Procurement Bidding in First-Price and Second-Price. Sealed Bid Common Value Auctions

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1 Procurement Bddng n Frst-Prce and Second-Prce Sealed Bd Common Value Auctons Anders Lunander * Abstract Wthn the framework of the common value model, we examne the magntude of the dfference n expected outcome between frst-prce and second-prce sealed bd auctons. The study s lmted to two emprcal specfcatons of bdders sgnals: Webull and normal dstrbuton. The optmal bd functons and the expected procurer s cost under both aucton formats are derved. Smulatons are undertaken to analyze the mpact that random draws of sgnals have on the dfferences n outcome from the two aucton formats. Usng estmates from structural estmaton n prevous emprcal work on frst-prce aucton data, where Webull and normal dstrbutons of sgnals have been appled, the hypothetcal expected gan from swtchng from a frst-prce sealed bd aucton to a second-prce sealed bd aucton mechansm s computed. Keywords: Common value auctons, procurement, Vckrey aucton JEL Classfcaton: D44 (auctons) * I am ndebted to Sören Blomqust, Matas Eklöf, Andreas Westermark and Anders Ågren for ther valuable comments. Ths work has been funded by grants from the Swedsh Transport & Communcatons Research Board (Kommunkatonsforsknngsberednngen) and the Swedsh Competton Authorty (Konkurrensverket).They are gratefully acknowledged.

2 . Introducton Government procurement, whch represents an mportant share of total government expendture (typcally 0 5% of GDP), s regulated by nternatonally agreed trade rules under the World Trade Organzaton. Members of the European Communty (EC) must also follow EC procurement drectves. A cautous nterpretaton of these agreements and rules s that they restrct bddng to a sealed-bd procedure and do not allow the procurer to use open or sequental bddng mechansms, such as the Englsh aucton. The domnant bddng mechansm overall seems to be the frst-prce sealed bd aucton (frst-prce aucton henceforth). From aucton theory, however, we know that n some envronments the second-prce sealed bd aucton (second-prce aucton henceforth) s at least as good or even better than the frst-prce aucton n terms of the procurer s expected cost. Mlgrom and Weber (98) show that f an afflated aucton model, such as the symmetrc common value model, s used to descrbe a partcular bddng envronment, then the second-prce aucton yelds a lower expected procurement cost than the frst-prce aucton mechansm. In vew of ths theoretcal result, the nfrequent use of the second-prce aucton procedure n procurement auctons s somewhat puzzlng. It s true that f we remove the assumptons of rskneutral bdders (frms), symmetry, and non-cooperatve behavor, the frst-prce aucton may be preferable to the second-prce aucton, but t s unlkely that ths alone explans why secondprce auctons are rare. Rothkopf et al. (990) consder varous possble reasons for the scarcty of the second-prce aucton, some of whch they fnd to be plausble. One such reason s that a bdder may be reluctant to bd hs reservaton cost f he fears that the procurer n a second-prce aucton wll cheat by puttng n magnary bds to force the prce below the Artcle XIII: 3 of the Agreement on Government Procurement (WTO) states that the opportuntes that may be gven to tenderers to correct unntentonal errors of form between the openng of tenders and the awardng of the contract shall not be permtted to gve rse to any dscrmnatory practce, and that all tenders solcted... shall be receved and opened under procedures and condtons guaranteeng the regularty of the openngs.

