Budget Optimization in Search-Based Advertising Auctions

Transcription

1 Budgt Optimization in Sarch-Basd Advrtising Auctions ABSTRACT Jon Fldman Googl, Inc. Nw York, NY Martin Pál Googl, Inc. Nw York, NY Intrnt sarch companis sll advrtismnt slots basd on usrs sarch quris via an auction. Whil thr has bn prvious work on th auction procss and its gam-thortic aspcts, most of it focuss on th Intrnt company. In this work, w focus on th advrtisrs, who must solv a complx optimization problm to dcid how to plac bids on kywords to maximiz thir rturn (th numbr of usr clicks on thir ads) for a givn budgt. W modl th ntir procss and study this budgt optimization problm. Whil most variants ar NP-hard, w show, prhaps surprisingly, that simply randomizing btwn two uniform stratgis that bid qually on all th kywords works wll. Mor prcisly, this stratgy gts at last a 1 1/ fraction of th maximum clicks possibl. As our prliminary xprimnts show, such uniform stratgis ar likly to b practical. W also prsnt inapproximability rsults, and optimal algorithms for variants of th budgt optimization problm. Catgoris and Subjct Dscriptors F.2 [Thory of Computation]: Analysis of Algorithms and Problm Complxity; J.4 [Computr Applications]: Social and Bhavioral Scincs Economics Gnral Trms Algorithms, Economics, Thory. Kywords Sponsord Sarch, Optimization, Auctions, Bidding. 1. INTRODUCTION Onlin sarch is now ubiquitous and Intrnt sarch companis such as Googl, Yahoo! and MSN lt companis and Work don whil visiting Googl, Inc., Nw York, NY. Prmission to mak digital or hard copis of all or part of this work for prsonal or classroom us is grantd without f providd that copis ar not mad or distributd for profit or commrcial advantag and that copis bar this notic and th full citation on th first pag. To copy othrwis, to rpublish, to post on srvrs or to rdistribut to lists, rquirs prior spcific prmission and/or a f. E C 7, J un 11 15, 27, San D igo, California, USA. Copyright 27 ACM /7/6...$5.. S. Muthukrishnan Googl, Inc. Nw York, NY [email protected] Cliff Stin Dpartmnt of IEOR Columbia Univrsity [email protected] individuals advrtis basd on sarch quris posd by usrs. Convntional mdia outlts, such as TV stations or nwspaprs, pric thir ad slots individually, and th advrtisrs buy th ons thy can afford. In contrast, Intrnt sarch companis find it difficult to st a pric xplicitly for th advrtismnts thy plac in rspons to usr quris. This difficulty ariss bcaus supply (and dmand) varis widly and unprdictably across th usr quris, and thy must pric slots for billions of such quris in ral tim. Thus, thy rly on th markt to dtrmin suitabl prics by using auctions amongst th advrtisrs. It is a challnging problm to st up th auction in ordr to ffct a stabl markt in which all th partis (th advrtisrs, usrs as wll as th Intrnt sarch company) ar adquatly satisfid. Rcntly thr has bn systmatic study of th issus involvd in th gam thory of th auctions [5, 1, 2], rvnu maximization [1], tc. Th prspctiv in this papr is not of th Intrnt sarch company that displays th advrtismnts, but rathr of th advrtisrs. Th challng from an advrtisr s point of viw is to undrstand and intract with th auction mchanism. Th advrtisr dtrmins a st of kywords of thir intrst and thn must crat ads, st th bids for ach kyword, and provid a total (oftn daily) budgt. Whn a usr poss a sarch qury, th Intrnt sarch company dtrmins th advrtisrs whos kywords match th qury and who still hav budgt lft ovr, runs an auction amongst thm, and prsnts th st of ads corrsponding to th advrtisrs who win th auction. Th advrtisr whos ad appars pays th Intrnt sarch company if th usr clicks on th ad. Th focus in this papr is on how th advrtisrs bid. For th particular choic of kywords of thir intrst 1,an advrtisr wants to optimiz th ovrall ffct of th advrtising campaign. Whil th ffct of an ad campaign in any mdium is a complicatd phnomnon to quantify, on commonly accptd (and asily quantifid) notion in sarchbasd advrtising on th Intrnt is to maximiz th numbr of clicks. Th Intrnt sarch companis ar supportiv to- 1 Th choic of kywords is rlatd to th domain-knowldg of th advrtisr, usr bhavior and stratgic considrations. Intrnt sarch companis provid th advrtisrs with summaris of th qury traffic which is usful for thm to optimiz thir kyword choics intractivly. W do not dirctly addrss th choic of kywords in this papr, which is addrssd lswhr [13]. 4

