OPTIMAL RESERVE PRICE FOR THE GENERALIZED SECOND-PRICE AUCTION IN SPONSORED SEARCH ADVERTISING

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1 o et.: Optm Reseve Pce fo the Geezed Secod-Pce Aucto Sposoed Sech Advetsg OPTIMAL RSRV PRIC OR TH GNRALIZD SCOND-PRIC AUCTION IN SPONSORD SARCH ADVRTISING Bchu o Deptmet of Mgemet, Schoo of Busess Log Isd Uvesty t C.W. Post, Boove, NY 548, USA bchu.o@u.edu We Yg Deptmet of Mgemet, Schoo of Busess Log Isd Uvesty t C.W. Post, Boove, NY 548, USA we.yg@u.edu Ju L Schoo of coomcs d Mgemet Southwest Jotog Uvesty, ChegDu 63 Ch u@home.swtu.edu.c ABSTRACT I sposoed sech dvetsg dvetses bd d s petg to eywod fom sech eges. Ths ppe pesets pcg mode fo sposoed sech dvetsg dymc fmewo. It focuses o the geezed secod-pce ucto, whch s wdey used by mo sech eges cudg Googe d Yhoo!. Ue the sseto the tetue tht the umbe of dvetses d the umbe of d s hve o mpct o the seecto of eseve pce, ou esut s otceby dffeet. We show tht the optm eseve pce s ffected by both fctos. I ptcu, ude set of md codtos, the optm eseve pce s equ to the epected vue of some ode sttstc of dvetse s pe-cc vues. Smutos bsed o the cotuous-tme bddg pocess cofm ou theoetc fdgs. Keywods: sposoed sech dvetsg; ocy evy-fee equbum; eveue mgemet; geezed secodpce ucto; eseve pce.. Itoducto Iteet dvetsg hs gow pdy ove the st decde. Amog vous types of teet dvetsg e.g., be, ch med, etc. sposoed sech dvetsg s oe of the fstest gowg souces of eveue. Googe epoted tot eveue of $6.6 bo ove the fsc ye of 7, d sposoed sech dvetsemet cotbuted $6.4 bo [Googe 8]. y teet dvetsemets e sod o pe-mpesso bss. Advetses py ft fees fo fed umbe of uts usuy thousd mpessos. The fees e egotted o cse-by-cse bss. Ths mechsm s epced by the geezed fst-pce GP uctos fo sposoed sech dvetsg toduced 997. I GP uctos, ech dvetse submts bd fo ptcu eywod. Whe teet use ccs o the d ssocted wth the eywod, the dvetse s chged fo wht he bds. The method s esy to use d the ety cost s ow fo dvetses. Howeve, thee s o pue sttegy equbum fo the GP d dvetses ectg to othe bddes moves pompty hve sgfct dvtge. The mechsm ecouges suffcet vestmet d esuts vote bddg pces [dem d Ostovsy 7]. To emedy the effcecy, Googe toduced ew system, ced AdWods Seect. The system stputes tht dvetse posto pys pce pe cc tht equs the bddg pce of the dvetse posto + pus mmum cemet. The mechsm s ced the geezed secod-pce GSP ucto. It hs become domt mechsm sposoed sech dvetsg eve sce. o cet eywod, Googe povdes ech dvetse efeece pce pe cc to w the fst befoe he bds hs mmum wgess to py, o cost pe cc CPC. Afte the bd, the dvetse eceves fomto of Pge 4

2 Jou of ectoc Commece Resech, VOL, NO 3, 9 the tcpted umbe of ccs d opque of hs bd. The s opque becuse t s estmted bsed o cuet met codto cudg the poo of dvetses d the bd pces. As d postos the bottom g few o o ccs, the cpcty of sposoed sech dvetsg s mted. gue shows the fot pge of Googe whe the eywod Hybd Vehce s eteed. Thee e 3 s o the top d 8 s o the ght sde. I gee, Googe s fst eeve s e most vube to the dvetse. Uess the dvetse s bd s mog the top eeve, hs d w ot be dspyed the fst pge. If comg dvetse offes pce hghe th oe of the occupts, the st posto o cuet st w be dspced. Sech eges ofte use eseve pces to swy dvetses bddg. Whe eseve pce s mposed, y bd beow the sceeg eve s ot dmtted to the ucto. As eseve pce s esy to dust, t hs become commo too fo sech eges to mge eveue. Obvousy f the eseve pce s set too hgh, t w tu wy dvetses d dvesey ffect the eveue stem becuse uoccuped d postos e wsted. O the othe hd, f the eseve pce s too ow, sech eges my fogo the oppotuty of ftg eveue whe dvetses e wg to py moe. Ntuy the optm eve of eseve pce bsed o demd, the umbe of d postos d othe fctos s of get coce to sech eges. Thee hve bee mted studes o the optm eseve pce the sposoed sech dvetsg tetue. Amog othe mttos, these studes ssume fed umbe of bddes d cm tht the optm eseve pce s oy depedet o the dstbuto of bddes pe-cc vue. Sce 8 Yhoo! hs o oge used fed mmum bd of $. d stted mposg vbe mmum bds fo some eywods [Yhoo! Sech Metg Hep 9]. Lte Googe toduced the fst pge bd estmte to ppomte the mmum CPC eeded to show sposoed the fst pge [Googe Adwods Hep 9]. Both mmum bd d fst pge bd estmte e posted d updted eguy. The pmy teest of ths esech, hece, s to vestgte wht s the optm eseve pce fo the sech ege to mmze ts epected eveue te fom ech sposoed sech ucto f fed eseve pce s o oge used pctce. gue : Sposoed dvetsg s fo eywod "Hybd Vehce" Googe Spshot ceted o J. 6th, 9.. Ltetue Revew Much of the ucto tetue studes how vous fctos ffect the see s eveue gg fom we detemto ue, bddes pefeece to eseve pce. Myeso [98], Rey d Smueso [98] study sgetem uctos. They dscove tht the tem shoud be wded to the bdde who hs the hghest wgess to py f d oy f t eceeds ctc esevto vue. Ms d Rey [989] eted the yss to uctos wth Pge 5