3 3 second-lowest bd. Another s that by bddng ts true cost a frm reveals valuable nformaton whch t mght be mportant to keep secret f the frm antcpates that, n the event of wnnng the aucton, t wll have to negotate wth subcontractors, labor unons, or fnancal nsttutons. The authors fnd the argument that the rarty of the second-prce aucton could be due to nerta unpersuasve. Although nsttutons are slow to mplement nnovatons, t seems unlkely that Vckrey s result from the sxtes could have passed unnotced, wthout any measures beng taken. Assumng that nsttutons have long been famlar wth the revenue-rankng of frst- and second-prce auctons, ths paper sets out to explan part of the passvty of nsttutons n regard to adoptng the second-prce aucton by examnng the cardnal dfference between the aucton formats. It s mportant to consder the magntude of the expected gan from swtchng aucton procedures and not focus solely on the ordnal rankng of aucton procedures. Although a swtch from a frst-prce to a second-prce aucton yelds an expected lower procurement cost, ths gan may be regarded as too small to cover the mplementaton cost. The benefts have to be substantal n order to gve up a well-tested and relable procurement method for an untred mechansm. A number of theoretcal and emprcal studes have focused on bddng behavor n the symmetrc frst-prce common value aucton, gven dfferent emprcal specfcatons of the dstrbuton of bdders sgnals and the dstrbuton of the true cost [e.g. Rothkopf (969), Smley (979), Thel (988), Levn and Smth (99), Paarsch (99), Wlson (99)]. Lttle work has, however, been devoted to dervng the explct bd functons for the second-prce aucton and comparng the predcted outcome wth that from the frst-prce aucton, gven the same underlyng structure of cost and sgnals. In ths paper we map the general symmetrc bd functons of the frst-prce and second-prce auctons onto two dfferent assumptons regardng the dstrbuton of the bdders sgnals: a

4 4 normal dstrbuton and a Webull dstrbuton. These are the only dstrbutons that to our knowledge have been appled to and estmated for frst-prce aucton data wth farly good ft (Paarsch, 99). Gven the assumptons of the common value aucton model, we can use these structural estmates to derve hypothetcally the predcted gan of usng the second-prce aucton nstead. The paper s organzed n the followng manner: Secton presents the bddng model for the frst-prce and second-prce aucton. In Secton 3 and 4 the theoretcal models are mapped onto the emprcal specfcatons, and the optmal bd functons and the expected procurer s cost are derved. In Secton 5 we use estmates from the structural estmaton n Paarsch (99) on frstprce aucton data to llustrate the procurer s expected cost reducton by swtchng from a frstprce aucton to a second-prce aucton procedure. Fnally, Secton 6 summarzes and concludes the paper.. The Model It s assumed that there are n rsk-neutral bdders bddng for a partcular contract, where the cost of carryng out the contract, c, s dentcal but unknown to all bdders pror to bddng. The bdder who submts the lowest bd s selected as the wnner and awarded the contract. Before the aucton, each bdder receves a prvate sgnal, z, concernng the cost of the contract, whch he uses to form an unbased estmate of c, that s E(c z ). No bdder knows the estmate of any other bdder. The bdders sgnals, z I, are postvely correlated (afflated) wth a cumulatve dstrbuton functon. F( z c). The bdders have pror belefs about the true cost, c, whch s characterzed by the cumulatve dstrbuton functon, G( c ). The number of bdders and the dstrbuton functons of z and c are assumed to be common knowledge. In dervng the equlbrum strateges n both auctons, I focus on bdder.

5 5. Frst-Prce Sealed Bd Aucton In the frst-prce aucton, where the wnnng bdder s pad the amount of hs bd to perform the task, the bdder, gven hs sgnal z, sets hs bd b to maxmze the expected payoff where n ( ( )) ( ) ( ) ( ) ( ) Π b, z b c F β b c g c z dc () ( ) g c z ( ) ( ) ( ) ( ) f z c g c f z c g c dc () s the posteror dstrbuton of c; β ( b ) s the nverse of the equlbrum strategy functon of ( ( )) the n other bdders; and ( ) F β b c n s the probablty that bdder wns the contract. Substtutng () nto (), the maxmzaton problem can be wrtten as n ( ( )) ( ) ( ) ( ) ( ) max b c F β b c f z c g c dc (3) b whch yelds the frst-order condton n * ( ( β ( ) )) ( ) ( ) F b c f z c g c dc n ( ( )) ( β ( ) ) * ( ) ( b c)( n ) F ( b ) c f b c d β b * * * β f ( z c) g( c) dc 0. db (4) Imposng symmetry among bdders, b ( ) z * β and rearrangng the terms gves the frst-order dfferental equaton z u p( u) du z p( t ) dt β( z) e e q( u) du + C (5) where the constant, C, s determned by the approprate boundary condtons.. Second-Prce Sealed Bd Aucton 3 The dervaton n ths secton s based on Paarsch (99). Se also Wlson (977). 3 The dervaton s based on Mlgrom and Weber (98).

