MATHEMATIC O OPERA ION RtSKAR I Vol  No I Februar  Primed in U
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MATHEMATIC O OPERA ION RtSKAR I Vol No I Februar Primed in U

SA OPTIMA AUCTIO DESIGN ROGE B MYERSO Northwestern Universitv Thi pape consider th proble face b a selle wh ha a singi objec t sel t on o severa possibl buyers whe th selle ha imperfec informatio abou ho muc th buyer migh b willin t pa fo th object T

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MATHEMATIC O OPERA ION RtSKAR I Vol No I Februar Primed in U




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MATHEMATIC O OPERA ION RtSKAR I Vol 5 No I Februar 198 Primed in U.S.A. OPTIMA AUCTIO DESIGN* ROGE B MYERSO Northwestern Universitv Thi pape consider th proble face b a selle wh ha a singi objec t sel t on o severa possibl buyers whe th selle ha imperfec informatio abou ho muc th buyer migh b willin t pa fo th object Th seller' proble i t desig a auctio gam whic ha a Nas equilibriu givin hi th highes possibl expecte utility Optima auction ar derive i this pape fo a wid clas o auctio desig problems Introduction Conside th proble face b someon wh ha a objec t sell an wh doe no

kno ho muc hi prospectiv buyer migh b willin t pa fo th object Thi selle woul lik t fin som auctio procedur whic ca giv hi th highes expecte revenu o utilit amon al th differen kind o auction know (progressiv auctions Dutc auctions seale bi auctions discriminator auctions etc.) I thi paper w wil construc suc optima auction fo a wid clas sellers auctio desig problems Althoug thes auction generall sel th objec a discoun belo wha th highes bidde i willin t pay an sometime the d no eve sel t highes bidder w shal prov tha n othe auctio mechanis ca giv highe expecte utilit t th seller analyz th

potentia performanc o differen kind o auctions w follo Vickre [11 an stud th auction a noncooperativ game wit imperfec informa tion (Se Harsany [3 fo mor o thi subject. Noncooperativ equilibri o specifi auction hav bee studie i severa papers suc a Griesmer Levitan an Shubi [1] Ortega-Reicher [7] Wilso [12] [13] Wilso [14 an Milgro [5 hav show asymptoti optimalit propertie fo sealed-bi auction a th numbe o bidder goe infinity Harri an Ravi [2 hav foun optima auction fo a clas o symmetri two-bidde auctio problems Independen wor o optima auction ha als bee don b Rile an Samuelso [8 an Maski an

Rile [4] A genera bibliograph o th literatur o competitiv biddin ha bee collecte b Rothkop an Star [10] Th genera pla o thi pape i a follows § present th basi assumption an notatio neede t describ th clas o auctio desig problem whic w wil study §3 w characteriz th se o feasibl auctio mechanism an sho ho t formulat th auctio desig proble a a mathematica optimizatio problem Tw lemmas neede t analyz an solv th auctio desig problem ar presente i §4 describe a clas o optima auction fo auctio desig problem satisfyin a regulator condition Thi solutio i the extende t th genera cas i §6 I §7 a exampl i

presente t sho th kind o counter-intuitiv auction whic ma b optima whe bidders valu estimate ar no stochasticall independent A fe concludin comment abou implementatio ar pu fort i §8 •Receive Januar 29 1979 revise Octobe 15 1979 AMS 1980 subject classification. Primar 90D45 Secondar 90C10 lAOR 1973 subject classification. Main Games OR/MS Index 1978 subject classification. Primary 23 games/grou decisions/noncooperative Key words. Auctions expecte revenue direc revelatio mechanisms +Th autho gratefull acknowledge helpfu conversation wit Pau Milgrom Michae Rothkopf an especiall Rober Wilson wh

suggeste this problem Thi pape wa writte whil th autho wa a visito th Zentru fu interdisriplinar Forschung Bielefeld Germany 0364-765X/81/0601/0058$01.2 Copyrigh - 1981 Th Institut o Managemen Science
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OPTIMA AUCTIO DESIG 5 Basi definition an assumptions T begin w mus develo ou basi defini tion an assumptions t describ th clas o auctio desig problem whic thi pape wil consider W assum tha ther i on selle wh ha a singl objec t sell face n bidders o potentia buyers numbere 1,2 ...,« W le N represen th se o bidders s tha ^={1,...,«} (2.1 wil us ( and t represen typica bidder i A'

Th seller' proble derive fro th fac tha h doe no kno ho muc th variou bidder ar willin t pa fo th object Tha is fo eac bidde / ther i som quantit ? whic i /' value estimate fo th object an whic represent th maximu amoun whic / woul b willin t pa fo th objec give hi curren informatio abou it shal assum tha th seller' uncertaint abou th valu estimat o bidde i ca describe b a continuou probabilit distributio ove a finit interval Specifi cally w le a represen th lowes possibl valu whic i migh assig t th object le bf represen th highes possibl valu whic / migh assig t th object an w le / .

