Wars of attrition and all-pay auctions with stochastic competition

Transcription

1 MPRA Munch Personl RePEc Archve Wrs of ttrton nd ll-py uctons wth stochstc competton Olver Bos Unversty Pnthéon-Asss, LEM 17. November 11 Onlne t MPRA Pper No. 3481, posted 18. November 11 :4 UTC

2 Wrs of Attrton nd All-Py Auctons wth Stochstc Competton Olver Bos Unversty Pnthéon-Asss (Prs ) November 11 Abstrct We extend the wr of ttrton nd ll-py ucton nlyss of Krshn nd Morgn (1997) to stochstc competton settng. We determne the exstence of equlbrum bddng strteges nd dscuss the potentl shpe of these strteges. Results for the wr of ttrton contrst wth the chrcterzton of the bddng equlbrum strteges n the frst-prce ll-py ucton s well s the wnner-py uctons. Furthermore we nvestgte the expected revenue comprsons mong the wr of ttrton, the ll-py ucton nd the wnner-py uctons nd dscuss the Lnkge Prncple s well. Our fndngs re pplcble to future works on contests nd chrty uctons. Keywords: All-py ucton, wr of ttrton, number of bdders JEL Clssfcton: D44, D8 1 Introducton The wde nd growng lterture on ll-py uctons ssumes tht the number of bdders s common knowledge. Yet, n mny stutons where ll-py uctons llustrte economc, socl nd poltcl ssues, prtcpnts do not know the number of ther opponents. Indeed, n lobbyng contests, R&D rces or bttles to control some mrkets, gents do not know the exct number of ther rvls. In lobbyng contest, some groups of nterest gve brbe to the decson mker n order to obtn mrket or poltcl fvor. In R&D rces, frms compete ech other to be the frst one to obtn ptent. The money spent n ths rce s A prevous verson of ths pper crculted under the ttle Wrs of ttrton wth stochstc competton. I would lke to thnk Pedro Jr-Moron, Phlppe Jehel nd Ron Hrstd for helpful dscussons. I lso thnk John Morgn for e-ml converstons. I m grtefulled to Clude d Aspremont, Gbrelle Demnge, Frnk Redel nd n nonymous referee whose comments mproved the qulty of ths work. All errors re mne. Address: Unversty Pnthéon-Asss (Prs ), LEM, 5/7 venue Vvn, 756 Prs, Frnce. E-ml: olver.bos@u-prs.fr.

3 not refundble. More generlly, the effect of n unknown number of bdders s n mportnt queston n ucton theory (see the recent ppers of Hrstd, Pekec, nd Tsetln (8) nd Pekec nd Tsetln (8)). However, to our knowledge there s no nlyss of ll-py uctons wth n uncertn number of bdders. Krshn nd Morgn (1997) nlyzed these ucton desgns wth fflted sgnls where the number of bdders s fxed nd common knowledge. In ths pper, we extend ther nlyss to stochstc competton frmework. In the followng we cll ll-py ucton the frstprce ll-py ucton nd wr of ttrton the second-prce ll-py ucton. We focus on equlbrum bddng strteges nlyss nd expected revenue comprsons s most of prevous ppers on wnner-py uctons wth uncertn number of bdders. McAfee nd McMlln (1987) nd Mtthews (1987) studed frst-prce uctons wth stochstc number of bdders. They determned whether t s better to concel or to revel the nformton bout the number of bdders for frst nd second-prce wnner-py uctons n dfferent frmeworks. 1 However, they dd not chrcterze the equlbrum strteges. Usng model à l Mlgrom nd Weber (198) wth ndependent prvte sgnls nsted of fflted ones, Hrstd, Kgel, nd Levn (199) estblshed tht equlbrum bds wth stochstc competton re weghted verges of the equlbrum bds n uctons where the number of bdders s common knowledge. Krshn () nvestgted ths result n nother wy wth n ndependent prvte vlue model. In recent pper Hrstd, Pekec, nd Tsetln (8) found the sme result n mult-unt wnner-py uctons wth common vlue. Pekec nd Tsetln (8) lso nvestgte mult-unt uctons wth unknown number of bdders. Indeed they determne the rnkng of the expected revenues for unform nd dscrmntory uctons. In ddton they compre the expected revenues for ech ucton desgn when the number of bdders s known nd unknown. In ths pper we determne the equlbrum strteges for the ll-py ucton nd the wr of ttrton under monotoncty ssumpton when the number of bdders s unknown. Indeed we ssume the Byesn ssessment of the bdder s vlue tmes hzrd rte gven stochstc number of bdders s n ncresng functon n the bdder s sgnl. It s generlzton of n ssumpton of Krshn nd Morgn (1997) when the number of bdders s fxed nd common knowledge. The consstency of ths ssumpton s dscussed through n exmple. The equlbrum strteges of the ll-py ucton, s well s wnner-py uctons (Hrstd, Kgel, nd Levn, 199), s weghted verge of equlbrum strteges tht would be chosen for ech number of bdders. However, t s not obvous for the wr of ttrton. Indeed, contrry to the frst nd second-prce wnner-py uctons, t does not drectly follow from the frst order condton tht the equlbrum strtegy should be equl to weghted 1 Mtthews (1987) consdered bdders wth n ncresng, decresng or constnt bsolute rsk-verson nd McAfee nd McMlln (1987) focused only on the rsk-verse bdders nd determned the optml ucton. In ther frmework, the number of dentcl przes s proportonl to the number of bdders. They showed tht n unknown number of bdders could chnge the results on nformton ggregton. Common knowledge of the proportonl rto llows to fnd the results on nformton ggregton when the number of bdders s suffcently hgh.

