b s stctly ceasg to the ght (fom the left at b = β, the β s a best espose fo $ = φ ( β. s Lemma 7: If φ ( b s stctly ceasg to the ght (fom the left at b = β > b, G ( b s ght (left dffeetable at β. Moeove, the ght (left devatve satsfes (-5 G ' ( u G b u β ( β, φ ( ( (, ( β + β β φ β = 0 7

10 u9d.doc Uqueess Lemma 8: φ ( b s ght (left dffeetable fo all b > b ad all. Suppose φ( b ( φ ( b,..., φ ( b s stctly ceasg at b. It follows fom Lemmas 7 ad 8 that φ( b satsfes (-6 F ' ( φ dφ = = F ( φ b u ( b, φ u ( b, φ We ca ewte ths matx fom as follows. 0.. ' F ( d (, b u b φ φ φ 0. (-7 A [ ] = [ ] whee A = 0.. F ( φ u ( b, φ Lemma 9: Edpot codto f o bdde has a postve pobablty of wg at the mmum bd. Suppose that F ( s = 0 ad u ( b, s = 0, =,...,. Defe e s F ' ( s F ( s The f the vecto of equlbum vese bd fuctos φ( b ( φ ( b,..., φ ( b satsfes the edpot codto φ ( b = s, =,...,, ad s stctly ceasg at b, 8

11 u9d.doc Uqueess b u b s ' (, (-7 φ ( b = ( + u ( b, s e = 3. UNIQUENESS It s ow helpful to tasfom vaables ad defe (3- z = l F ( s, s [ s, s ]. Sce ths fucto s stctly ceasg ove ts doma we ca vet ad defe the stctly ceasg fucto z (3- h ( z = F ( e, z [ z, 0 ], whee z l F ( s. Also defe v ( b, z l u ( b, h ( z. By Lemmas 7 ad 8, f φ ( s ceasg at b, the b s the soluto to the followg maxmzato poblem: Max U ( x, φ ( b = F x u x b = ( φ ( (, φ ( x Moeove the fst ode codto (3-3 F ' ( φ dφ = = F ( φ b u ( b, φ u ( b, φ must be satsfed, whee t s udestood that the devatves ae ethe left o ght devatves. The afte tasfomg the vaables, b s the soluto to the maxmzato poblem Max V ( x, z ( b = z ( x + v ( x, z ( b x = ad must satsfy the fst ode codtos: 9

12 u9d.doc Uqueess dz + b v x z b (, ( = 0, =,...,. = To smplfy otato we also defe P b z b v x z (, = (,, =,...,. If Assumptos ad 3 hold, P (3-4 v > 0 < 0 ad P > 0. z b The fst ode codtos ca the be ewtte as: (3-5 dz = P ( b, z, =,...,. = Lemma 0: Cosde solutos ( z ( b,..., z ( b ad ($ z ( b,...,$ z ( b to the system of dffeetal equatos (3-6 dz = P ( b, z, =,...,. = o some teval [ b', b "] ove whch, fo all =,..,, z ( b < 0 ad P b z (, > 0. Suppose that z$ ( b" z ( b" > 0 fo all =,...,. The z$ ( b' z ( b' > 0, =,, Moeove, Poof: d [ z$ ( b z ( b] < 0, b [ b', b"]. = Let z ( b, α, =,..., be a soluto to the system of dffeetal equatos satsfyg the edpot codto z ( b", α = ( α z ( b" + αz$ ( b" 30

13 u9d.doc Uqueess The α z ( b", α = z$ ( b" z ( b" > 0, =,...,. Rewtg (3-6 matx fom we have ' (3-7 A[ z ( b] = [ P ( b, z ] whee A s as defed (-7 except that t s ow a that A s vetble ad that matx. It s eadly cofmed B A = γ.. γ.. γ γ whee γ= ( Ivetg (3-7 we obta dz (3-8 [ ] = B[ P ( b, z ] I patcula, dz (3-9 = B [ P ( b, z ] = ( P ( P whee B s the th ow of B. = Summg ove, dz = = = P Dffeetatg by α, (3-0 d z = α = = P z z α By costucto z > 0 at b", =,,. α Defe b$ f{ b z = > 0, fo all =,..., } α 3