[a;,fe,]-^R b th probabilit densit functio fo I' valu estimat f, W assum that - o < a < i < + oo /(?, > 0 Vr e [a,,ft,] an /( i a continuou functio o [a,,^,] /) [a, fc,]^[0,1] wil denot th cumulativ distributio functio correspondin t th density/(•) s tha 'ddsr (2-2 Thu Fj(ti) i th seller' assessmen o th probabilit tha bidde / ha a valu estimat o ? o less wil le T denot th se o al possibl combination o bidders valu estimates tha is r=[a,,6,] x[a„,ft„] (2.3 Fo an bidde / w le r_ denot th se o al possibl combination o valu estimate whic migh b hel b bidder othe tha i, s tha ^M (2-4 Unti §7 w wil

assum tha th valu estimate o th n bidder ar stochasticall independen rando variables Thus th join densit functio o T fo th vecto •= (tf, . . , , tj of individua valu estimate i /(o urn- jN course bidde / consider hi ow valu estimat t b a know quantity no a rando variable However w assum tha bidde / assesse th probabilit distribu tion fo th othe bidders valu estimate i th sam way a th selle does Tha is bot th selle an bidde i asses th join densit functio o r_ fo th vecto (_ = (?,,... f, I,(;+ ,...,( o value fo al bidder othe tha ; t b /-/('-,) n fjCj)- A Th seller' persona valu estimat fo th

object i h wer t kee i an no sel i an o th n bidders wil b denote b t^. W assum tha th selle ha n privat informatio abou th object s tha / i know t al th bidders
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ROCiF B MYfiRSO Ther ar tw genera reason wh on bidder' valu estimate ma b unknow t th selle an th othe bidders First th bidder' persona preference migh b unknow t th othe agent (fo example i th objec i a painting th other migh no kno ho muc h reall enjoy lookin a th painting) Second th bidde migh hav som specia informatio abou th intrinsi qualit o th objec (h migh kno i th paintin i a ol maste o a copy) W ma refe t

thes tw factor a preference uncertainty an quality uncertainty.^ Thi distinctio i ver important I ther ar onl preferenc uncertainties the informin bidde / abou bidde /' valu estimat shoul no caus / t revis hi valuation (Thi doe no mea tha / migh no revis hi biddin strateg i a auctio i h kne j valu estimate thi mean onl tha /' hones preference fo havin mone versu havin th objec shoul no change. However i ther ar qualit uncertainties the bidde / migh ten t revis hi valuatio o th objec afte learnin abou othe bidders valu estimates Tha is / learne tha t^ wa ver low suggestin that ha receive

discouragin informa tio abou th qualit o th object, the ; migh honestl revis downwar hi assessmen o ho muc h shoul b willin t pa fo tb object muc o th literatur o auction (se [11] fo example) onl th specia cas o pur preferenc uncertaint i considered I thi paper w shal conside a mor genera clas o problems allowin fo certai form o qualit uncertai a well Specifically w shal assum tha ther exis n revision effect functions e^ [a,,6,]-^( suc that i anothe bidde / learne tha t^ wa /' valu estimat fo th object the ; woul revis hi ow valuatio b ei(t-). Thus i bidde / learne tha t = (t^. . . . , tJ wa

th vecto o valu estimate initiall hel b th n bidders the / woul revis hi ow valuatio o th objec t Similarly w shal assum tha th selle woul reasses hi persona valuatio o th objec t /G. h learne tha t wa th vecto o valu estimate initiall bel b th bidders I th cas o pur preferenc uncertainty w woul simpl hav ej{tj) = 0 (T justif ou interpretatio o r a i's initia estimat o th valu o tb object w shoul assum tha thes revisio effect hav expected-valu zero s tha However thi assumptio i no actuall necessar fo an o th result i thi paper withou it onl th interpretatio o th f woul change, Feasibl auctio

medianisnis Give th densit function / an tb revisio effec function e an u a above th seller' proble i t selec a auctio mechanis t maximiz hi ow expecte utility W mus no develo th notatio describ th auctio mechanism whic h migh select T begin w shal restric ou attentio t a specia clas o auctio mechanisms tb direct revelation mecha- nisms. a direc revelatio mechanism th bidder simultaneousl an confidentiall announc thei valu estimate t th seller an th selle the determine wh get I a indebte t Pau Milgro fo pointin ou thi distinction
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OPTIMA AUCTIO DESIG 6 th objec an ho muc eac

bidde mus pay a som function o th vecto o announce valu estimate ? = (/,,... /„) Thus a direc revelatio mechanis i describe b a pai o outcome functions (p,x) (o th form •.T->W an x -.T-^R") suc that i t i th vecto o announce valu estimate the /?,(/ i th probabilit tha i get th objec an Xj(t) i th expecte amoun o mone whic bidde / mus pa t th seller (Notic tha w allo fo th possibilit tha a bidde migh hav t pa somethin eve i h doe no get th object. shal assum throughou thi pape tha th selle an th bidder ar ris neutra an hav additivel separabl utilit function fo mone an th objec bein sold Thus i