4 verge. Usng n exmple, ths result s dscussed. Moreover n nswer for the ndependentprvte-vlues model s provded. Expected revenues re not only compred for the wr of ttrton nd the ll-py ucton but lso mong ll-py nd wnner-py mechnsms. Then, we show tht the stochstc competton does not ffect the rnkng of the expected revenues nd the Lnkge Prncple s well. It s not n ntutve result. Indeed, we prove tht the unknown number of bdders ffects bddng strteges dfferently for the wr of ttrton, the ll-py ucton nd the wnner-py uctons. Moreover bddng strtegy comprsons re provded mong the ll-py nd wnner-py mechnsms. The pper s orgnzed s follows. The model nd prelmnres re descrbed n Secton. The nlyss of the wr of ttrton nd the ll-py uctons re gven n Sectons 3 nd 4. Secton 5 compres expected revenues nd bddng strteges. Some computtonl detls re provded n Appendx. Model wth Stochstc Competton The model follows nd generlzes the prelmnres of Krshn nd Morgn (1997) (henceforth K-M) n stochstc competton settng (s McAfee nd McMlln (1987) nd Hrstd, Kgel, nd Levn (199) used n the study of wnner-py uctons). There s n ndvsble object tht cn be llocted to N = {1,,..., n} potentl bdders, wth n <. Every potentl bdder s rsk neutrl. Frstly, we consder set of bdders A N. Denote A = the crdnlty of set A. Pror to the ucton, ech bdder observes rel-vlued sgnl X [, x]. The vlue of the object to bdder, whch depends on hs sgnl nd those of the other bdders, s denoted by V, = V, (X) = V (X, X ) where V, whch s the sme functon for ll bdders, s symmetrc n the opponent bdders sgnls X = (X 1,..., X 1, X +1,..., X ). It s ssumed tht V s non-negtve, contnuous, nd non-decresng n ech rgument. Moreover, the bdders vluton for the object s supposed bounded for ll : EV, <. Let f be the jont densty of X 1, X,..., X, symmetrc functon n the bdders sgnls. Besdes, for ny -tuple y, z [, x] wth m = {mx(y, z )} =1 nd m = {mn(y, z )} =1, f stsfes the fflton nequlty f( m)f(m) f(y)f(z). Afflton s strong form of postve correlton s dscussed by Mlgrom nd Weber (198). It mens tht f bdder s sgnl s hgh, then other bdders sgnls re lkely hgh too. As consequence, the competton s lkely to be strong. Let F Y 1 (. x) be the condtonl dstrbuton of Y 1, where Y 1 = mx{x j } j=, gven X 1 = x nd f Y 1 (. x) the correspondng densty functon. 3

5 When the number of potentl bdders s common knowledge, we cn defne v (x, y) = E(V,1 X 1 = x, Y 1 = y), (1) the Byesn ssessment of bdder 1 when hs prvte sgnl s x nd the mxml sgnl of hs opponents s y. As n K-M, we ssume tht v (x, y) s ncresng. 3 We consder the stuton n whch bdders do not know the number of ther rvls when they choose ther strtegy. For ny subset A of N, we denote π A the probblty tht A s the set of ctve bdders. Moreover, the probbltes π A re ndependent of the bdders denttes nd ucton rules. Sets wth equl crdnlty hve equl probbltes. Therefore, the ex nte probblty to hve prtcpnts n the ucton s the sum of probbltes wth the sme crdnl : s := A =,A N Let p bdder s updted probblty tht there re bdders condtonl upon the event tht he s n ctve bdder. We suppose tht these probbltes re common knowledge nd symmetrc such s p = p. Therefore 4 p := A =, A N B N 3 Anlyss of the Wr of Attrton π A π A nd p = p = s π B n s In ths secton we determne the equlbrum strteges for the wr of ttrton wth fflted sgnls. It s not cler from the frst order condton tht the equlbrum strteges re weghted verge of the equlbrum strteges tht would be chosen for ech number of bdders. Then we consder n ndependent-prvte-vlues model to nvestgte further ths queston. 3.1 Generl Cse wth Afflted Sgnls Assume tht the number of bdders s common knowledge nd ech bdder bds n mount b. Thus, the pyoff of the bdder f b s the vector of bds s V, (X) mx b j f b > mx b j j j 1 U, (b, X) = #Q(b) V,(X) b f b = mx b j j b f b < mx b j j 3 As Mlgrom nd Weber (198) nd K-M remrk, snce X 1 nd Y 1 functon of ts rguments. But they dopted the sme ssumpton. 4 For detl, see McAfee nd McMlln (1987). =1 re fflted, v (x, y) s non-decresng 4