14 u9d.doc Uqueess The, fom (3-0 d z < 0 α = o ( b $, b"]. Hece fo some =,..., (3- z ( b $ z > ( b" α α > 0. Dffeetatg (3-9 by α (3- d z = b P z P z ( ( z b z b = By costucto, thee must be some such that z b α α ($, = 0. Thus, fo ths, the fal tem o the ght had sde of (3- appoaches zeo as b b $. Moeove, fom (3- z b α α ($, > 0 fo at least oe othe. The, sce P < 0, the ght had sde z z of (3- s stctly less tha zeo some ght eghbohood of b $. Hece s stctly α deceasg ths ght eghbohood of. But the z b α α ($, caot be zeo afte all. We z coclude that fo all =,...,, ad all b < b ", > 0 α. Ths poves the fst clam. The secod clam follows mmedately fom (3-0. Q.E.D. The poof of uqueess fo the case of two buyes s ow elatvely staghtfowad. Poposto : Uqueess wth two buyes If Assumpto holds, equlbum s uque. Poof: Lemma 3 uquely defes the lowe suppot of each buye's bd dstbuto, b. By Lemma 4 the suppot must be a teval, [ b, b ]. Wth oly a lttle futhe wo, the poof of uqueess also povdes a alteatve poof of exstece fo the buye case. 3

15 u9d.doc Uqueess Case (: Fo some, F ( φ ( b > 0. I ths case, fo some, the lowe suppot fo z, z = l F ( s s bouded fom below. By Lemma 8, buyes' equlbum vese bd fuctos satsfy (3-3 ad hece (3-6 must also hold. The by Lemma 0, thee s a uque b such that the pa of dffeetal equatos ( z( b, z ( b satsfyg z ( b = 0, =,, also satsfes the lowe bouday codto z ( 0 = z, =,. Case (: Fo all, F ( φ ( b = 0. Sce both equlbum vese bd fuctos must be stctly ceasg we ca apply Lemma 9. That s, ay equlbum bd fuctos fo buye must have the same slope at b. Let γ be the maxmum bd oe equlbum ad let $γ < γ be the maxmum bd aothe. Let φ ( b, γ ad φ ( b, γ $ be coespodg equlbum vese bd fuctos. The F ( φ ( γ $, γ $ = > F ( φ ( γ $, γ, =,. By Lemma 0, φ ( b, γ $ > φ ( b, γ fo all b > b. Fom Lemma 9, F ( ( b F ( $ φ φ ( b = O(( b b Sce F'( s stctly postve ad F ( φ ( b = F ( φ ( b = 0, t also follows fom Lemma 9 that Thus F ( φ ( b = O( b b. F ( φ $ ( b F ( φ ( b F ( φ ( b = O( b b It follows that fo ay ε > 0, thee exsts δ> 0 such that 33

16 u9d.doc Uqueess F ( φ $ ( b F ( φ ( b F ( φ ( b < ε, fo all b [ b, b + δ ], Reaagg ad tag logs l F ( φ $ ( b < l F ( φ ( b + l( + ε Summg ove, F b F b < + = l ( φ $ ( l ( φ ( l( ε, b [ b, b + δ ]. Moeove, by Lemma 9, ths dffeece s deceasg b. The fo all b, F b F b < + = l ( φ $ ( l ( φ ( l( ε. By costucto the fst sum s zeo at b. The l F ( φ ( b > l( + ε. = We have theefoe show that > F ( φ ( b $, γ > = ( + ε But ths must hold fo all ε > 0. Thus γ = γ $ ad so aga equlbum s uque. Q.E.D. Fo moe tha buyes, establshg uqueess s sgfcatly moe complcated sce t s o loge ecessaly the case that all buyes have equlbum bd dstbutos wth the same suppot. It s tutvely clea that buyes may ot have the same maxmum bd. Fo f buye 3's maxmum esevato pce s fa lowe tha that of buye ad buye, t s lely that competto betwee the latte buyes wll push the maxmum bd above aythg buye 3 s wllg to pay. Whle ths complcato ca be dealt wth, thee s a futhe poblem. I geeal thee s o easo to suppose that the suppot of each buye's equlbum bd 34