2 wards advrtisrs and provid statistics about th history of click volums and prdiction about th futur prformanc of various kywords. Still, this is a complx problm for th following rasons (among othrs): Individual kywords hav significantly diffrnt charactristics from ach othr;.g., whil fishing is a broad kyword that matchs many usr quris and has many compting advrtisrs, human fishing bait is a nich kyword that matchs only a fw quris, but might hav lss comptition. Thr ar complx intractions btwn kywords bcaus a usr qury may match two or mor kywords, sinc th advrtisr is trying to covr all th possibl kywords in som domain. In ffct th advrtisr nds up compting with hrslf. As a rsult, th advrtisrs fac a challnging optimization problm. Th focus of this papr is to solv this optimization problm. 1.1 Th Budgt Optimization Problm W prsnt a short discussion and formulation of th optimization problm facd by advrtisrs; a mor dtaild dscription is in Sction 2. A givn advrtisr ss th stat of th auctions for sarchbasd advrtising as follows. Thr is a st K of kywords of intrst; in practic, vn small advrtisrs typically hav a larg st K. Thr is a st Q of quris posd by th usrs. For ach qury q Q, thr ar functions giving th clicks q(b) andcost q(b) that rsult from bidding a particular amount b in th auction for that qury, which w modl mor formally in th nxt sction. Thr is a bipartit graph G on th two vrtx sts rprsnting K and Q. For any qury q Q, th nighbors of q in K ar th kywords that ar said to match th qury q. 2 Th budgt optimization problm is as follows. Givn graph G togthr with th functions clicks q( ) andcost q( ) on th quris, as wll as a budgt U, dtrmin th bids b k for ach kyword k K such that P q clicksq(bq) ismaximizd subjct to P q costq(bq) U, whr th ffctiv bid b q on a qury is som function of th kyword bids in th nighborhood of q. Whil w can cast this problm as a traditional optimization problm, thr ar diffrnt challngs in practic dpnding on th advrtisr s accss to th qury and graph information, and indd th rliability of this information (.g., it could b basd on unstabl historical data). Thus it is important to find solutions to this problm that not only gt many clicks, but ar also simpl, robust and lss rliant on th information. In this papr w dfin th notion of a uniform stratgy which is ssntially a stratgy that bids uniformly on all kywords. Sinc this typ of stratgy obviats th nd to know anything about th particulars of th graph, and ffctivly aggrgats th click and cost functions on th quris, it is quit robust, and thus dsirabl in practic. What is surprising is that uniform stratgy actually prforms wll, which w will prov. 2 Th particulars of th matching rul ar dtrmind by th Intrnt sarch company; hr w trat th function as arbitrary. 1.2 Our Main Rsults and Tchnical Ovrviw W prsnt positiv and ngativ rsults for th budgt optimization problm. In particular, w show: Narly all formulations of th problm ar NP-Hard. In cass slightly mor gnral than th formulation abov, whr th clicks hav wights, th problm is inapproximabl bttr than a factor of 1 1, unlss P=NP. W giv a (1 1/)-approximation algorithm for th budgt optimization problm. Th stratgy found by th algorithm is a two-bid uniform stratgy, which mans that it randomizs btwn bidding som valu b 1 on all kywords, and bidding som othr valu b 2 on all kywords until th budgt is xhaustd 3. W show that this approximation ratio is tight for uniform stratgis. W also giv a (1/2)-approximation algorithm that offrs a singl-bid uniform stratgy, onlyusing on valu b 1. (This is tight for singl-bid uniform stratgis.) Ths stratgis can b computd in tim narly linar in Q + K, th input siz. Uniform stratgis may appar to b naiv in first considration bcaus th kywords vary significantly in thir click and cost functions, and thr may b complx intraction btwn thm whn multipl kywords ar rlvant to a qury. Aftr all, th optimum can