6 where j nd Q(b) := {rgmx b } s the collecton of the hghest bds. Strteges t the symmetrc equlbrum re noted β when the number of bdders s known. K-M show tht the bddng equlbrum strtegy when the bdders re nformed bout the number of bdders s β (x) = v (y, y)λ(y y, )dt () where λ(y x, ) = f Y 1(y x) nd wth the followng boundry condtons: 1 F Y 1 (y x) β () = nd lm x x β (x) =. Let us ssume the sme mechnsm for stochstc number of bdders nd denoted β : [, x] R + bdder s pure strtegy, mppng sgnls nto bds. As we consder only the symmetrc equlbr, we focus on the symmetrc nd ncresng pure strteges β β 1 = β =... = β. As the number of bdders s stochstc, the defnton of the equlbrum strtegy concerns bdders belefs bout the number of ctve bdders. Strtegy β s clled equlbrum strtegy f for ll bdders β(x) rgmx b E E[U, (b, β(x ), X) X = x] x [, x] (3) where β(x ) = (β(x 1 ),...β(x 1 ), β(x +1 ),..., β(x )) nd E s the expectton opertor wth respect to the dstrbuton of the bdders belefs. The uncertn number of bdders enters the expected utlty through the vlue of the object for the bdder nd the sze of the vector of bds b. 5. Assume tht ll bdders except bdder 1 follow symmetrc nd dfferentble equlbrum strtegy. Bdder 1 receves sgnl x nd bds n mount b. The expected utlty of bdder 1 s Π W (b, x) = E E[U,1 (b, β(x 1 ), X) X 1 = x] = E E{[V,1 β(y 1 )]1 β(y 1 ) b b1 β(y 1 )>b X 1 = x} = E E{E{[V,1 β(y 1 )]1 β(y 1 ) b b X 1, Y 1 } X 1 = x} = [ β 1 (b) p [v (x, y) β(y))]f Y 1 (y x)dy b 1 p F Y 1 (β 1 (b) x) ] (4) wth β 1 (.) the nverse functon of β(.). The mxmzton of (4) wth respect to b leds to: [ p v (x, β 1 (b))f Y 1 (β 1 1 (b) x) β (β 1 (b)) 1 ] p F Y 1 (β 1 (b) x) = (5) At the symmetrc equlbrum b = β(x), thus (5) yelds β (x) = = p v (x, x)f Y 1 (x x) 1 p F Y 1 (x x) w (x)β (x) (6) 5 It lso enters through the collecton of the hghest bds Q(b). Yet, when #Q(b) > 1 the vlue of the ntegrl s zero: t lest one of the support s n tom. Thus, we do not need to consder t. 5

7 wth the weghts w (x) = p (1 F Y 1 (x x)) 1 p (7) F Y 1(x x) By () nd (6) we know tht β(.) s ncresng. It follows tht n equlbrum strtegy must be gven by β(x) = w (x)β (x) w (t)β (t)dt (8) Thus, we hve necessry condton bout the shpe of β. We prove tht t s ndeed n equlbrum strtegy under n ddtonl ssumpton, s stted n the next theorem. Ths ssumpton provdes suffcent condton for the exstence of the symmetrc monotonc equlbrum bddng strteges. Defnton 1. Let φ : R R be defned by φ(x, y ) = v (x, y) λ(y x, ) where λ(y x, ) = f Y 1 (y x) 1 p F Y 1(y x). φ(., y ) s the product of v (., y), n ncresng functon, nd λ(y x, ), non-ncresng functon. 6 Besdes, φ s equvlent to v (x, y)λ(y x, ) defned by K-M when the number of gents s common knowledge. Assumpton 1. φ(x, y ) s ncresng n x for ll y. Theorem 1. Under ssumpton 1, symmetrc equlbrum n wr of ttrton s represented by β(x) = w (x)β (x) wth β (t) nd w (t) gven by () nd (7). w (t)β (t)dt Proof. Frst, β(.) s contnuous nd dfferentble functon. Indeed, by K-M we know tht β (.) s contnuous nd dfferentble functon. We hve to verfy the optmlty of β(z) when bdder 1 s sgnl s x. Usng equton (5), we fnd tht Π W β(z) (β(z), x) = = 1 β (z) 1 p v (x, z)f Y 1 (z x) β (z) 1 + p F Y 1 (z x) [ p v (x, z)f Y 1 (z x) p v (z, z) λ(z z, )(1 = 1 β (z) (1 p F Y 1 (z x)) p [φ(x, z ) φ(z, z )] ] p F Y 1(z x)) When x > z, s φ(x y, ) s ncresng n x, t follows tht ΠW (β(z), x) >. In smlr β(z) Π W mnner, when x < z, β(z) (β(z), x) <. Thus, Π W (β(x), x) =. As result, the β(z) mxmum of Π W (β(z), x) s cheved for z = x. 6 Ths fct cn be proved n smlr wy tht the hzrd rte λ(y x, ) of the dstrbuton F Y 1 (y x) s non-ncresng n x. 6