17 u9d.doc Uqueess dstbuto s a teval. Istead thee may be "gaps", that s, tevals ove whch a buye does ot bd. As we shall see, such possbltes caot ase f the followg assumpto o pefeeces also holds. Assumpto 4: b u b s (, u ( b, s b u b s b u b s b u b s (, (, (, > l( > l( u ( b, s b u ( b, s b u ( b, s It ca be eadly cofmed that Assumpto 4 holds f o u ( b, s = e A( s b u ( b, s = ( w + s b θ θ ( w, 0 < θ, that s, all buyes ae s eutal o all have the same costat degee of absolute o elatve s aveso. Thus, the case of detcal pefeeces, the assumpto s elatvely mld. O the othe had, Assumpto 4 fals geecally f pefeeces dffe. Note also that afte tasfomg vaables, Assumpto 4 becomes P ( b, z > P ( b, z l P ( b, z > l P ( b, z b b Lemma : Suppose equlbum vese bd fuctos ae dffeetable o [ b', b "]. If the logathm of buye 's expected payoff, V ( b, z, >, s o-ceasg at b' ad Assumptos -4 hold, V ( b, z s deceasg o [ b', b "]. Poof: Suppose t s buyes,..., who have stctly ceasg vese bd fuctos, φ ( b,..., φ ( b. By Lemma 8, these vese bd fuctos ae dffeetable o [ b', b "]. Fom 35

18 u9d.doc Uqueess (3-6 t follows that (3-3 dz P ( b, z = 0, =,...,. = Totally dffeetatg by b, (3-4 d z P P z = < 0, =,...,. b z b = Summg (3-3 ove fom to, dz (3-5 ( P ( b, z = 0. = = Smlaly, summg (3-4 ove fom to, d z P (3-6 ( < 0. b = = Cosde the logathm of buye 's expected payoff, V = l U ( b, φ ( b. Hece fom (3-5 b V b s dz (, = = P. (3-7 Also b V b s (, = = P P b V b s d z P (, = < b = = P P b b Hece (3-8 Suppose P b V ( b, s P b = b P ( = b b V b s (, 0. Fom (3-7 36

19 u9d.doc Uqueess P = P ( 0. (3-9 It follows fom (3-8 that Fom (3-3 ad (3-5 b V b s P (, < b = P b P b P P ( P < P, =,..., m. = The, fom (3-7 f V = l U ( b, s, P < P, =,..., m. b 0 the Appealg to Assumpto we obta 0 b V b s (, <. Q.E.D. Appealg to Lemma we have the followg mpotat esult. Lemma : No Gaps If Assumpto 4 holds, the suppot of buye 's equlbum bd dstbuto s a teval [ b, b ], =,...,. Poof: Suppose that oly z,..., z ae stctly ceasg (fom the left at b $. The, fom (3-3, dz P ( b, z = 0, =,...,. = I matx fom, dz (3-0 A [ ] = [ P ]. 37

20 u9d.doc Uqueess Let A m m be a matx composed of the fst m ows ad colums of A. Fom (3-0, dz (3- A m m m m m [ ] + α[ ] = [ P ]. dz whee α = > 0, uless dz = m+ = 0, = m +,...,. Reaagg we obta: dz A m m m m [ ] = [ P α]. Ivetg ths expesso, we obta dz m m m m m m m α [ ] = B [ P α] = B [ P ]. m Suppose some subset of the buyes bd o the teval ( b $, b '. Wthout loss of geealty we may elabel these buyes m+,...,. The, fom (3-0, the ght devatves of z ( b,..., z ( m b satsfy dz A m m m m [ ] = [ P ]. Compag ths wth (3- t follows mmedately that the ght devatves ae stctly lage tha the left devatves uless dz ($ b = 0, = m +,...,. 3 The z z (,..., ( ae all dffeetable at b $. ' " ' " Let [ b, b ] be the fst gap fo buye, =,...,. Suppose bm = M b. Fom the " above agumet, t follows that z ( b,..., z ( b s dffeetable o [ b, b m ]. Sce equlbum expected utlty ceases cotuously wth type, ad type z m ' ' ( b chooses b m m, t must be the case that ' " " " " (3- V ( b, z ( b = V ( b, z ( b V ( b, z ( b, b b' m m m m m m m m m m m 38