3 o et.: Optm Reseve Pce fo the Geezed Secod-Pce Aucto Sposoed Sech Advetsg mutpe detc tems. gebecht-wggs [987] shows tht f the umbe of bddes s fed, thee ests optm eseve pce fo sge-tem uctos. Whe the umbe of bddes ves, howeve, deceses the epected umbe of bddes my outwegh beefts esutg fom the eseve pce. Hece, zeo eseve pce my become desbe ths cse. Buow d Robets [989] dopt the mg eveue yss to ssocte ucto theoy wth ecoomc fudmets, usg moe tutve method to deve the optm eseve pce peseted Myeso [98], d Rey d Smueso [98]. They et d f efe to the pobbty dstbuto d desty fuctos of bddes wgess to py, espectvey. They show tht f the moopostc see sets te-to-eve-t pce, the the epected qutty of ses woud be q. The see shoud dsqufy those bddes wth egtve mg eveues, d the vue tht edes zeo mg eveue s the optm eseve pce. pcty, t s the souto of d MR q. dq f Recet theoetc d empc wos eted to sposoed sech uctos c be foud Asdem [6], Aggw d Hte [5], Meht et. [5], Szyms d Lee [6], d Böges et. [6]. These ppes deveop e-optm mechsms fo pcg d cpcty octo. eg et. [7] compe seve ucto mechsms fo d sots though umec tests. A epemets, howeve, e coducted fo sttc uctos tht e competed oe shot. dem et. [7] vestgte GSP uctos used by sech eges to se sposoed s. They fd tht GSP ucto geey does ot hve equbum domt stteges, d tuthteg s ot equbum sttegy. They toduce the ocy evy-fee equbum of smuteous gme duced by GSP whee ech dvetse cot be bette off by swppg bds wth the dvetse ed oe posto bove hm. Sm esuts c be foud depedet wo by V [7]. He pots out tht the d posto ucto s cosey eted to the ssgmet gme studed the tetue [Roth d Sotomyo 99], d ts equbum c be epcty ccuted. dem d Schwz [6] study the mpct of eseve pce d the oe of met depth GSP uctos. They effm the optmty codto d cm tht the optm eseve pce s depedet of the umbe of bddes. eg et. [7b] show tht smuteous pooed ucto f the vue fucto of bddes c be defed s poduct of two fuctos whee ech etes to the d posto o vue pe cc septey, the optm eseve pce fo the sech ege s depedet of the umbe of bddes, o the umbe of d postos. Numecy they show tht t s ot ecessy optm to set the th hghest pce s the eseve pce... Motvto d Cotbuto To ou owedge, fctos oud eseve pce fo sposoed sech dvetsemet hve ot bee fuy ddessed the tetue. Ced out o cotuous-tme bss, sposoed sech uctos e ftey epeted sequet gmes wth vey compe equbum. To smpfy the pobem, most theoetc d empc studes the tetue focus o spshot of typc tme peod. The spshot ssumes ech tme tev hs the sme t codto d bddg pmetes. The sme umbe of dvetses smuteousy bd the sme tot umbe of d postos vbe to them. o empe, seve studes [dem d Schwz 6; eg et. 7b] show tht the optm eseve pce sposoed sech ucto shoud be the sme s the oe defed. Such optm eseve pce s oy detemed by the dstbuto of vue pe cc, wth o coecto to the umbe of bddes o the umbe of obects o se. I ety, howeve, pt of o postos my hve bee te whe ew bddg s eceved. Some pevous wg bddes e ey to hod the postos uess they u out of budget o e outbd by ew comes. Not oy competg mog themseves, ew comes so compete wth cumbets wth wg postos. The spshot ppoch msses the dymc tue of the ucto pocess d the tedepedece betwee cosecutve tme peods. mpe Assume tht secod-pce ucto wth sge d posto sts fo two peods. Thee e bddes ech peod whose vues pe cc foow the ufom dstbuto o [, ]. If the posto s te peod, the we w sty the posto uess he s outbd peod. Accodg to, the best eseve pce shoud be /. Howeve, t c be vefed tht whe =, the optm eseve pce s.55. uthemoe, s ceses, the optm eseve pce so goes up. The ede s efeed to the pped fo dets. A tutve epto s tht whe thee e moe dvetses the ucto, the umbe of bddes who bd ove / w popotoy cese. As esut, the secod hghest bd, whch s wht the we pys, ppes to be gete th /. The esech coducted ths ppe s motvted by the c of dymc fetue of estg modes. We cosde dvetsg cpcty s peshbe poduct d oe ucto s cotuous-tme pocess. Sech Pge 6