8 K-M dscussed ssumpton 1 when the number of bdders s common knowledge. Ths ssumpton mens tht v (., y) ncreses fster thn λ(y x, ) decreses. However, s n the wr of ttrton wth fxed number of bdders, ths s not problem. Indeed, ths ssumpton holds f the fflton between X nd Y 1 s not so strong. We gve n exmple below to llustrte ths dscusson wth stochstc number of bdders. 7 Exmple 1. Let f(x) = denote f Y (x, y 1, y,..., y 1 ) the jont densty of (X 1, Y 1, Y +1 (1 + order sttstc of (X,..., X ) such s Y 1 Y Therefore, =1 x ) on [, 1] wth X bdder s sgnls nd let us,..., Y 1 ) wth Y k the k th -hghest... Y 1. Let us consder {, 3}. f Y (x, y) = 4 5 (1 + xy) on [, 1] f Y3 (x, y 1, y ) = 16 9 (1 + xy 1y )1 y1 y on [, 1] 3 Frst of ll, we cn esly verfy tht the fflton nequlty gven holds. We lso ssume tht v (x, y) = (x + y). Then computtons led to f Y 1 (y x) = 1 + xy + x f Y 1 3 (y x) = 4y + xy 4 + x nd F Y 1 (y x) = y + xy + x nd F Y 1 3 (y x) = y 4 + xy 4 + x We cn lso verfy tht F Y 1 (y x) s non-ncresng n x. We obtn (1 + xy)(x + 4) φ(x, y ) = (x + y) (x + 4)(x + ) p y( + xy)(4 + x) p 3 y (4 + xy )( + x) 4y( + xy )( + x) φ(x, y 3) = 3(x + y) (x + 4)(x + ) p y( + xy)(4 + x) p 3 y (4 + xy )( + x) Thus, ssumpton 1 holds (some detls re gven n ppendx). Usng the results where the number of bdders s common knowledge, the boundry condton β() = follows. Thus, f the expected vlue s bounded whtever the number of potentl bdders, then the bddng strtegy wll be bounded too. Followng the sme logc thn K-M, we could determne tht lm β(x) =. Indeed, n ths stuton, x x β(x) p ( 1 v (y, y) λ(y y, )dy + mn v (z, z) ln p ) F Y 1 (z z) 1 p F Y 1 (x z) Hrstd, Kgel, nd Levn (199) nd Hrstd, Pekec, nd Tsetln (8) show tht the form of the equlbrum strteges for wnner-py uctons s such tht β(x) = w (x)β (x). However, ths result s not obvous for the wr of ttrton. Indeed, contrry to wnner-py uctons nd the ll-py ucton (cf nfr.), n the cse of the wr ttrton, t s not drect result of the frst order condton tht the equlbrum strtegy should be equl to weghted verge. Yet, the followng exmple llustrtes n smple cse tht the bddng strtegy n the wr of ttrton wth stochstc competton could be wrtten s weghted verge of the bddng strteges tht would hve been chosen for ech number of compettors. 7 Ths exmple generlzes n exmple of K-M wth two fxed bdders. 7

9 Exmple. Let f(x) = =1 x on [, 1] wth X bdder s sgnls nd let {, 3}. As n Exmple 1 we ssume tht v (x, y) = (x + y). Therefore, f Y (x, y) = 4xy on [, 1] f Y3 (x, y 1, y ) = 16xy 1 y 1 y1 y on [, 1] 3 We cn esly verfy tht the fflton nequlty nd the ssumpton 1 hold. equlbrum strteges for fxed number of bdders re gven by β (x) = 8 y 1 y dy nd = 4 toto When the number of bdders s stochstc nd p = p 3 =.5 = 8x + 4 ln 1 + x 1 x β(x) = 8 β 3 (x) = 4 y 1 + 3x x x 4 dy y 4 1 y 4 dy ( x ln 1 + x ) 1 x + rctn x Then the = 8 y 3 y y y y y + dy = 1x ln 1 + x 1 x + 16 rctn x 3 All these bddng strteges re depcted n Fgure 1. The bddng strtegy wth stochstc number of bdders β (sold lne) s lwys hgher thn the bddng strtegy wth bdders (long dshed lne) nd lower thn the bddng strtegy wth 3 bdders (short dshed lne) for ll vlue of x. Then we cn fnd vector of weghts such s the bddng strtegy wth stochstc competton would be wrtten s weghted verge of the bddng strteges wth fxed number of bdders. Β x, Β3 x, Β x x Fgure 1: Bddng strteges β, β 3 nd β. 3. An Exmple: Independent-Prvte-Vlues Model As we hve seen prevously, nd despte Exmple, t s not obvous tht the equlbrum strtegy n the wr of ttrton s equl to weghted verge such tht β(x) = w (x)β (x). 8

10 In ths secton, we provde n nswer for the IPV model. Let us consder tht ech bdder ssgns vlue X to the object, ndependently dstrbuted on [, x] from the dentclly dstrbuton F. Therefore, the bddng strtegy where the number of bdders s common knowledge s β (x) = ( 1) yf(y)f (y) 1 F 1 (y) dy nd the bddng strtegy wth stochstc competton s gven by. β(x) = p ( 1) yf(y)f (y) 1 p F 1 (y) dy Lemm 1. The equlbrum strtegy n wr of ttrton s decresng n for ll. Proof. β x (x) = yf(y)f (y) (1 F 1 (y)) [1 F 1 (y) + ( 1) ln F (y)]dy As 1 F 1 (y) + ( 1) ln F (y) s negtve for ll, y, the result follows. If β(x) [β (x), βā(x)] for ll x wth β (x) = mn {β (x) N s > } nd βā(x) = mx {β (x) N s > } then we cn fnd vector of weghts (z (.)) wth z (.) = 1, z (.) for ll x such tht β(x) = z (x)β (x). Thus, we stte: Proposton 1. In n IPV model, the equlbrum strtegy n the wr of ttrton wth stochstc competton s weghted verge of equlbrum strteges where the number of bdders s common knowledge. Proof. We hve to dstngush two cses. Indeed from Lemm 1 ether p 1 = nd then βā(x) = β (x) or p 1 > nd β (x) = β n (x). β(x) β (x) = [ yf(y) [1 p F 1 p ( 1)F (y) (y)][1 F (y)] As p ( 1)F (y) p ( )F 1 (y) 1 s negtve, β(x) β (x). If p 1 > β (x) = β 1 (x) = then the result follows. However f p 1 = : β(x) β n (x) = yf(y) [1 >1 p F 1 (y)][1 F n 1 (y)] p k(y, )dy p ( )F 1 (y) 1 where k(y, ) = ( 1)F (y) + (n )F n+ 3 (y) (n 1)F n (y) s postve for ll nd y. Thus n both cses, p 1 = nd p 1 >, β(x) [β (x), βā(x)] for ll x nd the equlbrum strtegy wth stochstc competton cn be wrtten s weghted verge of equlbrum strteges wth fxed number of bdders. >1 ] dy 9