bidde / know tha hi valu estimat i /, the hi expecte utilit fro auctio mechanis describe b {p,x) i (t_,)dt_, (3.1 wher dt__i = dt^ . . . dti_idti^i . . . dt^. Similarly th expecte utilit fo th selle fro thi auctio mechanis i wher dt = dt^ . . . dt^. No ever pai o function {p,x) represent a feasibl auctio mechanism however Ther ar thre type o constraint whic mus b impose o {p,x). First sinc ther i onl on objec t b allocated th functio p mus satisf th followin probabilit conditions Pj{t) < 1 an pi{t) > 0 V e A' V G T. (3.3 jeN Second w assum tha th selle canno forc a bidde t participat i a

auctio whic offer hi les expecte utilit the h coul get o hi own I h di no participat i th auction th bidde coul no get th object bu als woul no pa an money s hi utilit payof woul b zero Thus t guarante tha th bidder wil participat i th auction th followin individual-rationality condition mus b satisfied U,{p,x,t,)>0, \/ieN, V,e[a,,Z.,] (3.4 Third w assum tha th selle coul no preven an bidde fro lyin abou hi valu estimate i th bidde expecte t gai fro lying Thu th revelatio mecha nis ca b implemente onl i n bidde eve expect t gai fro lying Tha is hones response mus for a Nas equilibriu i th

auctio game I bidde ; claime tha .? wa hi valu estimat whe / wa hi tru valu estimate the hi expecte utilit woul b wher ( _,,.5, = (r, . . . , r, ,,5,,? + , . . . , /„) Thus t guarante tha n bidde ha an incentiv t li abou hi valu estimate th followin incentive-compatibility condition mus b satisfied {v,{t)p^(t^,,s,)-x,{t^,,s,))f_,(t_,)dt_, (3,5
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ROGB B MYERSO sa tha (p,x) is feasible (o tha (p,x) represent a feasibl auctio mechanism if (3.3), (3.4), an (3.5 ar al satisfied Tha is i th selle plan t allocat th objec accordin t p an t deman monetar payment fro bidder accordin t x,

the th schem ca b implemented wit al bidder willin t participat honestly i an onl i (3.3)-(3.5 ar satisfied Thu far w hav onl considere direc revelatio mechanisms i whic th bidder ar suppose t honestl revea thei valu estimates However th selle coul desig othe kind o auctio games I a genera auction game, eac bidde ha som se o strateg option 9, an ther ar outcom function : e X X e, ~^M an x ; (>? X X e -»M" whic describe ho th allocatio o th objec an th bidders fee depen o th bidders strategies (Tha is i ^ = (S, . , . , 9^) wer th vecto o strategie use b th bidde i th auctio game then^,(^ woul b

th probabilit o / gettin th objec an Xi{9) woul b th expecte paymen fro / t th seller. auction mechanism i an suc auctio gam togethe wit a descriptio o th strategi plan whic th bidder ar expecte t us i playin th game Formally a strategic plan ca b represente b a functio ^, [a,, 6,]-^©, suc tha ^,(/, i th strateg whic / i expecte t us i th auctio gam i hi valu estimat i f, I thi genera notation ou direc revelatio mechanism ar simpl thos auctio mecha nism i whic 0 = [a,,6, an ^(r, = /, thi genera framework a feasibl auctio mechanis mus satisf constraint whic generaliz (3.3)-(3.5) Sinc ther i onl

on object th probabilitie ^,(^ mus nonnegativ an su t on o less fo an 9. Th auctio mechanis mus offe nonnegativ expecte utilit t eac bidder give an possibl valu estimate o els woul no participat i th auction Th strategi plan mus for a Nas equilibriu i th auctio game o els som bidde woul revis hi plans migh see tha proble o optima auctio desig mus b quit unmanageable becaus ther i n boun o th siz o complexit o th strateg space 0 whic th selle ma us i constructin th auctio game Th basi insigh whic enable u t solv auctio desig problem i tha ther i reall n los o generalit i considerin onl direc

revelatio mechanisms Thi follow fro th followin fact LEMM 1 (TH REVELATIO PRINCIPLE. Given any feasible auction mechanism, there exists an equivalent feasible direct revelation mechanism which gives to the seller and all bidders the same expected utilities as in the given mechanism. Thi revelatio principl ha bee prove i th mor genera contex o Bayesia collectiv choic problems a Theore 2 i [6] T se wh i i true suppos tha w ar give a feasibl auctio mechanis wit arbitrar strateg space 0, wit outcom function p an x, an wit strategi plan 9j, a above The conside th direc revelatio mechanis represente

b th functions/? T-^W an x:T^W suc tha Tha is i th direc revelatio mechanis (p,x), th selle firs ask eac bidde t announc hi type an the compute th strateg whic th bidde woul hav use accordin t th strategi plan i th give auctio mechanism an finall imple ment th outcome prescribe i th give auctio gam fo thes strategies Thus th direc revelatio mechanis {p,x) alway yield th sam outcome a th give auctio mechanism s al agent get th sam expecte utilitie i bot mechanisms An {p,x) mus satisf th incentive-compatibiht constraint (3.5), becaus th
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OPTIMA AUCTIO DESIG 6 Strategi plan forme

a equilibriu i th give feasibl mechanism (I an bidde coul gai b lyin t th selle i th revelatio game the h coul hav gaine b "lyin t himself o revisin hi strategi pla i th give mechanism, Thus (p, x) i feasible Usin tb revelatio principle w ma assume withou los o generality tha th selle onl consider auctio mechanism i th clas o feasibl direc revelatio mechanisms Tha is w ma hencefort identif th se o feasibl auctio mecha nism wit th se o al outcom function {p,x) whic satisf th constraint (3,3 throug (3,5) Th seller' auctio desig proble i t choos thes function p T -^U" an X : T-^W s a t maximiz