4 Jou of ectoc Commece Resech, VOL, NO 3, 9 eges ty to se s my vbe d postos s possbe t the hghest pces. We eme how the eseve pce ffects the see s epected eveue te d demostte how the optm eseve pce dffes fom those obted the cuet tetue. Ou mode ttempts to mmze the og-tem epected eveue te wth eseve pce s coto vbe. We show tht gve tot of d postos d dvetses, the optm eseve pce GSP ucto fo sposoed sech dvetsemet s the th gest epected vue pe cc mog dvetses, o ts + th ode sttstc. I othe wods, the optm eseve pce depeds o the umbe of d postos, the umbe of ucto ptcpts ddto to the dstbuto of the pe-cc vues. Ou esut dffes fom wht s cmed dem d Schwz [6] d othes. A smuto s coducted to mmc the dymc sposoed sech ucto. Smuto esuts cofm tht the eseve pce tht equs the th gest epected pe-cc vue cheves the hghest eveue te fo the sech ege. The est of the ppe s ogzed s foows. I Secto, we evew the GSP ucto mechsm fo sposoed sech dvetsg. Bsed o the cocept of ocy evy-fee equbum, we deve the optm eseve pce fo GSP ucto Secto 3. The secto so dscusses mttos of the mode. Secto 4 pesets smuto to mmc the e pctce sposoed sech dvetsg. Cocudg ems e pced Secto 5.. qubum Geezed Secod-Pce Aucto GSP s the pmy mechsm fo sposoed sech dvetsg. Afte web use submts sech tem quey, the sech ege dspys cet umbe of eevt sposoed s o esut pges ddto to sech esuts. If the use ccs o y of these s, he s dected to dvetse s web ste. The dvetse woud be chged by the sech ege fo sedg the use to hs web pge. ch dvetse hs hs esevto vue o such cc. o the eywod submtted, dvetses specfy the mmum wgess to py fo cc by teet uses. The posto ech d ppes o the esut pge s detemed by the of bds decesg ode. I GSP ucto ech dvetse pys the et hghest dvetse s bd. Thee e mted umbe of dvetsg postos tht the sech ege c povde o ech esut pge, d s pced dffeet postos hve dffeet pobbtes to be cced. I gee, pced t the top of pge s moe ey to be cced th show t the bottom. Howeve, use who s dected to the dvetse s web ste fte ccg the show dffeet postos s ssumed to hve the sme pobbty to me puchse. Boos [4] shows oy modete dffeeces puchse pobbtes whe ds e pced t dffeet postos. Whe sech eges dvetses o the esut pge, they so eed to cosde tht dffeet dvetses my eceve dffeet umbe of ccs, o cc though tes CTR, eve f the ds e pced the sme posto. Ths s becuse dvetses o top postos my ot ecessy geete hghe eveue. Yhoo! dsegds the dffeece d smpy s dvetses the descedg ode of the bds. Googe s bddes ccodg to umbe, esuted by mutpyg ech bd wth ts quty scoe. The quty scoe s bsed o CTR, eevcy of the eywod d othe fctos. ch dvetse pys mout tht esues hs umbe to supss the et dvetse. Cosde sposoed sech ucto wth d postos d dvetses. oowg dem et. [7], we ssume tht the dstbutos of dvetses pe-cc vues e depedet d detc. Advetses e seut. The epected CTR eceved by d posto s. Sce top d posto eceves hghe CTR th tht of sot t owe posto, we hve fo d fo. Let dvetse s bd be b d hs pe-cc vue be v. It s ce tht v b. Note tht v my vy fo dffeet dvetses, but t ems the sme fo bdde eve f hs d s pced dffeet postos. Let b d g be the bd pce d detty of the th hghest bd espectvey. If b s gete th the eseve pce, GSP octes the th posto to the dvetse wth the th hghest bd,,...,m,. Whe the th s cced by use, g, whee the dvetse g pys the sech ege mout tht equs to the et hghest bd, eveue eceved by the sech ege fom the th posto s b. Hece, the d the pyoff to dvetse g s b p v b. 3 g Pge 7