21 u9d.doc Uqueess By Lemma, sce V m s oceasg at b m ' t must be the case that V m s deceasg o [ b ', " m bm ]. But ths cotadcts (3-. The thee ca be o such teval. Q.E.D. We the have the followg uqueess esult fo buyes. Poposto : Uqueess wth detcal pefeeces ad suppots Suppose that all buyes have the same payoff fucto u ( b, s = u( b, s ad buye types ae all daws fom dstbutos wth the same suppot [ α, s ]. The f Assumptos -4 hold, the equlbum bd fuctos ae uque. Poof: Sce pefeeces ae detcal, the lowest type wllg to pay b fo the tem φ ( b = φ( b, =,...,. We wll cosde oly the case whch, fo some, F ( φ ( b > 0. The, fo some, the lowe suppot fo z, z s bouded fom below. By Lemma, the lowe suppot of each buye's equlbum bd dstbuto s b = φ ( s. We ow show that ude ou hypotheses, the uppe suppot of each buye's bd dstbuto s the same. Suppose these uppe suppots ae b b... b. Sce at least buyes must bd ay subteval of [ b, b ], b = b. Suppose the fo some >, b = b = b... b > b Sce b s optmal fo s, Pob{ bds less tha b } u( b, s u( b, s. = Hece The poof fo the case whch, fo all =,...,, F ( φ ( b = 0 follows vey closely case ( Poposto. 39

22 u9d.doc Uqueess Pob{ bds less tha b } u( b, s < u( b, s. = But the buye s bette off bddg b tha b whe hs type s s. Thus b caot, afte all, be less tha b. By Lemma, t follows that equlbum vese bd fuctos φ ( b ae stctly ceasg o [ b, b ]. By Lemmas 6-8, the vese bd fuctos ae cotuously dffeetable o [ b, b ], hece must satsfy (3-3. Afte tasfomg vaables, t follows that (3-6 must hold, that s, = dz = P ( b, z, =,...,. The appealg to Lemma 0, thee s a uque soluto to ths dffeetal equato satsfyg the edpot codtos. Q.E.D. We ext show that ths esult ca be exteded to the case of dffeet suppots. Cosde ay sequece 0 P... P. Suppose that fo some, (3-7 ( P + > P = Addg P + to both sdes P + + > = P The, sce P + P +, P + + > = P 4 Thus f (3-7 holds fo =m t holds fo all > m. Clealy (3-7 does ot hold fo =. Thus thee s a uque m, m such that 40

23 u9d.doc Uqueess (3-8 ( - P + P, m = + = (3-9 ( - P > P, > m. Lemma 3: Maxmum equlbum bds Suppose that b s the uppe suppot of the equlbum bd dstbuto. Suppose that buyes ae labelled so that P ( b, 0... P ( b, 0. Defe m to satsfy (3-8 ad (3-9. The b = b f ad oly f m. Moeove, afte appopate elabellg so that b (3-30 ( P ( b, 0 = P ( b, z ( b, > m. Poof: = Suppose fst that fo < m b... b < b = b m + The, fom (3-5, ove the teval [ b, b ], +... b, m dz - P = 0 = 5, =,..., Summg ove, fom to, dz ( - - P = 0 = = Hece b V b dz (,0 = - P + + = = P P + - = 0 by (3-5 By Lemma t follows that b V + ( b,0 > 0 o [ b, b ]. + 4

24 u9d.doc Uqueess But the V + (b,0 does ot tae o ts maxmum at b + Suppose ext that buyes,..., have b maxmum bds. Agug as above,. Hece b = b afte all. = b whle all othe buyes have lowe = dz - P = 0, =,..., Summg ove, fom to, Hece ( - dz - P = 0 = = ( - P P = 0 = But ths cotadcts (3-9. Suppose we elabel buyes so that b... < b = b. m+ m V m+ (b,0 must tae o ts maxmum at b m+. Hece m b V b dz P m (,0 = - + m+ = = m P Pm + m - = = 0, at b b m = +. By Lemma, gve Assumpto 3, thee ca be at most oe such tug pot. Poceedg exactly the same mae we coclude that (3-7 uquely defes b We ow ote that fo ay b, (3-8-(3-30 uquely defe b,..., b. m+ Q.E.D. b,..., as fuctos of the maxmum bd b. Thus, fo ay b thee s a uque soluto to the system of dffeetal equatos though the edpots z ( b = 0. Appealg oce aga to Lemma 0, t follows 4