configur arbitrary bids on ach of th kywords. Evn for th simpl cas whn th graph is a matching, th optimal algorithm involvs placing diffrnt bids on diffrnt kywords via a knapsack-lik packing (Sction 2). So, it might b surprising that a simpl two-bid uniform stratgy is 63% or mor ffctiv compard to th optimum. In fact, our proof is strongr, showing that this stratgy is 63% ffctiv against a strictly mor powrful advrsary who can bid indpndntly on th individual quris, i.., not b constraind by th intraction imposd by th graph G. Our proof of th 1 1/ approximation ratio rlis on an advrsarial analysis. W dfin a factor-rvaling LP (Sction 4) whr primal solutions corrspond to possibl instancs, and dual solutions corrspond to distributions ovr bidding stratgis. By driving th optimal solution to this LP, w obtain both th proof of th approximation ratio, and a tight worst-cas instanc. W hav conductd simulations using ral auction data from Googl. Th rsults of ths simulations, which ar highlightd at th nd of Sction 4, suggst that uniform bidding stratgis could b usful in practic. Howvr, important qustions rmain about (among othr things) altrnat bidding goals, on-lin or stochastic bidding modls [11], and gam-thortic concrns [3], which w brifly discuss in Sction MODELING A KEYWORD AUCTION W dscrib an auction from an advrtisr s point of viw. An advrtisr bids on a kyword, which w can think of as a word or st of words. Usrs of th sarch ngin submit quris. If th qury matchs a kyword that has bn bid on by an advrtisr, thn th advrtisr is ntrd into an auction for th availabl ad slots on th rsults pag. What constituts a match varis dpnding on th sarch ngin. 3 This typ of stratgy can also b intrprtd as bidding on valu (on all kywords) for part of th day, and a diffrnt valu for th rst of th day. 41

3 An advrtisr maks a singl bid for a kyword that rmains in ffct for a priod of tim, say on day. Th kyword could match many diffrnt usr quris throughout th day. Each usr qury might hav a diffrnt st of advrtisrs compting for clicks. Th advrtisr could also bid diffrnt amounts on multipl kywords, ach matching a (possibly ovrlapping) st of usr quris. Th ultimat goal of an advrtisr is to maximiz traffic to thir wbsit, givn a crtain advrtising budgt. W now formaliz a modl of kyword bidding and dfin an optimization problm that capturs this goal. 2.1 Landscaps W bgin by considring th cas of a singl kyword that matchs a singl usr qury. In this sction w dfin th notion of a qury landscap that dscribs th rlationship btwn th advrtisr s bid and what will happn on this qury as a rsult of this bid[9]. This dfinition will b cntral to th discussion as w continu to mor gnral cass Positions, bids and click-through rat Th sarch rsults pag for a qury contains p possibl positions in which our ad can appar. W dnot th highst (most favorabl) position by 1 and lowst by p. Associatd with ach position i is a valu α[i] that dnots th click-through rat (ctr) of th ad in position i. Thctris a masur of how likly it is that our ad will rciv a click if placd in position i. Th ctr can b masurd mpirically using past history. W assum throughout this work that that α[i] α[j] ifj<i, that is, highr positions rciv at last as many clicks as lowr positions. In ordr to plac an ad on this pag, w must ntr th auction that is carrid out among all advrtisrs that hav submittd a bid on a kyword that matchs th usr s qury. W will rfr to such an auction as a qury auction, tomphasiz that thr is an auction for ach qury rathr than for ach kyword. W assum that th auction is a gnralizd scond pric (GSP) auction [5, 7]: th advrtisrs ar rankd in dcrasing ordr of bid, and ach advrtisr is assignd a pric qual to th amount bid by th advrtisr blow thm in th ranking. 