11 The next exmple consders unform dstrbutons nd t most three bdders. Then n explct shpe of the vector of weghts s determned. Even n ths smple cse, ths vector cnnot be wrtten s esly s for the wnner-py uctons. Exmple 3. Let us consder the vlue X s gven by unform dstrbuton on [, 1] nd the number of bdders could be or 3. Then the equlbrum strteges for fxed number of bdders re gven by y β (x) = 1 y dy nd = x ln(1 x) toto When the number of bdders s stochstc β 3 (x) = y 1 y dy = x + ln 1 + x 1 x β(x) = = x p y + p 3 y 1 p y p 3 y dy p y p 3 (y 1)(y y o ) dy = x 1 p 3 p 1 y o ln(1 x) + 1 p 3 p y o 1 y o ln[ y o (x y o )] where y o = p p + 4p 3 nd belongs to (, 1]. p 3 Usng Proposton 1 there exsts vector of weghts (z (.), z 3 (.)) such tht z (x)β (x) + z 3 (x)β 3 (x) = β(x) for ll x (, 1]. It follows tht p y x + ln(1 x) p z 3 (x) = 3 (y 1)(y y o ) dy nd z (x) = 1 z 3 (x) for ll x (, 1]. x + ln(1 + x) Remrk tht f p = then z 3 (x) = 1 for ll x. 8 z 3 (x) [, 1]. Moreover t s routne to verfy tht 4 Anlyss of the All-Py Aucton As before ssume the number of bdders s common knowledge nd ech bdder bds n mount b. Thus, the pyoff of the bdder s V, (X) b f b > mx b j j 1 U, (b, X) = #Q(b) V,(X) b f b = mx b j j b f b < mx b j j where j nd Q(b) := {rgmx b } s the collecton of the hghest bds. Strteges t the symmetrc equlbrum re noted α when the number of bdders s known. K-M show tht 8 Indeed p y p 3(y 1)(y y o) dy = dy 1 y. 1

12 the bddng equlbrum strtegy when the bdders re nformed bout the number of bdders s α (x) = wth the followng boundry condtons: v (t, t)f Y 1 (t t)dt (9) α () = nd lm x x α (x) = lm x x v (x, x). (1) As for the wr of ttrton, we focus only on the symmetrc pure strteges α : [, x] R +, clled n equlbrum strtegy f for ll bdders (such tht ) α(x) rgmx b E E[U, (b, α(x ), X) X = x] x [, x] where α(x ) = (α(x 1 ),...α(x 1 ), α(x +1 ),..., α(x )). Assume tht ll bdders except bdder 1 follow symmetrc nd dfferentble equlbrum strtegy. Bdder 1 receves sgnl x nd bds n mount b. The expected utlty of bdder 1 s Π A (b, x) = E E[U,1 (b, α(x 1 ), X) X 1 = x] = E E[V,1 1 α(y 1 ) b b X 1 = x] = E E[E[V,1 1 α(y 1 ) b b X 1, Y 1 ] X 1 = x] = α 1 (b) p [v (x, y) α(y))]f Y 1 (y x)dy b (11) wth α 1 (.) the nverse functon of α(.). The mxmston of (11) wth respect to b leds, t the symmetrc equlbrum b = α(x), to α (x) = p α (x) (1) By (9) nd (1) the bddng strtegy α(.) s n ncresng functon. It follows from the boundry condton (1) tht n equlbrum strtegy must be gven by α(x) = p α (x) (13) Once gn, we hve only necessry condton bout the shpe of the equlbrum strtegy. Under ssumpton 9 1 we prove tht α(.) s ndeed n equlbrum strtegy, s stted n the next theorem. 9 Indeed, ths ssumpton mples tht v (., y)f Y 1 (y.) s ncresng for ll y. The proof s smlr to the proof of Proposton 3 of K-M. 11