Uo(p,x) subjec t (3,3)-(3,5) Notic tha w hav no use (2,7 o (2.8 anywher i thi section Thu (3,3)-(3,5 characteriz th se o al feasibl auctio mechanism eve whe th bidder comput thei revise valuation t;,(/ usin function t), T->R, whic ar no o th specia additiv for (2,7) However i th nex thre sections t deriv a exphci solutio th proble o optima auctio design w shal hav t restric ou attentio t th clas o problem i whic (2.7 an (2.8 hold Anaiysi o tii probiem Give a auctio mechanis (p, x) w defin {t-ddt-, (4.1 fo an bidde / an an valu estimat ;, S Qi(p,tj) i th conditiona probabiht tha bidde i wil ge

th objec fro th auctio mechanis ip,x) give tha hi valu estimat i ?, Ou firs resul i a simplifie characterizatio o th feasibl auctio mechanisms LEMM 2 (p, x) is feasible if and only if the following conditions hold: ifs, then Qi(p,s,) < Q,{p,t^), y/iGN, V5,,r e[a,,fe,.] (4,2 mp,x,t,) = mp,x,a,) -( ['•Q,(p,s,)ds,, V G N, V/ s[a,,bA; (4,3 -'a Ui(p,x,ai)>0, y/ieN; (4,4 and Pj{t) < 1 and p,{t) > 0 V G iV V G T. (3.3 PROOF Usin (2.8) ou specia assumptio abou th for o t),(?) w ge Thus th incentive-compatibilit constrain (3.5 i equivalen t ,t,) > U,{p,x,s,) + (? - 5, Q>(p,s,), \/i G N, Vr,,^ G [a,,*,]

(4.6 Thu {p,x) i feasibl i an onl i (3.3) (3.4) an (4.6 hold W wil no sho tha (3.4 an (4.6 impl (4.2)-(4.4)
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ROGE B MYERSO Usin (4,6 twic (onc wit th role of,? an / switched) w ge The (4,2 follows whe ,s < r, Thes inequalitie ca b rewritte fo an S > 0 ,s^) < Q,{p,s, + 6)5 Sinc Qi{p,s,) i increasin i s^, i i Rieman integrable So ''Qi(P^s.)dSi= Ui{p,x,t,)- U,{p,x,a,), whic give u (4,3), course (4,4 follow directl fro (3,4), s al th condition i Lemm 2 follo fro feasibility No w mus sho tha th condition i Lemm 2 als impl (3,4 an (4,6), Sinc Qi{p,s^) > 0 b (3,3), (3,4 follow fro

(4,3 an (4,4) sho (4,6), suppos ,y < ;, the (4,2 an (4,3 giv us ^(p,x,t,)= U,ip,x,s,)+ {'•Q:{p,r,)dr, U,ip,x,s,)+f''Q,.(p,s,)dr, Similarly i s^ > ? the U,{p,x,t,)= U,(p,x,s,)- f''Q^(p,r,) ,s^)- (''Q,(p,s,)dr, Thu (4,6 follow fro (4,2 an (4,3) S th condition i Lemm 2 als impl feasibility Thi prove th lemma (p,x) represent a optima auctio i an onl i i maximize UQ(P,X) subjec (4.2)-(4,4 an (3,3), Ou nex lemm offer som simple condition fo optimality LEMM 3 Suppose that p: T-^W maximizes subject to the constraints (4,2 arui (3,3) Suppose also that x,{t) = p,(t)v,{t) - [''p,(t._,,s^)ds,, \/i EN, V G

r (4,8 Then (p,x) represents an optimal auction. PROOF Recallin (3,2), w ma writ th seller' objectiv functio a \p^(t)(^{t)-va{t))f{t)dt dt. (4,9 <= ,
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OPTIMA AUCTIO DESIG 6 But usin Lemm 2 w kno tha fo an feasibl {p,x): U,(p,x,a,)- U,{p,x,a) - \\ Ja, Fro (2.7 an (2.8 w ge f,( - M =' -' - ^,(',) (4-11 Substitutin (4.10 an (4.11 int (4.9 give us {VQ{t)f{t)dt- 2 U,{p,x,a). (4.12 •'T ieN th seller' proble i t maximiz (^4.12 subjec t th constraint (4.2), (4.3), (4.4) an (3.3 fro Lemm 2 I thi formulation x appear onl i th las ter o th objectiv functio an i th constraint (4.3 an (4.4)