4 In sponsord sarch auctions, this advrtisr pays only if th usr actually clicks on th ad. Lt (b[1],...,b[p]) dnot th bids of th top p advrtisrs in this qury auction. For notational convninc, w assum that b[] = and b[p] =α[p] =. Sinc th auction is a gnralizd scond pric auction, highr bids win highr positions; i.. b[i] b[i + 1]. Suppos that w bid b on som kyword that matchs th usr s qury, thn our position is dfind by th largst b[i] thatisatmostb,thatis, pos(b) = arg max(b[i] :b[i] b). (1) i Sinc w only pay if th usr clicks (and that happns with probability α[i]), our xpctd cost for winning position i 4 Googl, Yahoo! and MSN all us som variant of th GSP auction. In th Googl auction, th advrtisrs bids ar multiplid by a quality scor bfor thy ar rankd; our rsults carry ovr to this cas as wll, which w omit from this papr for clarity. Also, othr auctions bsids GSP hav bn considrd;.g., th Vickry Clark Grovs (VCG) auction [14, 4, 7]. Each auction mchanism will rsult in a diffrnt sort of optimization problm. In th conclusion w point out that for th VCG auction, th bidding optimization problm bcoms quit asy. would b cost[i] = α[i] b[i], whr i = pos(b). W us cost q(b) and clicks q(b) to dnot th xpctd cost and clicks that rsult from having a bid b that qualifis for a qury auction q, and thus cost q(b) =α[i] b[i] whr i =pos(b), (2) clicks q(b) =α[i] whr i =pos(b). (3) Th following obsrvations about cost and clicks follow immdiatly from th dfinitions and quations (1), (2) and (3). W us R + to dnot th nonngativ rals. Obsrvation 1. For b R +, 1. (cost q(b), clicks q(b)) can only tak on on of a finit st of valus V q = {(cost[1],α[1]),...,(cost[p],α[p])}. 2. Both cost q(b) and clicks q(b) ar non-dcrasing functions of b. Also, cost-pr-click (cpc) cost q(b)/clicks q(b) is non-dcrasing in b. 3. cost q(b)/clicks q(b) b. For bids (b[1],...,b[p]) that corrspond to th bids of othr advrtisrs, w hav: cost q(b[i])/clicks q(b[i]) = b[i], i [p]. Whn th contxt is clar, w drop th subscript q Qury Landscaps W can summariz th data containd in th functions cost(b) and clicks(b) as a collction of points in a plot of cost vs. clicks, which w rfr to as a landscap. For xampl, for a qury with four slots, a landscap might look lik Tabl 1. bid rang cost pr click cost clicks [$2.6, ) $2.6 $1.3.5 [$2.,$2.6) $2. $.9.45 [$1.6,$2.) $1.6 $.4.25 [$.5,$1.6) $.5 $.1.2 [$,$.5) $ $ Tabl 1: A landscap for a qury It is convnint to rprsnt this data graphically as in Figur 1 (ignor th dashd lin for now). Hr w graph clicks as a function of cost. Obsrv that in this graph, th cpc (cost(b)/clicks(b)) of ach point is th rciprocal of th slop of th lin from th origin to th point. Sinc cost(b), clicks(b) andcost(b)/clicks(b) ar non-dcrasing, th slop of th lin from th origin to succssiv points on th plot dcrass. This condition is slightly wakr than concavity. Suppos w would lik to solv th budgt optimization problm for a singl qury landscap. 5 As w incras our bid from zro, our cost incrass and our xpctd numbr of clicks incrass, and so w simply submit th highst bid such that w rmain within our budgt. On problm w s right away is that sinc thr ar only a finit st of points in this landscap, w may not b abl to targt arbitrary budgts fficintly. Suppos in th xampl from Tabl 1 and Figur 1 that w had a budgt 5 Of cours it is a bit unralistic to imagin that an advrtisr would hav to worry about a budgt if only on usr qury was bing considrd; howvr on could imagin multipl instancs of th sam qury and th problm scals. 42

4 Clicks $.5 $1. $1.5 Cost Figur 1: A bid landscap. of $1.. Bidding btwn $2. and $2.6 uss only $.9, and so w ar undr-spnding. Bidding mor than $2.6 is not an option, sinc w would thn incur a cost of $1.3 and ovrspnd our budgt Randomizd stratgis To rctify this problm and bttr utiliz our availabl budgt, w allow randomizd bidding stratgis. Lt B b a distribution on bids b R +. Now w dfin cost(b) = E b B [cost(b)] and clicks(b) =E b B [clicks(b)]. Graphically, th possibl valus of (cost(b), clicks(b)) li in th convx hull of th landscap points. This is rprsntd in Figur 1 by th dashd lin. To find a bid distribution B that maximizs clicks subjct to a budgt, w simply draw a vrtical lin on th plot whr th cost is qual to th budgt, and find th highst point on this lin in th convx hull. This point will always b th convx combination of at most two original landscap points which thmslvs li on th convx hull. Thus, givn th point on th convx hull, it is asy to comput a distribution on two bids which ld to this point. Summarizing, Lmma 1. If an advrtisr is bidding on on