13 Theorem. Under ssumpton 1, symmetrc equlbrum n n ll-py ucton, denoted α(.), s weghted verge of equlbrum strteges, denoted α (.), tht would be chosen for ech number of bdders such tht α(x) = p α (x). Proof. To prove tht α s optml, we follow the sme wy tht for the wr of ttrton. α(.) s contnuous nd dfferentble functon. Indeed, by K-M we know tht α (.) s contnuous nd dfferentble functon. We verfy the optmlty of α(z) when bdder 1 s sgnl s x. Usng equton (1), we fnd tht Π A α(z) (α(z), x) = 1 p v (x, z)f Y 1 (z x) α (z) 1 = 1 α p [v (x, z)f (z) Y 1 (z x) v (z, z)f Y 1 (z z)] As we sd before, ssumpton 1 mples tht v (x, y)f Y 1 (y x) s ncresng n x for ll y. When x > z, t follows tht ΠA α(z) (α(z), x) >. In smlr mnner, when x < z, Π A (α(z), x) < α(z) Π A. Thus, (α(x), x) =. α(z) As result, the mxmum of ΠA (α(z), x) s cheved for z = x. Usng the results where the number of bdders s common knowledge, the boundry condton α() = follows. Thus, f the expected vlue s bounded whtever the number of potentl bdders, then the bddng strtegy wll be bounded too. Followng the sme logc thn K-M, we could determne tht lm α(x) = lm mx v (x, x). x x x x Thus, the bdders belefs bout the number of compettors s crucl to determne the equlbrum strteges. Indeed, the stochstc number of bdders does not ffect the bdders strteges t the equlbrum of the ll-py ucton nd the wr of ttrton n the sme wy. 5 Bddng Strtegy nd Revenue Comprsons In ths secton we nvestgte the expected revenue comprsons for the wr of ttrton nd the ll-py ucton. We lso compre the expected revenues nd the equlbrum strteges obtned from the ll-py nd wnner-py mechnsms. Fnlly the Lnkge Prncple s dscussed. 1 The probblty tht potentl bdder s tkng prt of the ucton s gven by A π A. Let us denote e d (.) the expected pyment of the current bdder n n ucton desgn d. Then the expected revenue s n =1 [ A π A]Ee d (X). 5.1 Wr of Attrton versus All-Py Aucton K-M show tht the expected revenue from the wr of ttrton s greter thn the expected revenue from the ll-py ucton when the number of bdders s known nd sgnls fflted. 1 Note tht the proofs of the expected revenue comprsons use the sme logc thn the proofs of K-M. 1

14 In our stochstc settng, t s not obvous tht ths result stll holds. Indeed, the uncertnty bout the number of bdders hs vrous consequences on the bdders strteges t the equlbrum. As opposed to the ll-py ucton, the equlbrum bddng strtegy n the wr of ttrton s not verge wth weght p of the bddng strteges for ech fxed number of bdders. Intutvely t s dffcult to determne from the equlbrum bddng strteges how the stochstc competton modfes the rnkng of the expected revenues. However, s we stte n the next proposton, the stochstc competton does not ffect the rnkng of the expected revenues. Proposton. Under ssumpton 1, the expected revenue from the wr of ttrton s greter thn or equl to the expected revenue from the ll-py ucton. Proof. Denote e A (.), the bdders expected pyment n the ll-py ucton t the symmetrc equlbrum nd e W (.) n the wr of ttrton. Then, under ssumpton 1, e W (x) = = β(x) = = = α(x) β(y) p p f Y 1 (x x)dy + β(x)(1 β (y) w (y)β (y)dy w (y)β (y)(1 p F Y 1 (y x)dy w (y)β (y) p F Y 1 (y x))dy v (y, y)f Y 1 (y y) 1 p F Y 1(y x) 1 p F Y 1(y y) dy p F Y 1 (y x)) p F Y 1 (y x)dy As e A (x) = α(x) nd F Y 1(y.) s non-ncresng functon for ll y, the wr of ttrton outperforms the ll-py ucton. 5. Wr of Attrton versus Second-Prce Aucton Our second result descrbes, under Assumpton 1, the rnkng of the equlbrum strteges from the wr of ttrton nd the second-prce ucton. Proposton 3. Under ssumpton 1, the equlbrum strteges from the wr of ttrton nd the second-prce ucton ntersect t lest once. Proof. Denote ω II (.), the bddng strtegy t the symmetrc equlbrum n the second-prce wnner-py ucton. Followng Hrstd, Kgel, nd Levn (199) the equlbrum strtegy s gven by ω II (x) = p f Y 1 (x x) p f Y 1(x x) v (x, x). Then, 13

15 E[ω II (Y ) X 1 = x, Y 1 < x] = In ddton, p v (y, y)f Y 1 (y y) p f Y 1 (y x) p f Y 1 (y y) p F Y 1 (x x) dy E[β(Y ) X 1 = x, Y 1 < x] = β(y) p f Y 1 (y x)dy p F Y 1(x x) = β(x) = = = β (y) p F Y 1 (y x) p F Y 1 From the fflton nequlty t follows for ll y x tht f x s suffcently low nd y w (y)β (y)dy x w (y)β (y) p F Y 1(y x) p F Y 1(x x) dy w (y)β (y) p F Y 1(x x) p F Y 1(y x) p dy F Y 1(x x) x p v (y, y)f Y 1 (y y) p F Y 1(x x) p F Y 1(y x) (1 p F Y 1(y y)) p F Y 1(x x) dy p f Y 1 (t x)dt p f Y 1 (y x) > x y (x x) dy y p f Y 1 (t x)dt p f Y 1 (y x) p f Y 1 (t y)dt p f Y 1 (y y) < x y f x suffcently hgh. p f Y 1 (t y)dt p f Y 1 (y y) It follows tht E[β(Y ) X 1 = x, Y 1 < x] < E[ω II (Y ) X 1 = x, Y 1 < x] f x s suffcently low nd E[β(Y ) X 1 = x, Y 1 < x] > E[ω II (Y ) X 1 = x, Y 1 < x] f x s suffcently hgh. K-M lso show tht the expected revenue from the wr of ttrton s greter thn the expected revenue from the second-prce wnner-py ucton when the number of bdders s known nd sgnls fflted. For smlr resons thn bove, t s not obvous tht ths result stll holds here. Yet, s we stte n the next proposton, the stochstc competton stll does not ffect the rnkng of the expected revenues. Proposton 4. Under ssumpton 1, the expected revenue from the wr of ttrton s greter thn or equl to the expected revenue from the second-prce ucton. Proof. Denote e II (.) the expected pyment t the symmetrc equlbrum n the second-prce wnner-py ucton such s e II (x) = p F Y 1 (x x)e[ω II (Y ) X 1 = x, Y 1 < x] 14