Thes tw constraint ma rewritte a mp,X, a,) > 0 V G N, V? e [ a,,b,] th selle choose x accordin t (4.8), the h satisfie bot (4.3 an (4.4), an h get whic i th bes possibl valu fo thi ter i (4.12) Thu usin (4.8) w ca dro x fro th seller' proble entirely Furthermore th secon ter o th righ sid o (4.12 i a constant independen o ip,x). S th objectiv functio ca b simplifie t (4.7), an (4.2 an (3.3 ar th onl constraint lef t b satisfied Thi complete th proo o th lemma Equatio (4.12 als ha a importan implicatio whic i wort statin a a theore i it ow right COROLLAR (TH REVENUE-EQUIVALENC THEOREM) The

seller's expected utility from a feasible auction mechanism is completely determined by the probability function p and the numbers Ujip,x,aj) for all i. That is, once we know who gets the object in each possible situation (as specified by p) and how much expected utility each bidder would get if his value estimate were at its
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ROGE B MYERSO lowest possible level a-, then the seller's expected utility from the auction does not depend on the payment function x. Thus, for example, the seller must get the same expected utility from any two auction mechanisms which have the

properties that (1 ihe object always goes to the bidder with the highest value estimate above / and (2 every bidder would expect zero utility if his value estimate were at its lowest possible level. If the bidders are symmetric and all e = 0 and a^ = 0 then the Dutch auctions and progressive auctions studied in [11 both have these two properties, so Vickrey's equivalence results may be viewed as a corollary of our equation (4,12) However, we shall see that Vickrey's auctions are not in general optimal for the seller. 5. Optima auction i th regula case Wit a simpl regularit assumption w ca

comput optima auctio mechanism directl fro Lemm 3 ma sa tha ou proble i regular i th functio id) (?,)-—^ (5,1 a monoton strictl increasin functio o /, fo ever i i A^ Tha is th proble i regula i c,(5',) c,(/, wheneve a < ,?, < / < bj. (Recal tha w ar assumin (r, > 0 fo al t i [a,,/?,] s tha c,(?, i alway wel define an continuous. No conside a auctio mechanis i whic th selle keep th objec i tQ > max, gjv(c, (/,)) an h give i t th bidde wit th highes c,(/, otherwise I c,(?, = Cj{tj) = max^^g (cyi.(^/j) > tQ, the th selle ma brea th ti b givin t th lower-numbere player o b som othe arbitrar rule

(Tie wil onl happe wit probabilit zer i th regula case. Thus fo thi auctio mechanism Pi(t) > 0 implie c,(/, = Tmx{Cj(t^)) > tQ. (5,2 Fo al t i T, thi mechanis maximize th su subjec t th constraint tha Pj{t) < 1 an pi{t) > 0 V/ Thus/ maximize (4.7 subjec t th probabilit conditio (3,3) T chec tha i als satisfie (4.2 w nee t us regularity Suppos 5 < r, The c,(,s, < c,(/,) an s wheneve bidde i coul wi th objec b submittin a valu estimat o s^, h coul als wi i h change t r, Tha is/>,(?„,,,$, < p^{t^i,t>), fo al /_^, S Qi{p,ti), th probabilit o / winnin th objec give tha f i hi valu estimate i indee

a increasin functio o ?, a (4.2 requires Thus/ satisfie al th condition o Lemm complet th constnictio o ou optima auction w le x b a i (4,8) Thi formul ma b rewritte mor intuitively a follows Fo an vecto t _ o valu estimate fro bidder othe tha / le z,(t_, = inf{5 |c,(5, > tQ an c,(5, > c^(?,) Vy ^ /} (5.3
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OPTIMA AUCTIO DESIG 6 The z,(r_, i th infimu o al winnin bid fo / agains f_, s if5,.>z (/_,) 'Thi give u ri i t: - Z:(t :) i ? > z, Finally (4,8 become (5.6 ifp,( = o Tha is bidde / mus pa onl whe h get th object an the h pay t3,(f_, Zi(t_i)), th amoun whic th objec woul hav

bee wort t hi i h ha submitte hi lowes possibl winnin bid al th revisio effec function ar identicall zer (tha is e,(/, = 0) an i al bidder ar symmetri (a = a, fe, = bj, fi-) = fj{-)) an regular the w ge max c,- (?o),max/, j (5.7 Tha is ou optima auctio become a modifie Vickre auctio [11] i whic th selle himsel submit a bi equa t c," (to) (notic tha al c = Cj i thi symmetri case an regularit guarantee tha c i invertible an the sell th objec t th highes bidde a th secon highes price Thi conclusio onl holds however whe th bidder ar symmetri an th c,(- function ar strictl increasing Fo example

suppos t^ = 0 eac a = 0 Z> = 1(K) e,(^, = 0 and/(r; = 1/100 fo ever / an ever f betwee 0 an 100 The straightforwar computation giv u <:,(?, = 2/ - 100 whic i increasin i tj. S th selle shoul sel t th highes bidde th secon highes price excep tha h himsel shoul submi a bi o c, '(0 = 0 + 100/ = 50 B announcin a reservatio pric o 50 th selle risk a probabilit (1/2) o keepin th objec eve thoug som bidde i willin t pa mor tha t^ fo it bu th selle als increase hi expecte revenue becaus h ca comman a highe pric whe th objec i sold Thu th optima auctio ma no b expos efficient T se mor clearl wh thi ca