5 o et.: Optm Reseve Pce fo the Geezed Secod-Pce Aucto Sposoed Sech Advetsg To eme the equbum ssue GSP uctos, we beg wth the oto defed by V [7]. Defto I symmetc Nsh equbum SN bd pces stsfy v b v b, d. g g 4 V [7] futhe shows f set of bd pces stsfes the SN codto 4 fo posto + d, the t stsfes 4 fo. Hece, dvetse g oy eeds to compe hs cuet posto wth hs two dcet postos. Sce he pys the et hghest bd pce dvetse g c dust hs bd pce b sghty wthout ffectg hs posto o pymet, d thee s ge of bd pces tht stsfy the SN codto. Modes bsed o such equbum wth vous mes c be foud Hs d Rvv [98], Wso [993], V [7] d dem et. [7]. Specfcy dem et. [7] c the SN wth = codto 4 ocy evyfee equbum. I ths ppe we foow the coveto d focus o the boudy sttes of the ocy evy-fee equbum. Defto I ocy evy-fee equbum, bd pces stsfy v, m,. 5 b b g The ocy evy-fee equbum s tutve: f the dvetse posto wts to move up to posto, the ddto vege cost he hs to py shoud be equvet to the vue of et ccs he woud hve eceved. I such equbum bd pces c be detemed epcty usg the ecusve fomu 5 I ths ppe we defe pcg sttegy fo sposoed sech dvetsg bsed o the ocy evy-fee equbum wth eseve pce. If thee e o eough bddes competg fo d postos,.e.,, the the dvetse the owest g pys the eseve pce; othewse, the dvetse t the th posto pys the eseve pce. Advetses whose ds ppe hghe postos bd ccodg to equto 5. Hece, the eve of eseve pce set by the sech ege decty ffects ts eveue. 3. The Optm Reseve Pce Sposoed Sech Advetsg 3.. M Resuts To the best of ou owedge, few theoetc yss of optm eseve pces cosde the dymc fetue d vetoy peshbty sposoed sech dvetsg. It s uce wht eseve pce mmzes sech ege epected eveue te. I ths secto, we povde theoetc fmewo to tce the pobem ocy evy-fee equbum. Let,,..., be -dmeso dom vecto, d,,..., be -tupe ssumed by,,...,. Regg,,..., cesg ode so tht,...,, whee m,,...,, s the secod smest vue the -tupe, d so o. It s ce tht of,,...,, of vues ssumed by,,..., m,,...,. The fucto tht tes o the ech possbe sequece,..., s ow s the th ode sttstc. Let,,..., be detc d depedet dom vbes wth commo pobbty desty fucto f d commo dstbuto fucto. It s we ow tht the mg pobbty desty fucto of s gve by d ts epectto c be wtte s f! f!! 6 Pge 8

6 Jou of ectoc Commece Resech, VOL, NO 3, 9! f d f d.!! It s esy to deteme the bd pce wg the th d posto b ecusvey fom the ocy evy-fee equbum. Let. The dvetse t the st posto pys the eseve pce. o coveece, et b. o, b b. 7 Whe thee e postos offeed t eseve pce, eveue to the sech ege pe ut of tme s dom vbe gve by R, b. 8 I vew of 7, wth d bddes the ucto we hve b b, whch c be substtuted to 8 to fom e fucto of ode sttstcs:.... R, 9 Sce s ucet, the epected eveue te c be depcted s R,. Lemm Cosde GSP ucto fo sposoed sech dvetsg wth dvetses d d s. Assume tht ech dvetse s pe-cc vue s depedety d detcy dstbuted..d.. Let be eseve pce, the the epected eveue fo the sech ege s cesg fucto of. stsfyg The poof of Lemm d othe poofs e pced the pped. Lemm sttes tht the optm eseve pce shoud ot be set beow the epected vue of the +th ode sttstc of dvetse s pe-cc vues. Ths s tutvey stghtfowd. If two dstct eseve pces esut the sme epected umbe of d sots to be sod, the the hghe pce shoud ed to moe eveues s the th hghest bdde pys moe. Lemm Cosde GSP ucto fo sposoed sech dvetsg wth dvetses d d s. Assume tht ech dvetse s pe-cc vue s fte d..d.;,whee s the hghest pe-cc vue;. The fo y stsfyg, cot be the optm eseve pce fo the sech ege. We ote tht Lemm eeds seve codtos. Codto s esobe s o pe-cc vues c go to fty. Codto s vey md, especy whe the umbe of bddes s ge comped to the umbe of Pge 9