6 6 In the second-prce aucton, where the wnnng bdder s pad the amount of the second-lowest bd to perform the task, the bdder s expected payoff when ts sgnal s z and t bds an amount b s ( β( ) ( )) Y ( ) Π( b, z) y c z, y f y z dy β ( b) β ( b) ( (, ) (, )) Y ( ) c y y c z y f y z dy (6) where c( z, y) E[ C Z z, Y y], whch s the expectaton of the cost to bdder when the sgnal receved by hm s z and the lowest sgnal among the other bdders s y. The second-last term n equaton (6) denotes the condtonal probablty dstrbuton functon of the frst-order statstcs, that s, the smallest order statstcs among the sgnals ( Z j j ). Snce c s ncreasng n ts frst arguments [for all y < z, c( z, y) c( y, y) > 0 and for all y > z, c( z, y) c( y, y) < 0 ] the ntegral s maxmzed by choosng b so that β ( b) b β( z). Thus, symmetrc equlbrum n a second-prce aucton s defned as ( ) c( z z) β z z, or equvalently by choosng,. (7) Mlgrom (98) shows that ths can be wrtten as β( z) [ n ] ( ) [ n ] ( ) ( ) ( ) ( ) c n f z c F z g c dc ( ) ( ). (8) ( n ) f z c F z g c dc

7 7 3. Bddng under Webull Dstrbuted Sgnals The bdder s sgnal s drawn from a Webull dstrbuton, wth the probablty densty functon ( ) γ γ γ z f z c γ γ z e 0 < γ, γ and 0 z ( ) g c [ 0, ] U k l / c (9) where E[ z ] ( + γ ) Γ / c mplyng that γ c The dsperson of f ( z c) s related to the nverse of γ, that s, the hgher the values of γ, the more concentrated the dstrbuton. Further, t s assumed that g( c) γ. / c. (0) Smley (979) shows that f g( c ) s proportonal to / c l where l s a real number, then the symmetrc equlbrum bd functon s proportonal to the sgnal; that s, β( z ) ρ ( > ) s a constant of proportonalty. Substtutng (9) and (0) nto Bayes s rule, we obtan ρz, where [ ] E c z ( ) ( ) cf z c g c dc 0 z f z c g c dc 0 ( ) ( ) () Equaton () shows that the posteror expected value s just the sgnal tself, that s, nonnformatve. The bdder s pror expectaton about c does not shft the posteror expected value of c away from ts sgnal z. 4 4 Makng the assumpton that g( c) / c gves E[ c z ] Γ( + / γ ) Γ( / γ ) z

8 8 3. Optmal Bd Functons Substtutng the dstrbutonal specfcatons nto the general bd functons for the frst-prce and second-prce sealed bd auctons, (5) and (8), the optmal bd functons are computed as (frst-prce aucton) β( z ) γ γ ( n ) ( n ) n / γ z () (second-prce aucton) β ( z ) n γ / γ nd z γ +. (3) Fgures a and b plot the optmal bd functons aganst the number of bdders, gven γ 5, and aganst dfferent values of γ, gven n 8. The fgures llustrate two man predctons wthn the symmetrc common value aucton model: () the optmal bd n the frst-prce aucton ntally decreases wth the number of bdders due to competton, but then, as the fear of the wnner s curse domnates the compettve effect, the bd ncreases wth the number of bdders; () an ncreased dsperson of sgnals (lower value of γ ) gves rse to a hgher adverse selecton bas, whch causes the bdder to adjust hs bd upwards. Fgure a: Optmal Bd versus Number of Fgure b: Optmal Bd versus Dsperson Bdders (z,γ 5) of Sgnals (z, n5),9,8,8,6 Frst-prce aucton,7,6 Frst-prce aucton,5,4 Second-prce aucton,4,,3, Second-prce aucton, 0,8 Number of bdders Value of g