happen conside th exampl i th abov paragraph fo th cas whe n = 1 The th selle ha valu estimat ^ = 0 an th on bidde ha a valu estimat take fro a unifor distributio o [0,100] E pos efficienc woul requir tha th bidde mus alway ge th object a lon a hi valu estimat i positive Bu the th bidde woul neve admi t mor tha a infinitesima valu estimate sinc an positiv bi woul wi th object S th selle woul hav t expec zer revenu i h neve kep th object I fact th seller' optima poUc i t refus t sel th objec fo les tha 50 whic give hi expecte revenu 25 Mor generally whe th bidder ar asymmetric th optima auctio

ma some time eve sel t a bidde whos valu estimat i no th highest Fo example whe e,(/, = 0 an /(?,) 1/(6 - a, fo al ? betwee a an 6 (th genera uniform distributio cas wit n revisio effects w ge
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ROCJB B MYERSO whic i increasin i /, S i th optima auction th bidde wit th highes c,(/, wil ge th object I /? < b^, the / ma wi th objec eve i ; < t^, a lon a 2tj bj > 2tj bj. I effect th optima auctio discriminate agains bidder fo who th uppe bound o th valu estimate ar higher Thi discriminatio discourage suc bidder fro under-representin valu estimate clos t thei hig b, bounds Optima

auction i th genera case Withou regularity th auctio mecha nis propose i th precedin sectio woul no b feasible sinc i woul violat (4.2) T exten ou solutio t th genera case w nee som carefull chose definitions Th cumulativ distributio functio 7 : [a,,/j,]->[0,1 fo bidde / i continuou an strictl increasing sinc w assum tha th densit functio / i alway strictl positive Thu /^( ha a invers F ' :[0 l]^[a,,Z),] whic i als continuou an strictl increasing Fo eac bidde / w no defin fou function whic hav th uni interva [0. ] a thei domain First fo an q i [0,1] le (6,1 an le j%{r)dr. (6.2 Nex le

G,:[0,1]-> b th conve hul o th functio //,() i th notatio o Rockafella ([9 p 36] {w,/-|,r2 C[0,1 an wr + ( - w)r = ^} (6,3 Tha is G, ) i th highes conve functio o [0,1 suc tha Gi(q) < H;(q) fo ever a conve function G i continuousl differentiabl excep a countabl man points an it derivativ i monoton increasing W defin g :{0,1]^ s tha g^(q)=G;(q) (6.4 wheneve thi derivativ i defined an w exten gj(-) t al o [0,1 b right continuity defin c, [a,,6,]-> s tha (I i straightforwar t chec that i th regula cas whe c,(. i increasing w get Gi = ^, gi = ^; an c = c,.) Finally fo an vecto o valu estimate t,

le M(t) b th se o bidder fo who C;(tj) i maxima amon al bidder an i highe tha fg M(t) = I / / < c^i'i) = max (/)) (6,6
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OPTIMA AUCTIO DESIG 6 ca no stat ou mai result tha i a optima auction th objec shoul alway b sol t th bidde wit th highes c,(f,) provide thi i no les tha No nius w ma thin o c,(/, a the priority level fo bidde i whe hi valu estimat i if, i th seller' optima auction THEOREM Letp:T->W and x:T^W satisfy ifiGM{t), and f',s,)ds^ (6.8 for all i in N and t in T. Then {p,x) represents an optimal auction mechanism. PROOF First usin integratio b parts w deriv th

followin equations (6.9 Bu G i th conve hul o // o [0,1 an // i continuous s G,(0 = //^,(0 an G,(l = Hi{\). Thu th endpoin term i th las expressio abov ar zero Now recal th maximan (4.7 i Lemm 3 Usin (6.9 w get (6,10 2 i S P {H,{F,{t,))-G,{F,{t,)))dQ,{p,t,). No conside {p, x) a define i th theorem Observ tha p alway put al probabilit o bidder fo who (c,(?, - t^ i nonnegativ an maximal Thus fo any/ satisfyin (3.3) t)dt. cours p itsel doe satisf th probabilit conditio (3,3)
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ROGE B MYERSO Fo any/ whic satisfie (4,2 (tha is fo whic Qi(p, f, i a increasin functio o /,) w mus hav

f" (H,{F,(t,)) - G,{F,{tj)))dQj{p,t,) > 0 (6,12 .ft, a, sinc // > G, se tha p satisfie (4.2) observ firs tha c,(?, i a increasin functio o /, becaus / an g ar bot increasin functions Thusp,(r i increasin a a functio tj, fo an fixe r_, an s Qj(p, tj) i als a increasin functio o /, S p satisfie (4,2) Sinc G i th conve hul o H, w kno tha G mus b fla wheneve G < H; tha is i Gj(r) < Hj{r) the g;(r) = G/'(r) = 0. S i Hj(Fj{tj)) - G,(F,(?,) > 0 the c,(f, an Qj{p,t,) ar constan i som neighborhoo o tj. Thi implie tha Gj{Fj(tj)))dQj{p,tj) = 0 (6.13 Substitutin (6,11) (6,12) an (6.13 bac int (6,10) w ca