7 o et.: Optm Reseve Pce fo the Geezed Secod-Pce Aucto Sposoed Sech Advetsg postos vbe. Codto cey hods fo y symmetc dstbuto such s om d ufom dstbutos. It so hods fo some symmetc dstbutos. o empe, the dstbuto wth. The demm fced by the sech ege s whethe to se t most s t hghe sttg pce such s, o se s t owe pce. Lemm shows tht though sg the eseve pce my mpove the eveue coected fom ech posto, t fs to do so f the epected umbe of d sots sod s compomsed. Ths cocdes wth the obsevto by gebecht-wggs [987] who fds tht y decese the epected umbe of bddes huts the bd-te s epected eveue moe th y beefts fom otv esevto pce. Combg the bove two emms, we cm the foowg theoem. Theoem Assume tht ech dvetse s pe-cc vue s..d., the the optm eseve pce fo the sech. ege s Whe the sech ege ses the eseve pce, t does ot oy ffect the dvetse t the st posto, but so hve ppe effect to the est occupts. Ths c be eed fom equto 7. As ceses, the bdde wg the th posto hs to py hghe eseve pce. Ths tu w ffect b vew of the equbum b codto, d ech othe bd pce. As eseve pce ses fom zeo, the dvetse t the st posto hs to cese hs bd pce to vod emto. Ths ew pce my outbd those bove hm d ede hm hghe posto. To pevet osg hs cuet posto, the dvetse t the secod owest posto hs to bd hghe ccodgy, so does the oe t the thd owest posto, d so o. Whe the eseve pce eceeds some dvetse s esevto vue, he hs o choce but to qut the ucto, cusg eveue oss to the sech ege. Hece, the sech ege shoud se the eseve pce hgh eough whe esug the ses of d postos. The +th ode sttstc of pe-cc vue s the optm eseve pce GSP ucto fo sposoed sech dvetsemet. As spec cse, the optm eseve pce s f dvetses pe-cc vues foow ufom dstbuto o [,]. 3.. Lmttos of the Mode I the pevous secto we deved the optm eseve pce ude cet codtos. The theoetc deveopmets, howeve, hve the mttos. st, though we cosde the dymc tue of vous umbe of dvetses ptcptg the ucto d copote t to the optm eseve pce, ou mode fs to cptue some othe dymc spects the GSP uctos. The GSP uctos ety e fte epeted gmes whee the sets of equb c be vey ge d the stteges edg to such equb e compe. As dem et. [7] gue, t s ot esobe to epect dvetses to be be to eecute such stteges. Sce dvetses c dust d fze the bds fte evewg the estmted, CTR d othe pefomce feedbcs eesed by the sech ege, foowg dem et. [7] d V [7] we focus o sttc GSP ucto whch bds e stbzed. Secody, the umbe of dvetses ptcptg the ucto, though beg efected the eseve pce, s uow d eeds to be estmted. Ths coud be dffcut whe the umbe of dvetses chges costty. Thdy, ou m esut coves wde ge of pobbty dstbutos cudg symmetc dstbutos, t ems to be see f the optm eseve pce ppes to bty dstbutos. We ote tht dvetses budget costts w so ffect the esut of ou mode. o empe, whe the eseve pce s beyod dvetses budget, t w ffect the umbe of ptcpts the ucto, whch tu, mpct the optm eseve pce. Howeve, ths mes the mode much moe compe d s beyod the scope of ths esech. We eve t s futue esech questo. 4. Numec pemet We desg smuto to mmc the e pctce sposoed sech dvetsg. Dffeet fom the epemets coducted eg et. [7], d dem d Schwz [6] whee oy sttc ucto s tested, we smute dymc d cotuous ucto mg to swe the foowg questos: Wht s the best eseve pce GSP ucto fo sposoed sech dvetsemet? Wht s the eveue beeft to the sech ege of settg the eseve pce s the + th ode sttstcs of the pe-cc vue? The otv esevto pce mes the optm esevto pce bsed o the fed umbe of bddes. Pge

8 Jou of ectoc Commece Resech, VOL, NO 3, Desg of the Smuto The smuto s coducted ove hozo of,t. Thee e d postos vbe fo se. Isted of ssumg eogeousy ow umbe of dvetses, we eme stem of bd vs foowg Posso pocess wth desty. We eted the GSP ucto descbed dem et. [7] to dymc settg, whee the sposoed sech ucto pogesses ove the cuse of tme. ch ew bd tgges ew oud of e-octo of d postos. gue shows the tmee of the smuto. At tme t the fst bd b s submtted. If t s gete th the eseve pce, the fst d posto w be wded. Avg t tme t the th bd competes wth cuet occupts. If thee e postos ufed, oe posto w be wded to the ew qufed bdde bsed o hs. Othewse the ett hs to outbd oe of the cuet occupts to secue posto. Let b d t be the bd pce d detty of the th hghest bd t tme t. I ocy evy-fee equbum bd pces shoud stsfy whee,...,m,. v, b b t t gt Sce the CTR fo the th d posto s d the dvetse the sech ege fo ech cc, ove the te-v tme sech ege fom the th posto s d the vege pyoff to dvetse t s t g g t pys b to t t t the vege eveue eceved by the t, g t bt t p t v g bt t, whee b. t t st bd th bd +th bd : g pys : g pys : b fo the st d posto; b fo the th d posto; g pys fo the th d posto. Δt t t t gue : Tmee fo the smuto t T Pge