9 9 3. Expected and Smulated Procurement Cost In order to rank the two aucton formats n terms of lowest procurement cost, the expected wnnng bd E[ w ] n the frst-prce aucton s compared wth the expected second-lowest bd [ ( : ) ] E b n n the second-prce aucton. Makng use of ( : n) ( ( : n) ) f ( z( : n) ) where f ( z( : n) ) h b dz db ( : n) n! ( )!( n ) { ( )} ( ) n { } ( ) F z F z f z procurer s expected payment under both aucton regmes s derved as, the (frst-prce aucton) E[ w] γ γ ( n ) ( n ) c, (4) (second-prce aucton) E[ b( : n) ] n / γ γ γ n + ( n ) n c n. (5) γ / γ / In table the dfference between (4) and (5) s computed for varous number of bdders and two szes of the dsperson of sgnals. By lettng c, the procurer s expected payment equals the constant of proportonalty. Table : The procurer s expected cost (c) γ 5 γ 0 () () (3) (4) (5) (6) (7) n ( ) E w E ( b ) ( : n) Dff. ( ) E w E ( b ) ( : n) Dff.,5,08 0,69,,040 0,07 3,,045 0,067,053,0 0,03 4,07,03 0,04,034,05 0,09 5,053,03 0,09,06,0 0,04 0,03,0 0,0,0,005 0,006 5,04,007 0,007,007,003 0,004 0,0,005 0,005,005,003 0,003 The table shows that the dfference between the constant of proportonalty of the bd the procurer pays dmnshes rapdly as the number of bdders ncreases. Increasng the dsperson

10 0 of sgnals,.e., lowerng γ, drves up the dfference n the procurer s expected payment between the aucton formats, but the magntude of ths dfference s reduced to 3% when there are more than four bdders. To complete the pcture of the dfferences n outcome between the two aucton formats, a number of smulatons are carred out. For a gven number of n bdders, n sgnals z are randomly drawn from the Webull dstrbuton and evaluated n the optmal bd functons of the two aucton formats (equatons and 3). For each draw of n sgnals, we then compute the dfference between the lowest bd n the frst-prce aucton and the second-lowest bd n the second-prce aucton, that s β ( z(: n) ) β nd ( z( : n) ). (6) st Settng E[ c ] throughout the smulatons mples that γ s determned for a gven value of γ. I focus on two cases of dsperson of sgnals, ( ) and ( ) γ 5 γ γ 0 γ The number of bdders ranges from to 0. Usng the specfcatons above, each aucton s smulated tmes and the mean, the medan and a 90% confdence nterval are computed. The results from the smulatons are presented n fgures a and b. The computed mean values of the smulated dfferences for a gven number of bdders reflect the results n column four and column seven n table. Postve values show that the procurer s cost s hgher under the frst-prce aucton. The medan value of the dfference n procurement cost s located above the mean value, ndcatng that the probablty densty of dfferences s concentrated well above zero.

11 Fgures a-b: Smulated Dfferences n Procurement Costs under Webull Dstrbuted Sgnals a) γ 0 b) γ 5 0,5 0,5 0, 0,4 0,5 0, 0,05 0-0,05-0, Medan Upper 90% bound Mean Dfference n procurement cost 0,3 0, 0, 0-0, -0, 4 Upper 90% bound Medan Mean ,5-0, Lower 90% bound Number of bdders -0,3-0,4 Lower 90% bound Number of bdders Table llustrates the share of postve outcomes n the smulatons, that s, the cases when the second-lowest bd n the second-prce aucton s lower than the lowest bd n the frst-prce aucton. The decreasng rato of postve outcomes supports our prevous fndng that the second-prce aucton s most advantageous wth few bdders. Table : The probablty of obtanng a lower cost under the second-prce aucton than under the frst-prce aucton for varous number of bdders. Number of bdders β z > β z 76% 69% 66% 64% 6% 6% 6% ( (: n) ) nd ( ( : n) ) st Removng 5% of the smulated dfferences n procurement costs n each tal of the dstrbuton for every number of bdders, gves a 90% confdence nterval wth an upper and a lower bound.