see that maximize (4,7 subjec t (4,2 an (3,3) Thi fact togethe wit Lemm 3 prove th theorem ge som practica interpretatio fo thes importan c functions conside th specia cas o n = 1 tha is suppos ther i onl on bidder The ou optima auctio becomes Xi{ti)='Pxiti) min(5 |c,(5, > t^] Tha is th selle shoul offe t sel th objec a th pric an h shoul kee th objec i th bidde i unwillin t pa thi price Thus i bidde / wer th onl bidder the th selle woul sel th objec t / i an onl i c,(?, wer greate tha o equa t ^o I othe words c,(f, i th highes leve o tQ, th seller' persona valu estimate suc tha th selle woul

sel th objec t / a a pric o ? o lower i al othe bidder wer removed Th ind^^deac assmoftion Throughou thi pape w hav assume tha th bidders valu estimate ar stochasticall independent Independenc i a stron assumption s w no conside a exampl t sho wha optima action ma loo lik whe valu estimate ar no independent Fo simplicity w conside a discret example Suppos ther ar tw bidders eac who ma hav a valu estimat o ? = 1 o ? = 10 fo th object Le u assum tha th join probabilit distributio fo valu estimate (?, ^2 is Pr(10,10 = Pr(100,100 = ^ Pr(10,100 = Pr(100,10 Obviousl th tw valu estimate ar no

independent Le u als assum tha ther ar n revisio effect (e = 0) an t^ = 0. No conside th followin auctio mechanism I bot bidder hav hig valu estimate (r = t2 = 100) the sel th objec t on o the fo pric 100 randomizin equall t determin whic bidde buy th object I on bidde ha a hig valu estimat (100 an th othe ha a lo valu estimat (10) the sel th objec t th
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OPTIMA AUCTIO DESIG 7 1 hig bidde fo 100 an charg th lo bidde 3 (bu giv hi nothing) I bot bidder hav lo valu estimate (10) the giv 1 unit o mone t on o them an give 5 unit o mone an th objec t th other agai choosin th recipien

o th objec a random Th outcom function (p,x) o thi auctio mechanis are ;7(10,100 = (0,1) p(m, 10 = (1,0) A:(100 100 = (5O,5O),A;(1O 10 = ( 10 - 10) x(10,100 = (30 100),A;(100 10 = (100,30) Thi ma see lik a ver strang auction bu i fac i i optimal I i straightfor war t chec tha honest i a Nas equilibriu i thi auctio game i tha neithe bidde ha an incentiv t misrepresen hi valu estimat i h expect th othe bidde t b honest Furthermore th objec i alway delivere t a bidde wh value mos highly an ye eac bidders expecte utilit fro thi auctio mechanis i zero whethe hi valu i hig o low S thi auctio

mechanis i feasibl an i allow th selle t exploi th entir valu o th objec fro th bidders Thu thi i optima auctio mechanism an i give th selle expecte revenu Uo(p,x) = 1(100 + i(130 - 1(130 + i(-20 = 70 se wh thi auctio mechanis work s well observ tha th selle i reall doin tw things First h i sellin th objec t on o th highes bidder a th highes bidders valu estimate Second i a bidde say hi valu estimat i equa t 10 the tha bidde i force t accep a side-be o th followin form "pa 3 i th othe bidder' valu i 1(X) ge 1 i th othe bidder' valu i 10. Thi side-be ha expecte valu 0 t a bidde whos valu

estimat i trul 10 sinc the th conditiona probabilit i 1/ tha th othe ha valu 10 an 2/ tha th othe ha valu 10 Bu i a bidde wer t li an clai t hav valu estimat 10 whe 10 wa hi tru valu estimate the thi side-be woul hav expecte valu |(-30 + ^(10 ^ fo hi (sinc h woul no asses conditiona probabilitie | an ^ respectivel fo th event tha hi competito ha valu estimat 10 an 10) Thi negativ expecte valu o th side-be fo a lyin bidde exactl counterbalance th temptatio t misrepresen i orde t bu th objec a a lowe price Thes side-bet wer no possibl i th independen case becaus eac bidders conditio probabilit

distributio ove th others valu estimate wa constant Bu i th genera non-independen case w ma expec tha thi side-be phenomeno wil commonl arise Tha is th selle ca exploi th ful valu o th objec b alway sellin t th highes bidde a th highes bidders valuation an the b settin u side-bet whic hav zer expecte valu i a bidde i hones bu hav negativ expecte valu i h lies I th side-bet ar carefull designed the ca counterbal anc th incentiv t li t bu th objec a a lowe price course w hav mad heav us o th risk-neutralit assumptio i thi analysis Fo risk-avers bidders th optima auction migh b somewha les

extreme Also th auctio gam suggeste i ou exampl ha a unfortunat secon equilibriu i whic bot bidder alway clai t b o th lo type althoug othe optima auctio mechanism ca b designe i whic th hones equilibriu i unique. (Fo example ^Eri Maski an Joh Rile hav recentl studie condition unde whic suc uniquenes ca b guaranteed
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RfXlE B MYERSO chang x to (1(X) 100) ;c(10,10 = (-15 -15) x(10,100 = (40.0) x(100,10 = (0,40) keepin p a above, On migh as whethe ther ar an optima auction fo ou exampl whic d no hav thi strang propert o sometime tellin th selle t pa mone t th bidders Th answe i