9 o et.: Optm Reseve Pce fo the Geezed Secod-Pce Aucto Sposoed Sech Advetsg T J Let J be the st bd v befoe T whee J J t sup, whee t. The weghted vege eveue te eceved by the sech ege ove,t s J t, 3 T d the weghted vege pyoff te to dvetses s J p p t. 4 T I the foowg epemets we use d p to guge the mpct of eseve pce. 4.. pemet : Compso of Dymc d Sttc GSP Auctos Ths epemet compes the pefomce of dymc GSP ucto d sttc GSP ucto whe dffeet eves of eseve pce e tested. Let T 5 d 8. To be cosstet wth the umec tests eg et. [7] we ssume costt decyg tteto fcto c. 48 mog d postos, meg tht the th d posto eceves 4.8% moe ccs o vege th the + th posto. oowg the smuto methodoogy of dem d Schwz [6], we ssume tht the pe-cc vue of ech dvetse foows ogom dstbuto wth me. d stdd devto. 5. Reseve pces gg fom to 5 e tested. Bd vs foow Posso dstbuto wth desty 4 dymc ucto. To smute sttc ucto we eed to decde fed umbe of dvetses. The vue of the umbe of dvetses ech sttc ucto s estmted fom ts coespodg stce the dymc ucto: we dvde the testg hozo to T dscete peods. The umbe of dvetses of ech peod the equs the umbe of d postos octed to dvetses t the begg of ech peod pus the ew bd vs ove tht peod. The vege umbe of dvetses ove T peods gves the umbe of dvetses ech sttc ucto. A epoted esuts e veged ove smuted uctos. gue 3: pected Reveue Dymc Aucto d Sttc Aucto fo Sposoed Sech Pge

10 Jou of ectoc Commece Resech, VOL, NO 3, 9 gue 4: pected Numbe of Ad Postos Sod Dymc Aucto d Sttc Aucto fo Sposoed The epected eveue cuves both dymc d sttc uctos e potted gue 3. It suggests tht thee s optm eseve pce tht mmzes the epected eveue dymc ucto fo sposoed sech dvetsg, d the eve of such eseve pce s hghe th the oe sttc ucto. gue 4 pesets the ses of d postos. A ow eseve pce pomotes ses; howeve, t does ot cete dequte eveue to the see due to ow pymet eceved fom dvetses. As the eseve pce ppoches zeo, eght d sots e occuped, deveg the hghest pyoff to dvetses whe the owest eveue to the sech ege. O the othe hd, f the eseve pce s beyod some dvetses wgess to py, few d postos my be eft vct, esutg ost ses. I gue 4, fo stce, f the eseve pce s set s 4. oy hf d sots e epected to be octed dymc ucto. Hece, the eve of eseve pce decty ffects the epected eveue. I the equbum of GSP ucto, dvetses wth hghe pe-cc vues e ssged to top postos whe those wth owe wgess to py ppe bottom sots. The equbum bd pce c be detemed v ecusve equto defed 5. Sce thee e oy postos fo se, t s optm fo the fst ecuded dvetse.e., the oe s t +th posto to bd hs tue vue. The gumet s sm to the Vcey gue 5: pected Reveue wth Thee Reseve Pces Imposed Dymc GSP Pge 3

11 o et.: Optm Reseve Pce fo the Geezed Secod-Pce Aucto Sposoed Sech Advetsg gue 6: Aucto Pefomce Whe ucto [V 7]. If the dvetse s ecuded, the t s megess to bd owe th hs vue; but f he hppes to w posto due to possbe ets of hghe bddes, the the dvetse s be to me poft. Sce the dvetse occupyg the th sot s supposed to py the + th hghest bd pce, the sech ege shoud set the eseve pce s the th gest vue pe cc, o the + th ode sttstc, to etct the mm eveue pemet : Optm Reseve Pce Dymc GSP Aucto We test thee eseve pces: ; d 3 dymc GSP uctos f whe the Posso v te of bds ves fom 7 to. Othe smuto pmetes em the sme s secto 4. ecept tht the vue pe cc foows ufom dstbuto o [, ]. Thus, thee eseve pces ude test become ; d 3, espectvey. Ths epemet ms to vefy whch eseve pce s the best fo sposoed sech uctos. To obt the epected vue of we estmte the vege umbe of dvetses of dymc ucto usg the sme ppoch secto 4.: The smuto hozo s dvded to T peods. The vue of s the vege umbe of dvetse of ech peod,.e., the sum of tot umbe of d sots occuped t the begg of ech peod d the umbe of ew etts ove tht peod. gue 5 pots the epected eveue te dymc GSP uctos whe equs, / d, espectvey. It s ce tht zeo eseve pce eds to the owest eveue te becuse ech dvetse pys the mmum fo oe d sot. Whe v te 7, the vege umbe of dvetses s estmted to be 5 by smuto. Hece, d cete the sme eveue becuse they equ to ech othe whe 5 d 8. As demd ceses, howeve, usg cey geetes hghe eveue th fed eseve pce tht soey depeds o the vue dstbuto fucto. Pge 4