12 4. Bddng under Normally Dstrbuted Sgnals An alternatve approach to dervng a computatonally tractable soluton for β( z ) s to assume that the sgnals are normally dstrbuted wth mean c and varance σ, and that the bdders have dentcal and dffuse pror dstrbuton for the unknown cost c. Formally, the specfcatons of the dstrbutons are denoted (7) z c f ( z c) σ φ σ c [ ] ( ) U, g c k. c Agan, usng Bayes s rule, we see that the bdder s pror expectaton about c does not shft the posteror expected value of c away from ts sgnal z. z c σ c e kdc E[ c z ] σ π z z c. (8) σ e kdc σ π 4. Optmal Bd Functons Gven the assumptons above and, addtonally, the assumpton that the estmaton errors are normal and ndependent of the true cost, then Levn and Smth (99), n a comment on Thel (988), show that the optmal bd functon n the frst-prce aucton can be derved as where ( ) ( ) n exp ( : n) b β z z α σ + γ zξ σ (9) ξ n ( ) φ( ) ( :n) ( ) un Φ u u du < 0 (0)

13 3 and α n n ( Φ( )) φ( ) nu u u du < 0. () ξ ( : n) Levn and Smth show that ndvdual ratonalty mples that γ 0. Note that equaton (0) s the expectaton of the frst order statstc n a sample of sze n of the standard normal dstrbuton. The γ s a parameter ndexng the famly of Nash equlbra, where one (γ 0 ) s lnear n the estmate. In the case (γ 0 ), equaton (9) turns nto the lnear form ( z) z α σ b β. () n The correspondng optmal bd functon n the second-prce aucton can be derved as 5 β nd ( z ) α σ σ z n + ξ (: n). (3) In fgure 3 the optmal bds n both aucton formats are plotted aganst the number of bdders. Fgure 3: Optmal Bd versus Number of Bdders (z0, σ ),5 Frst-prce aucton Optmal bd,5 Second-prce aucton 0, Number of bdders Agan, the optmal bd n the frst-prce aucton ntally decreases wth the number of bdders due to competton, but then, as the fear of the wnner s curse domnates the compettve effect, 5 See appendx for proof.

14 4 the bd ncreases wth the number of bdders. From () and (3), t s also obvous that the optmal bd rses as the dsperson of sgnals ncreases Expected and Smulated Procurement Cost As n the prevous secton, the dfference n procurement cost between the frst-prce sealed bd and the second-prce sealed bd aucton s examned by comparng the expected wnnng bd n the frst-prce aucton wth the expected second-lowest bd n the second-prce aucton. The expected wnnng bd n the frst-prce aucton and the second-lowest bd n the second-prce aucton are where (frst-prce aucton) E[ w] c + σ ( ξ n αn ) [ ] (second-prce aucton) E b( : n) (: ) (4) c + σ + ξ( : n) α n (5) ξ ( ) ( ) ( : n) n ( ) ( ) ( ) ξ ( : n ) n n x F x F x f x dx (6) and ξ (: n ) α n 0, + ξ( ) α : n n 0 for n. ξ ( : n) Thus, the dfference between the two expected values can be expressed as a postve lnear functon n the dsperson of sgnals where the coeffcent s determned by the number of bdders. In table 3 we have computed the dfference between (4) and (5) for varous number of bdders, where c 0 and σ. 6 Note that α > /ξ (:n) for n >

15 5 For small numbers of bdders the dfference s sgnfcant, but the dfference decreases rapdly as the number of bdders ncreases. Table 3: The procurer s expected cost (c0 and σ ) n ( ) E w E ( b ) ( : n) Dff.,08 0,564 0, ,66 0,36 0, ,478 0,39 0,39 5 0,385 0,93 0,9 0 0,4 0, 0,3 5 0,73 0,085 0, ,49 0,07 0,078 Agan, we complete the pcture of the dfference n outcome by randomzng the sgnals and evaluatng the same set of sgnals n the optmal bd functons () and (3). For each gven number of bdders, the correspondng number of sgnals are randomly drawn from the standardzed normal dstrbuton. In each smulated aucton wth the same number of bdders, the lowest bd n the frst-prce aucton s compared wth the second-lowest bd n the secondprce aucton. Every aucton for a gven number of bdders s smulated tmes. Fgure 4: Smulated Dfferences n Procurement Costs under Normally Dstrbuted Sgnals,5 Upper 90% bound 0,5 Medan 0 Mean ,5 Lower 90% bound - -,5 Number of bdders