16 wth ω II (x) = p f Y 1 (x x) p f Y 1(x x) v (x, x). e W (x) = =e II (x) p v (y, y)f Y 1 (y y) 1 p F Y 1(y x) 1 p F Y 1(y y) dy p v (y, y)f Y 1 (y y) To get ths result remrk tht p f Y 1(y y) 1 p F Y 1(y y) p f Y 1(y x) p f Y 1(y y) dy p f Y 1 (y x) 1 p F Y 1 (y x) holds for ll y x All-Py Aucton versus Frst-Prce Aucton The next Proposton descrbes, under ssumpton 1, the rnkng of the equlbrum strteges from the ll-py ucton nd the frst-prce ucton. We show n n exmple tht these two bddng strteges re not strctly ordered for fxed number of bdders for ll rnge of x. Proposton 5. Under ssumpton 1, the equlbrum strteges from the ll-py ucton nd the frst-prce ucton ntersect t lest once. Proof. Denote ω I (.), the bddng equlbrum strtegy n the frst-prce wnner-py ucton such s (see Hrstd, Kgel, nd Levn (199)) ω I (x) = p F Y 1 (x x) p F Y 1(x x) ωi (x) wth ω(x) I = x v (y, y) f Y 1(y y) { } x F Y 1 (y y) exp f Y 1 (t t) F Y 1 (t t) dt dy. y Let us consder the Exmple 1 for v (x, y) = x. If bddng strteges cnnot be strctly ordered for p = 1 they cnnot be strctly ordered nether for p < 1. Computtons led to nd α (x) = y 1 + y + y dy = 4 3 x3 x + x 4 ln x + 11 Ths fct cn be proved n smlr wy tht the hzrd rte λ(y x, ) of the dstrbuton F Y 1 (y x) s non-ncresng n x. 15

17 ω I (x) = = 4 = 4 = 4 y 1 + y + y dy 1 + y + y exp ( 1 1 y x + y ( + y + x { y ) exp ) 1/ y x = 4 4 3x (x 1) + 3x( + x ) 1/ 1 + t } t + t 3 dt dy { ( 1 t + y ( + y ) 1/ ( + x dy ) 1/ t ) } + t dt dy As ω I(.15) =.15 > α (.15) =.9 nd ω I(.75) =.79 < α (.75) =.6 the result follows. Our next result compres the expected revenues obtned from the ll-py ucton nd the frst-prce ucton. Equlbrum bddng strteges n the frst-prce wnner-py ucton nd the ll-py ucton wth stochstc competton cn be wrtten s weghted verge of equlbrum strteges tht would be chosen for ech number of bdders. However the weght of the verge re dfferent nd cnnot be strctly rnked. Then once gn, t s not obvous tht results wth exogenous number of bdders stll holds. Yet, s we stte n the next proposton, the stochstc competton does not ffect the rnkng of the expected revenues. Proposton 6. Under ssumpton 1, the expected revenue from the ll-py ucton s greter thn or equl to the expected revenue from the frst-prce ucton. Proof. Denote e I (.) the expected pyment t the symmetrc equlbrum n the frst-prce wnner-py ucton such s e I (x) = p F Y 1 (x x)ω I (x) Then, e I (x) = = =e A (x) To get ths result remrk tht 1 exp p F Y 1 (x x) p F Y 1 (x x) p F Y (x x) ωi (x) 1 p v (y, y)f Y 1 (y y) F Y 1(x x) { } x F Y 1 (y y) exp f Y 1 (t t) y F Y 1 (t t) dt dy p v (y, y)f Y 1 (y y)dy { y } f Y 1 (t t) F Y 1 (t t) dt F Y 1(x x) F Y 1 (y y) for ll y x. 1 Ths fct s proved by K-M pge