12 Jou of ectoc Commece Resech, VOL, NO 3, 9 To futhe demostte why depct the ses of d postos ude s the optm eseve pce fo sposoed sech ucto, we d ts eveue g ove gue 6. The f hozot s epesets the coespodg + th ode sttstcs of vue pe cc s ves fom 7 to. It shows tht f the demd fo d sots s hgh eough, mposg hghe eseve pce of s be to octe peshbe vetoy d cete moe eveue fo the sech ege. As stted the tetue [eg et. 7; dem d Schwz 6], whe the umbe of dvetses ceses, the mpct of eseve pce dmshes due to moe competto ceted mog bddes. gue 6 shows tht s becomes moe sgfct. o stce, whe, demd ses, the eveue g of settg usg. 7 esuts 5% moe eveue th. The mg eveue beeft of f settg, howeve, s decesg demd, whch s udestdbe becuse s becomes much ge etvey to fed, ts mpct to the + th ode sttstcs s eggbe. The esuts bove e cosstet wth the empc evdeces show the US met whee the CPC fo eywods c ge fom few cets to $ depedg o the poputy [Moso 9]. I ddto, doptg sm sttegy s Googe d o oge fg the mmum bd t $., Yhoo! ow uses vbe eseve pces some eywod mets bsed o seve fctos such s the umbe of bddes d the bd mouts [Yhoo! Sech Metg Hep 9]. Such shft pcg sttegy coobotes ou theoetc fdgs. 5. Cocuso Sposoed sech dvetsg s mut-bo do busess. Sech eges such s Googe d Yhoo! ofte use eseve pce to fuece dvetse s bddg fo d s petg to eywod though the geezed secod-pce GSP ucto. Howeve, s dvetsg cpcty s peshbe d the ucto pocess s dymc, bty theshods my dvesey ffect eveues. Ths ppe studes the optm eseve pce sttc GSP ucto fo sposoed sech dvetsg. We pove tht gve tot of d postos d dvetses the eseve pce tht mmzes the sech ege s vege eveue te s the th gest epected vue pe cc, o ts +th ode sttstc mog dvetses. Hece, the optm eseve pce sposoed sech ucto depeds o the umbe of d postos, the umbe of dvetses d the pe-cc vues. Ou esut otceby dffes fom the estg oes the tetue whee the optm eseve pce sposoed sech ucto s cmed to be oy eted to bddes vue dstbuto. A smuto mmcg the e pctce sposoed sech dvetsg shows tht costt eseve pce fs to cptue the dymcs the ucto pocess. If the demd s ge, settg eseve pce s s be to octe d postos d cete hghe eveue te th othe tetves. Pctc mpemetto of vbe eseve pce Googe d Yhoo! coobotes ou fdgs. Seve possbe etesos to ths wo e woth beg metoed. st, the cuet souto s ppcbe to symmetc dstbutos of pe-cc vues d othe dstbutos wth cet codtos. Oe my eted t to gee vue dstbuto, epoe the stuctu popety of the optm eseve pce wth espect to the umbe of d postos d umbe of bddes. Secody, the optm eseve pce s deved sge-shot sttc GSP ucto. It my be teestg to cosde dymc mode vestgtg how the sech ege shoud updte the eseve pce bsed o the cumbet d potet ew ety mut-peod hozo. y, the mpct of dvetses budget costts o the optm eseve pce deseves beg emed. RRNCS Aggw, G. d J.D. Hte, Kpsc Auctos, st Woshop o Sposoed Sech Auctos, ACM ectoc Commece, 5. Asdem, K., Bddg Pttes Sech ge Auctos, Secod Woshop o Sposoed Sech Auctos, ACM ectoc Commece, 6. Böges, T., I. Co, M. Pesedofe d V. Petce, qubum Bds Sposoed Sech Auctos: Theoy d vdece, Wog Ppe, Deptmet of coomcs, Uvesty of Mchg, 6. Pge 5