No i w ad th constrain tha th selle shoul neve pa mone t th bidder (tha is al x,( > 0) the n feasibl auctio mechanis give th selle expecte utilit highe tha 6 f T prov thi fact observ tha th auctio desig proble i a linea programmin proble whe th numbe o possibl valu estimate finite a i thi example Th objectiv functio i th proble i UQ{P, X), whic i linea i p an x. A i §2 th feasibilit constraint ar o thre types probabilit constraint {p,{t) > 0,^ipM < 1) individual-rationalit constraint {Ui{p,x,tj) > 0) an incentive-compatibilit constraint (tha Uj{p,x,ti) mus b greate tha o equa th utilit whic /

woul expec fro actin a i 5 wer hi valu estimat whe t- wa true) Al o thes constraint ar linea i p an x. S w get a linea programmin problem an fo ou exampl it optima valu i 70 wit th optima solutio show above Bu i w ad th constraint A;,( > 0 fo al / an t, the th optima valu drop t 661 fo thi example T attai thi "second-best valu o 66 wit nonnegativ x. th selle shoul kee th objec i ? = / = 10 an otherwis th selle shoul sel th objec t a hig bidde fo 1(X) Implementation A fe remark abou th implementabilit o ou optima auction shoul no b made Onc th / an e function hav bee specified th onl

computation necessar t implemen ou optima auctio ar t comput th c function an t evaluat (6,8) Bu thes ar al straightforwar one-dimensiona problems Th equilibriu strategie fo th bidder ar als eas t comput i ou optima auction sinc eac bidder' optima strateg i t simpl revea hi tru valu estimate term o sensitivit analysis notic tha (6,8 guarantee tha ou auctio mechanis {p,x) wil b feasible an yet th densities d no appea i (6,8), S ou optima auctio wil satisf th individual-rationaht an incentive-compatibilit con straint ((3.4 an (3.5) eve i th densit function ar misspecifie fro th poin o vie o th

bidders Howeve th revision-effec function e d appea i (6.8 (throug Vj), s i ther ar error i specifyin th e function the bidder ma hav incentiv t bi dishonestl i th auctio w compute general w mus recogniz tha a auctio desig proble mus b treate lik an proble o decision-makin unde uncertainty N auctio mechanis ca guarante t th selle th ful realizatio o hi object' valu unde al circumstances Thus th selle mus mak hi bes assessmen o th probabilitie an choos th auctio desig whic offer hi th highes expecte utility o average Th usua "garbage-in garbage-out warnin mus appl here a i al operation research

bu carefu us o model an sensitivit analysi shoul enabl a selle t improv hi averag revenue wit optimall designe auctions Reference [1 GrieSmer J H. Levitan R E an Shubik M (1967) Toward a Stud o Biddin Processes Par Four Game wit Unknow Costs Naval Res. Logist. Quart. 1 415-433 [2 Harris M an Raviv A (1978) Allocatio Mechanis an th Desig o Auction Workin Paper Graduat Schoo o Industria Administration Carnegie-Mello University Pittsburgh PA
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OPTIMA AUCTIO DESIG 7 [3 Harsanyi J C (1967-1968) Game wit Incomplet Informatio Playe b "Bayesian Players Management Sci. 1 159-189 320-334

486-502 [4 Maskin E an Riley J G (1980) Auctionin a Indivisibl Object Discussio Pape No 87D Kenned Schoo o Govemment Harvar University [5 Milgrom P R (1979) A Convergenc Theore fo Competitiv Biddin wit Differentia Informa tion Econometrica. 4 679-688 [6 Myerson FI B (1979) Incentiv Compatibilit an th Bargainin Problem Econometrica. 4 61-73 [7 Ortega-Reichert A (1968) Model fo Competitiv Biddin unde Uncertainty Technica Repor No Departmen o Operation Research Stanfor University [8 Riley J G an Samuelson W F (t appear) Optima Auctions American Economic Review. [9 Rockafellar R T (1970) Convex

Analysis. Princeto Universit Press Princeton [10 Rothkopf M H an Stark R M (1979) Competitiv Bidding a Comprehensiv Bibliography OR. 364-390 [11 Vickrey W (1961) Counterspeculation Auction an Competitiv Seale Tenders Journal of Finance. 8-37 [12 Wilson R B (1967) Competitiv Biddin wit Asymmetrica Information Management Sci. 1 A816-A820 [13 , (1969) Competitiv Biddin wit Disparat Information Management Sci. 1 446-448 [14 . (1977) A Biddin Mode o Perfec Competition Review of Economic Studies 4 511-518 GRADUAT SCHOO O MANAGEMENT NORTHWESTER UNIVERSITY 200 SHERIDA ROAD EVANSTON ILLINOI 6020