25 u9d.doc Uqueess that ths soluto s a stctly deceasg fucto of the edpot b. Thus oce aga the equlbum bd fuctos ae uque. To summaze, we have poved: Poposto 3: Uqueess wth dffeg suppots fo each buye's equlbum bd dstbuto Suppose that each buye has the same utlty fucto u = u(b,s. The f Assumptos -4 hold, the equlbum bd fuctos ae uque. We coclude wth oe esult that does allow fo dffeeces pefeeces. Poposto 4: Uque equlbum wth ceasg bd shadg Suppose Assumptos ad hold ad u ( b, s = υ ( s b, =,..,, whee υ ( s cocave. The thee s a uque equlbum fo whch the payoff to the we υ (s -b (s s stctly ceasg fo all s. Poof: Tasfomg vaables, P ( b, z = ' υ ( h ( z - b υ ( h ( z - b Sce υ ( s cocave, t follows fom the hypothess of the Poposto that d P ( b, z ( b < 0. Hece, fom (3-6, d z < 0, =,...,. The = the fst ode codtos ae suffcet fo a maxmum. Thus we ca ague exactly as the poof of the pevous theoem. 43

26 u9d.doc Uqueess Q.E.D. What Poposto 4 tells us s that thee s at most oe equlbum whch hghe types always shade the bds moe. That s, d (3-3 ( s b ( s 0, =,..., ds As we shall see, ude a mld addtoal assumpto, ths s the case f thee ae o asymmetes. The f the asymmetes ae ot too lage, t s deed plausble that (3-3 wll hold. Lemma 4: Bd shadg a symmetc equlbum Suppose u (b,s = υ (s - b, that s s s buye 's esevato pce. Suppose also that each buye's esevato pce s a daw fom the same dstbuto wth c.d.f. F (. The f (3-3 d F( s ( ds F ( s ' > 0 bddes wth hghe esevato pces shade the bds moe. Poof: I the symmetc case the fst ode codtos, (3-3 become ' ' F ( φ dφ υ ( φ- b ( - = F( φ ν( φ- b Tasfomg vaables, the symmetc equlbum bd fucto must satsfy dz (3-33 ( - = ( ( - P h z b whee h ' z F( φ ( = ' F ( φ Dffeetatg by b, d z ' ' (3-34 ( - = P ( h( z - b( h ( z dz - 44

27 u9d.doc Uqueess Sce lm P (z- b d z =, z z < 0 fo all b suffcetly close to b. Let $ b be the smallest b such d z that = 0 ad d z s stctly ceasg at b $. Dffeetatg (3-34 by b we have at b = b $, 3 d (3-35 ( - ( $ = ( ( ( $ z b P h z z b 3 ' " ' P ' ( s egatve by Assumpto, ad by hypothess (3.3 holds ad so h(z s covex. Theefoe the ght had sde of (3-35 s stctly less tha zeo. But ths cotadcts ou d z tal hypothess. It follows that must be egatve eveywhee. Next ote that φ(b = h(z(b, whee z = l F(h Dffeetatg by b, ' ( = h ( z z ( b ad h ( z = φ ' b ' ' F( φ ' F ( φ d z Sce ' ' s egatve t follows fom (3-34 that h ( z z ( b >. Hece φ ' ( b > ad so b ' ( s <. Q.E.D. 4. Equlbum wth "Ovebddg" Thoughout the pevous sectos we have assumed that o buye eve bds moe tha hs esevato pce (Assumpto. Ths s ot qute as ocuous a assumpto as t may seem. The easo s that f a buye bds above hs esevato pce ove some age of types, equlbum hs oppoets may always bd hghe. If they do so, the ovebdde eve actually ws whe he bds above hs valuato. Whle we llustate the pot wth a smple example, the aalyss s qute geeal. 45