13 o et.: Optm Reseve Pce fo the Geezed Secod-Pce Aucto Sposoed Sech Advetsg Boos, N., The Ats R Repot: How Advetsemet ge R Impcts Tffc, Techc Repot, Ats Isttute, Juy 4. Buow, J. d J. Robets, The Smpe coomcs of Optm Auctos, Jou of Potc coomy, Vo. 97, No. 5:6-9, 989. dem, B., M. Ostovsy d M. Schwz, Iteet Advetsg d the Geezed Secod Pce Aucto: Seg Bos of Dos Woth of Keywods, Amec coomc Revew, Vo. 97, No. :4-59, 7. dem, B. d M. Schwz, Optm Aucto Desg Mut-ut vomet: The Cse of Sposoed Sech Auctos, Wog Ppe, Hvd Uvesty, 6. dem, B. d M. Ostovsy, Sttegc Bdde Behvo Sposoed Sech Auctos, Decso Suppot Systems, Vo. 43, No. :9-98, 7. gebecht-wggs, R., O Optm Resevto Pces Auctos, Mgemet Scece, Vo. 33, No. 6:763-77, 987. eg, J., H. K. Bhgv d D. M. Peoc, Impemetg Sposoed Sech Web Sech ges: Computto vuto of Atetve Mechsms, Ifoms Jou o Computg, Vo. 9, No. :37-48, 7. eg, J., Z.M. She d R.L. Zh, Red Items Auctos d Oe Advetsemet, Poducto d Opetos Mgemet, Vo. 6, No. 4:5-5, 7b. Googe Qutey gs Repot, -Q.pdf, 8. Googe Adwods Hep, st Pge Bd stmtes, 9. Hs, M. d A. Rvv, A Theoy of Moopoy Pcg Schemes wth Demd Ucetty, Amec coomc Revew, Vo. 7, No. 3: , 98. Ms,. d J. Rey, Optm Mut-Ut Auctos, Ofod Uvesty Pess, 989. Meht, A., A. Sbe, U. Vz d V. Vz, AdWods d Geezed O-e Mtchg, Poceedgs of the 5 46th Au I Symposum o oudtos of Compute Scece OCS5, 5. Moso, C., Googe Wts You, CNNMoey.com, My 6th, 9. Myeso, R.B., Optm Aucto Desg, Mthemtcs of Opetos Resech, Vo. 6, No. :58-73, 98. Rey, J.G. d W.. Smueso, Optm Auctos, Amec coomc Revew, Vo. 7, No. 3:38-39, 98. Roth, A. d M. Sotomyo, Two-Sded Mtchg, Cmbdge Uvesty Pess, 99. Szyms, B.K. d J.S. Lee, Impct of ROI o Bddg d Reveue Sposoed Sech Advetsemet Auctos, Secod Woshop o Sposoed Sech Auctos, ACM ectoc Commece, 6. V, H.R., Posto Auctos, Iteto Jou of Idust Ogzto, Vo. 5, No. 6:63-78, 7. Wso, R.B., Noe Pcg, Ofod Uvesty Pess, Ofod, UK, 993. Yhoo! Sech Metg Hep, Pcg d Mmum Bds, 9. Pge 6

14 Jou of ectoc Commece Resech, VOL, NO 3, 9 APPNDI Ayss of mpe : Let d be the hghest bd peod d espectvey. Let the eseve pce be d sech ege s eveue peod be,,. I vew of gebecht-wggs 987, we hve R R. Notce tht the hghest bdde peod w sty peod d s oed by othe dvetses. If R, the R R. Othewse we cosde the foowg cses. If, the R ; If, the R ; If, the R. As the pdf of d e d espectvey, t c be show tht d As esut, R R R R R d d d d, 3 R d[ R] It c be vefed tht d / eseve pce s betwee / d. 3.. d 5 d[ R] d fo suffcety ge. Hece, the optm d Poof of Lemm Let It ce tht t both d, the epected umbe of d postos the sech ege c se s d the espected eveue tes, epeseted by d, e d. Pge 7

15 o et.: Optm Reseve Pce fo the Geezed Secod-Pce Aucto Sposoed Sech Advetsg Pge 8 Hece,.. Poof of Lemm Sce, f s set up s eseve pce, the the epected umbe of d postos the sech ege c se s. Accodg to Lemm, y stsfyg w geete moe epected eveue th does. Let d, we show tht. Note tht whe the eseve pce s, the epected umbe of d sots the sech ege c se s. Hece, we hve. So t suffces to show, o equvety. It s ot dffcut to show the foowg equto fte ppyg tegto by pts:.!!! d Note tht.!!! d f As esut,. d f d 6 We show the ght hd of 6 s ess th, o

16 Jou of ectoc Commece Resech, VOL, NO 3, 9 Pge 9, d f d whee s the gest pe-cc vue mog dvetses. Obseve, d d f d f d d f whee, d the st equty esuts fom the ssumpto tht. Thus, f, o, we hve.