kyword, subjct to a budgt U, thn th optimal stratgy is to pick a convx combination of (at most) two bids which ar at th ndpoints of th lin on th convx hull at th highst point for cost U. Thr is on subtlty in this formulation. Givn any bidding stratgy, randomizd or othrwis, th rsulting cost is itslf a random variabl rprsnting th xpctd cost. Thus if our budgt constraint is a hard budgt, w hav to dal with th difficultis that aris if our stratgy would b ovr budgt. Thrfor, w think of our budgt constraint as soft, that is, w only rquir that our xpctd cost b lss than th budgt. In practic, th budgt is oftn an avrag daily budgt, and thus w don t worry if w xcd it on day, as long as w ar mting th budgt in xpctation. Furthr, ithr th advrtisr or th sarch ngin (possibly both), monitor th cost incurrd ovr th day; hnc, th advrtisr s bid can b changd to zro for part of th day, so that th budgt is not ovrspnt. 6 Thus in th rmain- 6 S py?answr=22183, for xampl. dr of this papr, w will formulat a budgt constraint that only nds to b rspctd in xpctation Multipl Quris: a Knapsack Problm As a warm-up, w will considr nxt th cas whn w hav a st of quris, ach which its own landscap. W want to bid on ach qury indpndntly subjct to our budgt: th rsulting optimization problm is a small gnralization of th fractional knapsack problm, and was solvd in [9]. Th first stp of th algorithm is to tak th convx hull of ach landscap, as in Figur 1, and rmov any landscap points not on th convx hull. Each picwis linar sction of th curv rprsnts th incrmntal numbr of clicks and cost incurrd by moving on s bid from on particular valu to anothr. W rgard ths pics as itms in an instanc of fractional knapsack with valu qual to th incrmntal numbr of clicks and siz qual to th incrmntal cost. Mor prcisly, for ach pic conncting two conscutiv bids b and b on th convx hull, w crat a knapsack itm with valu [clicks(b ) clicks(b )] and siz [cost(b ) cost(b )]. W thn mulat th grdy algorithm for knapsack, sorting by valu/siz (cost-pr-click), and choosing grdily until th budgt is xhaustd. In this rduction to knapsack w hav ignord th fact that som of th pics com from th sam landscap and cannot b tratd indpndntly. Howvr, sinc ach curv is concav, th pics that com from a particular qury curv ar in incrasing ordr of cost-pr-click; thus from ach landscap w hav chosn for our knapsack a st of pics that form a prfix of th curv. 2.2 Kyword Intraction In rality, sarch advrtisrs can bid on a larg st of kywords, ach of thm qualifying for a diffrnt (possibly ovrlapping) st of quris, but most sarch ngins do not allow an advrtisr to appar twic in th sam sarch rsults pag. 7 Thus, if an advrtisr has a bid on two diffrnt kywords that match th sam qury, this conflict must b rsolvd somhow. For xampl, if an advrtisr has a bid out on th kywords shos and high-hl, thn if a usr issus th qury high-hl shos, it will match on two diffrnt kywords. Th sarch ngin spcifis, in advanc, a rul for rsolution basd on th qury th kyword and th bid. A natural rul is to tak th kyword with th highst bid, which w adopt hr, but our rsults apply to othr rsolution ruls. W modl th kyword intraction problm using an undirctd bipartit graph G =(K Q, E) whrk is a st of kywords and Q is a st of quris. Each q Q has an associatd landscap, as dfind by cost q(b) and clicks q(b). An dg (k, q) E mans that kyword k matchs qury q. Th advrtisr can control thir individual kyword bid vctor a R K + spcifying a bid a k for ach kyword k K. (For now, w do not considr randomizd bids, but w will introduc that shortly.) Givn a particular bid vctor a on th kywords, w us th rsolution rul of taking th maximum to dfin th ffctiv bid on qury q as b q(a) = max a k. k:(k,q) E By submitting a bid vctor a, th advrtisr rcivs som 7 S py?answr=14179, for xampl. 43