28 u9d.doc Uqueess Suppose that thee ae ust two buyes, each wth a utlty fucto u (b,s = s - b. Thus buye 's type s also hs esevato pce. Fo a ovebddg equlbum t s ecessay that mmum esevato pces dffe. Suppose theefoe that buye 's esevato pces ae ufomly dstbuted ove [,] buye 's ove [0,]. By Lemma, the mmum bd by buye s 0.5 f thee s o ovebddg. Suppose that the selle sets a mmum pce [0.5,. Aga by Lemma, the mmum pce s. Let b (s ;, b (s ; be the uque equlbum of ths aucto wth o ovebddg. (Fo s <, buye stays out of the bddg. Now suppose that the selle dops hs mmum pce ad the ew bd fuctos ae b ( s = b ( s; αs + ( α, s < b ( s = b ( s;, s, whee 0<α < That s, buye ovebds f hs esevato pce s less tha. Sce b ( = = b (, buye ws wth zeo pobablty f hs type s less tha. Thus ovebddg by buye s a best espose. Fo all s, ad b <, U ( b, s = ( s b Pob{ b < b} = ( s b Pob{ αs + ( α < b} Hece = (s - b(+ b ᾱ b U b s (, = [ s + ( α b] α [ ( + α ], sce b < α [ ( + α ], sce s α 46

29 u9d.doc Uqueess 0 <, f α, sce Thus fo all, < <, thee exst cotuous stctly mootoc equlbum bd fuctos wth the popety that buye 's mmum bd s ad buye ovebds f ad oly f hs esevato pce s less tha. Thee ae two easos why such equlba ae cosdeably less teestg tha the uque equlbum wthout ovebddg. Fst, as log as thee s a postve pobablty that buyes wll mae a mstae ad bd less tha wth postve pobablty, buye s stctly wose off bddg above hs esevato pce. That s, ovebddg equlba ae ot temblg had pefect. Secod, t s ot dffcult to show that equlbum payoffs of all the buyes ae lowe whe thee s ovebddg. Thus the buyes Paeto pefe the o-ovebddg equlbum. 5. Cocludg Remas We have establshed a geeal uqueess esult fo the case of two buyes. Wth moe tha bddes, f dffeeces amog buyes ca be expessed puely as dffeeces belefs, we have a futhe stog uqueess esult. Fally, wth dffeeces both pefeeces ad belefs, we have show that thee ca be at most oe equlbum wth the popety that buyes shade the bds moe whe they have hghe esevato pces. We also ague that ths "mootoc shadg" assumpto s mld f asymmetes ae suffcetly small. Whe dffeeces utlty fuctos ad dstbutos of types ae lage, the aalyss s cosdeably moe complcated sce t s o loge ecessaly the case that the suppot of each buye's equlbum bd dstbuto s a teval. Ou coectue s that equlbum bd fuctos ae at least geecally uque. 47

30 u9d.doc Uqueess The stogest assumpto made the pape s that buyes' esevato values ae depedetly dstbuted. I Mas ad Rley (994 we establsh exstece of mootocally ceasg equlbum bddg stateges ude the weae assumpto that buyes' esevato values ae afflated. It emas ope as to whethe thee exst o-mootoc equlba o whethe thee s a (geecally uque mootoc equlbum. 48

31 u9d.doc Uqueess REFERENCES Bhchada, S. ad J.G Rley, "Equlbum Ope Commo Value Auctos," Joual of Ecoomc Theoy, 49 (99. Cox, J. C., V. L. Smth ad J.M. Wale, "Theoy ad Behavo of Fst Pce Auctos," Joual of Rs ad Ucetaty, ( Dasgupta, P. ad E.S. Mas, "Exstece of Equlbum Dscotuous Games, I: Theoy, II: Applcatos," Revew of Ecoomc Studes, 53 (986-6, 7-4. Mas, E.S. ad J.G. Rley, "Optmal Auctos wth Rs Aveage Buyes," Ecoometca, 5 (984: , "Asymmetc Auctos," evsed (993., "Equlbum Sealed hgh Bd Auctos, evsed (994. Mlgom, P., "Ratoal Expectatos, Ifomato Acqusto ad Compettve Bddg," Ecoometca, 50 (98: Mlgom, P. ad R.J. Webe, "A Theoy of Auctos ad Compettve Bddg," Ecoometca, 50 (Septembe, 98: 08-. Nalebuff, B.J., "Pzes ad Icetves," upublshed doctoal dssetato, Nuffeld College, Oxfod, 98., ad J.G. Rley, "Asymmetc Equlba the Wa of Attto," Joual of Theoetcal Bology, 3 (985: Rley, Joh G., "Stog Evolutoay Equlbum ad the Wa of Attto," Joual of Theoetcal Bology, 8 (980: Vcey, W., "Coutespeculato, Auctos ad Compettve Sealed Tedes," Joual of Face, 6 (96:

32 u9d.doc Uqueess Wlso, R., "Bddg," J. Eatwell, M. Mlgate ad. Newma (eds., The New Palgave 50

33 APPENDIX Lemma 5: Stct mootocty popety of bd dstbutos. Let G ( b be the c.d.f. of the maxmum bd of all 's oppoets. The fo ay b $ = b ($ s such that 0 < G ($ b <, ad fo ay ε > 0, G ( b $ ε < G ( b $. Poof: Let b$ = b ( s. If o othe buyes bd [ b $ ε, b $ ] wth postve pobablty, G ( b $ ε = G ( b $. The buye s stctly bette off bddg b $ ε tha b $, cotadctg the defto of b $. If oly oe othe buye bds [ b$ ε, b $ ] wth postve pobablty, buye must also. Fo othewse the othe buye (call hm buye has the same pobablty of wg f he bds b $ ε as f he bds [ b$ ε ', b $ ], fo ε'< ε. It follows that buye bds [ b $ ε', b $ ] wth zeo pobablty. But the o buye bds [ b $ ε ', b $ ] wth postve pobablty, cotadctg ou eale esult. Q.E.D. Lemma 6: If φ ( b s stctly ceasg to the ght (fom the left at b = β, the β s a best espose fo $ = φ ( β. s Poof: Sce both cases ae hadled the same way, we cosde oly the case whch φ ( b s stctly ceasg fom the ght. If φ ( b s also stctly ceasg fom the left, the lemma follows mmedately. The suppose that φ ( b = s$ f ad oly f b [ α, β ]. That s, fo some b$ [ α, β ], y ( b $ = s$. Sce φ ( b s stctly ceasg to the ght at β, thee exsts a deceasg sequece {b,...,b t,...} appoachg β ad a coespodg oceasg sequece {y (b,..., y (b t,...} appoachg $s. Sce b t s optmal fo paamete y (b t, we have

34 u9a.doc Uqueess t t t t (A. G ( b u ( b, y ( b G ( b $ u ( b $, y ( b 0, fo all t. Sce G ( ad u ae cotuous, we have the lmt, (A. G ( β u ( β,$ s G ( b $ u ( b $,$ s 0 Fom (A. t follows that buye, wth paamete $s, s at least as well off choosg β as $ b. Q.E.D. Lemma 7: If φ ( b s stctly ceasg to the ght (fom the left at b = β > b, G ( b s ght (left dffeetable at β. Moeove, the ght (left devatve satsfes (A-3 G ' ( u G b u β ( β, φ ( ( (, ( β + β β φ β = 0 Poof: Sce the two cases ae hadled the same way, we cosde oly the case whch φ ( b s stctly ceasg to the ght. We ow that φ ( b s cotuous. The at β thee exsts a deceasg sequece {b,b,...} appoachg β such that y (b t s defed fo all t ad appoaches s$ = ( β mootocally fom above. y Sce b t s optmal fo s t = y (b t we eque t t t t G ( β u ( β, y ( b G ( b u ( b, y ( b t t Subtactg G ( b u ( β, y ( b fom both sdes, we obta t t t t t t [ G ( β G ( b ] u ( β, y ( b G ( b [ u ( b, y ( b u ( β, y ( b ] Dvdg though by ( b t t β u ( β, y ( b we the obta