16 6 Fgure 4 llustrates the mean, the medan and a 90% confdence nterval from the smulatons. The procurer s expected dfference n cost between the aucton formats converges to zero as the number of bdders ncreases. 7 For each number of bdders, we also examne the procurer s probablty of gettng a lower cost wth the second-prce aucton by countng the number of postve outcomes from the smulatons. Table 4: The probablty of obtanng a lower cost under the second-prce aucton than under the frst-prce aucton for varous number of bdders Number of bdders β z > β z 79% 74% 7% 69% 68% 67% 67% ( (: n) ) nd ( ( : n) ) st Agan, where there are very few bdder the second-prce aucton s very lkely to generate a lower procurement cost than the frst-prce aucton does, but wth ncreased competton the superorty of the second-prce aucton decreases somewhat. 5. A Numercal Example Paarsch (99) examnes whether bddng behavor n tree-plantng contract auctons held n the provnce of Brtsh Columba, Canada, can be explaned by ether the prvate values or the common value model. Usng varous emprcal specfcatons, Paarsch rejects the prvate values model and fnds evdence consstent wth ratonal behavor wthn the common value model, where both the Webull dstrbuton and the normal dstrbuton specfcaton of sgnals seem to ft the data farly well. Gven that bdders actons are consstent wth theory when facng a second-prce sealed bd aucton, we may use the estmated structural parameters of the dstrbuton of the sgnals n 7 Gven the large number of smulatons, one may regard ( z( : n ) ) and β st ( z( : n )) as ndependent random β nd ( ) varables. The dfference n expected value then breaks down to σ ξ ( ) / ξ ( ) ξ : n : n ( : n), whch converges to zero as n.

17 7 Paarsch (99) to llustrate hypothetcally what could be predcted to have been ganed by the procurer by usng the second-prce sealed bd aucton nstead of the frst-prce sealed bd aucton. The maxmum lkelhood estmates of the structural parameters for the Webull dstrbuton and the normal dstrbuton c,γ and σ n Paarsch are reproduced n table 5. The estmates are based on the wnnng bds from 44 auctons. Table 5: Maxmum Lkelhood Estmates of Structural Parameters a Parameter Dstrbuton Webull Normal c 0,4 0,0 γ 3,6 σ 0,50 n Mean Mn. Max a The parameters are for costs measured n dollars per tree planted Makng use of equatons (4), (5), (), and (3) the dfference n expected costs between the frst-prce and the second-prce aucton for varous numbers of bdders s presented n fgure 5. Gven the Webull dstrbuton, the estmated gan from a swtch s relatvely low, about 4% for auctons wth few bdders and dmnshng rapdly as the number of bdders ncreases.

18 8 Fgure 5: Expected Cost under the Frst-Prce and Second-Prce Auctons (dollars per tree planted) Webull dstrbuton Normal dstrbuton 0,35 0,4 Frst-prce aucton 0,3 Frst-prce aucton Second-prce aucton 0,35 Second-prce aucton 0,3 0,5 0,5 0, 0, Number of bdders Number of bdders The dfference n expected costs under the normal dstrbuton exhbts a smlar pattern, although the expected gans from a swtch of aucton format are somewhat hgher. 6. Summary Ths analyss of the magntude of the expected dfference between a frst-prce sealed bd aucton and a second-prce sealed bd aucton s based on only two dstrbutons of sgnals and costs, whch of course lmts the general value of ths study. We have shown that the superorty of the second-prce aucton over the frst-prce aucton n terms of expected procurer s cost s related to changes n the number of bdders and the dsperson of sgnals. Gven our dstrbutons of sgnals and cost, the second-prce aucton s most benefcal when there are very few bdders or when there s hgh dsperson of bdders sgnals. The smulatons also ndcate that the probablty of gettng a better outcome when usng the second-prce aucton, gven a moderate number of bdders (>5), s about two-thrds. Makng use of the results from Paarsch (99), who fnds the common value model consstent wth observed behavor, we estmate that the predcted hypothetcal gan of swtchng to the second-prce aucton may be sgnfcant for a small number of bdders (<5) but s relatvely low for a larger number of bdders.