18 5.4 Lnkge Prncple When the number of bdders s common knowledge, Mlgrom nd Weber (198) nd K-M determne rnkng reltonshp n the expected revenue mong frst nd second-prce n wnner-py nd ll-py uctons. Tht derves from the comprson of the sttstcl lnkges between the bdder s expected pyment nd hs sgnl. Ths result, clled lnkge prncple, s bsed on the fflton. Let us consder bdder 1. Let e M (z, x) be hs expected pyment wth bd z nd sgnl x n the ucton mechnsm M nd e M (x, x) be the dervtve wth respect to the second rgument t z = x. Theorem 3 (K-M s Lnkge Prncple, 1997). Suppose M nd L re two ucton mechnsms wth symmetrc nd ncresng equlbr such tht e M (, ) = e L (, ) =. If for ll x, e M (x, x) el (x, x) then for ll x em (x, x) e L (x, x). The lnkge prncple s stll stsfed wth the stochstc competton. To see ths formlly, consder the ucton mechnsm M nd let Π M (z, x) be the expected pyoff of bdder wth bd z nd sgnl x. Then, Π M (z, x) = R(z, x) e M (z, x) = z p v (x, y)f Y 1 (y x)dy e M (z, x) The expected gn of wnnng s the sme n ll mechnsms wth stochstc competton (s n the cse of fxed number of bdders). Moreover the stochstc number of bdders s ntegrted n the expected pyment nd then does not ffect the lnkge prncple propertes. We could pply the lnkge prncple to compre the expected pyment between wnner-py nd ll-py mechnsms nd then get the sme results thn bove. 6 Concluson In ths pper we determne the equlbrum strteges n the wr of ttrton nd the ll-py ucton wth fflted vlues nd stochstc competton. We estblsh suffcent condton for the exstence of the monotonc equlbrum bddng strteges. We hve shown tht n the wr of ttrton, n opposte to the ll-py ucton nd the wnner-py uctons, t does not drectly follow from the frst order condton tht the equlbrum strtegy s equl to weghted verge. Even f stochstc competton ffects the ll-py ucton nd the wr of ttrton n dfferent wys, we prove tht t does not modfy the rnkng of the expected revenues nd the K-M s lnkge prncple. Our results cn be useful for mny pplctons of ll-py desgns such s n contest theory nd chrty uctons. Indeed, recent ppers compre ll-py nd wnner-py uctons to rse money for chrty nd suggest to use n ll-py desgn. In prtculr, Goeree, Mslnd, 17

19 Onderstl, nd Turner (5) show tht the second-prce ll-py ucton s better to rse money for chrty thn the frst-prce ll-py ucton nd the wnner-py uctons. Chrty uctons my be mplemented for specl events or on the Internet. A lrge number of chrty uctons tke plce whle potentl bdders do not know the number of compettors. 13 As we do not ntroduce externltes n the bdders pyoff, our results could not be ppled to chrty uctons. However, s they chnge some nsghts n the second-prce ll-py ucton ths work lets us open questons for future reserch on chrty uctons. 7 Appendx Boundry Condton of the Equlbrum Strtegy for the wr of ttrton. β(x) = = z p v (y, y) λ(y y, )dy + p v (y, y) λ(y y, )dy z z p v (y, y) λ(y y, )dy + p v (z, z) λ(y z, )dy z z p v (y, y) λ(y y, )dy + mn v (z, z) p λ(y z, )dy p z v (y, y) λ(y y, )dy + mn v (z, z) ln z ( 1 p F Y 1 (z z) 1 p F Y 1 (x z) Boundry Condton of the Equlbrum Strtegy for the ll-py ucton. α(x) = p v (y, y)f Y 1 (y y)dy p v (x, y)f Y 1 (y x)dy (14) mx v (x, x) mx v (x, x) (14) s consequence of ssumpton 1. Dervton of Exmple 1. p f Y 1 (y x)dy ) x φ1 (x, y ) = 4 (x + )(1 p F Y 1 (y x) p 3 F Y 1 3 (y x)) (x + y)(xy + 1) p3y4 x+4 + p 3(xy +4)y p y (x+4) 1 p F Y 1 (y x) p 3 F Y 1 3 (y x) [ y + xy (x + y)(xy + 1) + 1 x + x+ + p (xy +y) (x+) 13 They cn know the number of ther potentl opponents but not the number of ther ctve rvls. ] 18

20 x φ1 (x, y 3) = 1y (x + 4)(1 p F Y 1 (y x) p 3 F Y 1 3 (y x)) (x + y)(xy + ) p3y4 x+4 + p 3y (xy +4) p y (x+4) 1 p F Y 1 (y x) p 3 F Y 1 3 (y x) Computtons led to non-negtve dervtves. [ y 3 + xy + (x + y)(xy + ) x + 4 x+ + p y(xy+) (x+) ] References Goeree, J. K., E. Mslnd, S. Onderstl, nd J. L. Turner (5): How (not) to rse money for chrty, Journl of Poltcl Economy, 113(4), Hrstd, R., J. Kgel, nd D. Levn (199): Equlbrum bd functons for uctons wth n uncertn number of bdders, Economcs Letters, 33, Hrstd, R., A. Pekec, nd I. Tsetln (8): Informton Aggregton n Auctons wth n Unknown Number of Bdders, Gmes nd Economc Behvor, 6, Krshn, V. (): Aucton Theory. Acdemc Press. Krshn, V., nd J. Morgn (1997): An Anlyss of the Wr of Attrton nd the All-Py Aucton, Journl of Economc Theory, 7, Mtthews, S. (1987): Comprng Auctons for Rsk Averse Buyers: A Buyer s Pont of Vew, Econometrc, 55, McAfee, P., nd J. McMlln (1987): Auctons wth stochstc number of bdders, Journl of Economc Theory, 43, Mlgrom, P., nd R. Weber (198): A Theory of Auctons nd Compettve Bddng, Econometrc, 5, Pekec, A., nd I. Tsetln (8): Revenue Rnkng of Dscrmntory nd Unform Auctons wth n Unknown Number of Bdders, Mngement Scence, 54,