35 u9a.doc Uqueess (A.4 t t t t t G ( b G ( β G ( b u ( b, y ( b u ( β, y ( b t t t b b u ( β, y ( b b β By Lemma 6 β s optmal fo $ = φ ( β. The s t t G ( β u ( β,$ s G ( b u ( b,$ s fo all t. t Subtactg G ( b u ( β,$ s fom both sdes ad the dvdg by ( b t β u ( β,$ s we the obta (A.5 t t t G ( b G ( β G ( b u ( b,$ s u (,$ s β t t b b u ( β,$ s b β I the lmt as b t β the ght had sdes of (A.4 ad (A.5 cocde. The G ( b s ght dffeetable at b. Moeove the ght devatve satsfes (A.3. Lemma 8: φ ( b s ght (left dffeetable fo all b > b ad all. Q.E.D. Poof: Suppose φ ( b,..., φ ( b ae stctly ceasg at b $ ad that φ ( b,..., φ ( b ae costat at b $. By Lemma 5. By Lemma 6 G ( b s dffeetable at b $, =,..,. Also, sce b $ > bo, φ ( b $ > s. The F ( φ ( b $ > 0 ad we may tae the logathm of both sdes of (A.6 l G ( b = F ( φ ( b, b > b to obta o (A.7 lg = l F ( φ ( b + c whee = +

36 u9a.doc Uqueess c = l F ( φ ( b = + Subtactg c fom both sdes we ca expess (A.7 matx fom as follows: lg c... lg c l F ( φ ( b. = B.. l F ( φ ( b whee B s defed (3-8. It follows that l F ( φ ( b ad hece φ ( b 6s ght dffeetable. Lemma 9: Edpot codto f o bdde has a postve pobablty of wg at the mmum bd. Suppose that F ( s = 0 ad u ( b, s = 0, =,...,. Defe e s F ' ( s F ( s The f the vecto of equlbum vese bd fuctos φ( b ( φ ( b,..., φ ( b satsfes the edpot codto φ ( b = s, =,...,, ad s stctly ceasg at b, b u b s ' (, (-7 φ ( b = ( + u ( b, s e = 3

37 u9a.doc Uqueess Poof: Ivetg (-6, F b u b ' (, ( φ ' F ( φ φ φ = B u ( b, φ The pemultplyg the th compoet by φ φ φ F φ φ φ φ b u b ' ( ' (, = B F ( u ( b, φ whee B s the th ow of B. Applyg l'hoptal's $ Rule The e b u ' ' φ φ = B b u s u φ ' [ e ] b u m = B = B b u s u m + φ ' ' φ whee m b u b s (, s u b s (, > 0. Ivetg oce moe ad eaagg, we obta fally, φ ' ( s m ( = +. e = Q.E.D. 4

38 u9a.doc Uqueess Poposto: Uqueess whe buyes ca bd moe tha the esevato pces If the two hghest mmum esevato pces ae ethe o-postve o postve ad equal, thee s a uque dstbuto of wg bds. Poof: Cosde the case whch the hghest mmum esevato pce s o-postve. By Lemma 3, the lowe suppot of the dstbuto of wg bds, b = 0. The f buye has a postve esevato pce, he has a stctly postve expected payoff by bddg o the teval ( b, b ( s. Thus all such buyes ae stctly wose off ovebddg. Ad f buye has a esevato pce b ( s < 0, hs expected payoff s egatve f he submts a stctly postve bd. He s theefoe stctly bette off emag out of the aucto o possbly bddg 0. If two o moe buyes bd zeo wth postve pobablty, they w wth postve pobablty ad thus have a egatve expected payoff. The at most oe buye ca behave ths way. Thus the oly possble dffeece betwee a equlbum wth ovebddg ad a oovebddg equlbum s that oe buye may bd zeo wth postve pobablty. The all ou pevous agumets hold fo bds stctly geate tha zeo. It follows that oly bddes wth zeo esevato pces wll bd zeo equlbum. Such bddes ae dffeet betwee bddg zeo ad ot bddg. If they all choose the latte stategy the ovebddes bd of zeo eve ws. Thus ths s a equlbum. Howeve, the ew equlbum has the same dstbuto of wg bds as befoe. Q.E.D. 5

39 u9a.doc Uqueess 6