19 9 The frst-prce sealed bd aucton s a well-establshed mechansm n governmental procurement bddng. Ths mechansm was surely consdered when the natonal acts and the nternatonal agreements on publc procurement were formulated. Changng a procedure that s regarded as more or less formal requres resources. If the estmates of the costs and benefts of alterng the procedure ndcate that there s very lttle gan from swtchng methods, an nsttuton may very well refran from mplementng the second-prce aucton. Ths argument seems especally to hold when the procurer expects that the number of potental bdders wll be relatvely large and that the dsperson of ther sgnals wll be small.

20 0 References Levn, Dan; Smth, James L; (99) Some Evdence on the Wnner s Curse: Comment Amercan Economc Revew, Vol. 8, No., Mlgrom, Paul R; (98) Ratonal Expectatons, Informaton Acquston, and Compettve Bddng Econometrca, Vol. 49, No. 4, Mlgrom, Paul R; Weber, Robert J; (98) A Theory of Auctons and Compettve Bddng Econometrca, Vol. 50, No. 5, 089 Paarsch, Harry J; (99) Decdng Between the Common and Prvate Value Paradgms n Emprcal Models of Auctons Journal of Econometrcs, Vol. 5, 9 5 Rothkopf, Mchael H; (969) A Model of Ratonal Compettve Bddng Management Scence Vol. 5, No. 7, Rothkopf, Mchael H; Tesberg, Thomas J; Kahn, Edward P; (990) Why are Vckrey Auctons Rare, Journal of Poltcal Economy, Vol. 98, Smley, Albert, K; (979) Compettve Bddng under Uncertanty: The Case of Offshore Ol Cambrdge: Ballnger Thel, Stuart E; (988) Some Evdence on the Wnner s Curse Amercan Economc Revew, Vol. 78, No. 5, Wlson, Robert; (977) A Bddng Model of Perfect Competton Revew of Economc Studes, 44, 5 58 Wlson, Robert; (99) Strategc Analyss of Auctons n Handbook of Game Theory, Volume, edted by R.J. Aumann and S. Hart, North-Holland

21 Appendx Optmal bd functon wth normally dstrbuted sgnals 8 (In what follows, the bdder s sgnal s denoted x) ( ) β x [ n ] ( ) [ n ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) n cf x c F x c g c dc n f x c F x c g c dc Restrctons that guarantee the exstence of lnear strateges: ) Each bdder s pror dstrbuton of value s dffuse: g(c) s constant for all c. ) Estmaton errors are statstcally ndependent of the contract s true cost: ( ) ( ) ( ) F' X c c f X c c f X c. 3) Each bdder s estmate of the cost s unbased: E( X ) c. Note that x x c x c ( ) ( τ ) τ ( ) ( ) ( ) F x c f c d f t c dt f t dt F x c where t τ c. The thrd equalty s due to restrcton. β ( x) k( n ) ( x c x) ( ( )) [ ( )] σ σ f x c F x c dc σ ( ) [ ( )] ( ) σ ( ) n k n f x c F x c σ dc n Let z β ( x) x σ c and dx dz [ ] ( / σ ) ( ) ( ) n / σ ( ) [ ( )] n x / σ f ( z) [ F( z) ] dz n / σ f ( z) [ F( z) ] dz n ( ) [ ( )] x n / σ ( ) [ ( )] ( ) ( ) σ. Then, f z σf x c and dc σdz. n x z f z F z dz f z F z dz zf z F z dz f z F z dz n [ ] [ ] ( ) ( ) zf z F z dz n / σ f z F z dz ( ) ( ) 8 The dervaton s based on the dervaton of the correspondng bd functon under the frst-prce aucton (see Levn and Smth, 99).

22 [ ] ( ) ( ) ( ) ( ) ( ) [ ( )] n σn n zf z F z dz x n n n f z F z dz Assume that x s dstrbuted normally, wth mean c and varance σ. β ( z) x x x + x + [ ] ( ) ( ) n zf z F z dz [ ] ( ) ( ) n / σ f z F z dz [ ] ( ) ( ) ( ) n σn n zf z F z dz [ ] ( ) ( ) ( ) n n n f z F z dz [ ] ( ) ( ) ( ) n σn / n z f ( z) [ F( z) ] dz ξ x ασ + σ ξ n σn n zf z F z dz (: n) ξ (: n) (: n) where ξ (: n ) < 0 Q.E.D.