5 numbr of clicks and pays som cost on ach kyword. W us th trm spnd to dnot th total cost; similarly, w us th trm traffic to dnot th total numbr of clicks: spnd(a)= cost q(b q(a)); traffic(a)= clicks q(b q(a)) q Q q Q W also allow randomizd stratgis, whr an advrtisr givs a distribution A ovr bid vctors a R K +. Th rsulting spnd and traffic ar givn by spnd(a)=e a A[spnd(a)]; traffic(a)=e a A[traffic(a)] W can now stat th problm in its full gnrality: Budgt Optimization Input: a budgt U, a kyword-qury graph G =(K Q, E), and landscaps (cost q( ), clicks q( )) for ach q Q. Find: a distribution A ovr bid vctors a R K + such that spnd(a) U and traffic(a) is maximizd. W conclud this sction with a small xampl to illustrat som fatur of th budgt optimization problm. Suppos you hav two kywords K = {u, v} and two quris Q = {x, y} and dgs E = {(u, x), (u, y), (v, y)}. Suppos qury x has on position with ctr α x [1] = 1., and thr is on bid b x 1 = $1. Qury y has two positions with ctrs α y [1] = α y [2] = 1., and bids b y 1 =$ɛ and by 2 = $1 To gt any clicks from x, an advrtisr must bid at last $1 on u. Howvr, bcaus of th structur of th graph, if th advrtisr sts b u to $1, thn his ffctiv bid is $1 on both x and y. Thush must trad-off btwn gtting th clicks from x and gtting th bargain of a click for $ɛ that would b possibl othrwis. 3. UNIFORM BIDDING STRATEGIES As w will show in Sction 5, solving th Budgt Optimization problm in its full gnrality is difficult. In addition, it may b difficult to rason about stratgis that involv arbitrary distributions ovr arbitrary bid vctors. Advrtisrs gnrally prfr stratgis that ar asy to undrstand, valuat and us within thir largr goals. With this motivation, w look at rstrictd classs of stratgis that w can asily comput, xplain and analyz. W dfin a uniform bidding stratgy to b a distribution A ovr bid vctors a R K + whr ach bid vctor in th distribution is of th form (b,b,...,b) for som ral-valud bid b. In othr words, ach vctor in th distribution bids th sam valu on vry kyword. Uniform stratgis hav svral advantags. First, thy do not dpnd on th dgs of th intraction graph, sinc all ffctiv bids on quris ar th sam. Thus, thy ar ffctiv in th fac of limitd or noisy information about th kyword intraction graph. Scond, uniform stratgis ar also indpndnt of th priority rul bing usd. Third, any algorithm that givs an approximation guarant will thn b valid for any intraction graph ovr thos kywords and quris. W now show that w can comput th bst uniform stratgy fficintly. Suppos w hav a st of quris Q, whr th landscap V q for ach qury q is dfind by th st of points V q = {(cost q[1],α q[1]),...,(cost q[p],α q[p])}. W dfin th st of intrsting bids I q = {cost q[1]/α q[1],...,cost q[p]/α q[p]}, lt I = q QI q, and lt N = I. W can indx th points in I as b 1,...,b N in incrasing ordr. Th ith point in our aggrgat landscap V is found by summing, ovr th quris, th cost and clicks associatd with bid b i,thatis, V = N i=1( P q Q costq(bi), P q Q clicksq(bi)). For any possibl bid b, if w us th aggrgat landscap just as w would a rgular landscap, w xactly rprsnt th cost and clicks associatd with making that bid simultanously on all quris associatd with th aggrgat landscap. Thrfor, all th dfinitions and rsults of Sction 2 about landscaps can b xtndd to aggrgat landscaps, and w can apply Lmma 1 to comput th bst uniform stratgy (using th convx hull of th points in this aggrgat landscap). Th running tim is dominatd by th tim to comput th convx hull, which is O(N log N)[12]. Th rsulting stratgy is th convx combination of two points on th aggrgat landscap. Dfin a two-bid stratgy to b a uniform stratgy which puts non-zro wight on at most two bid vctors. W hav shown Lmma 2. Givn an instanc of Budgt Optimization in which thr ar a total of N points in all th landscaps, w can find th bst uniform stratgy in O(N log N) tim. Furthrmor, this stratgy will always b a two-bid stratgy. Putting ths idas togthr, w gt an O(N log N)-tim algorithm for Budgt Optimization, whrn is th total numbr of landscap points (w latr show that this is a (1 1 )-approximation algorithm): 1. Aggrgat all th points from th individual qury landscaps into a singl aggrgat landscap. 2. Find th convx hull of th points in th aggrgat landscap. 3. Comput th point on th convx hull for th givn budgt, which is th convx combination of two points α and β. 4. Output th stratgy which is th appropriat convx combination of th uniform bid vctors corrsponding to α and β. W will also considr a spcial cas of two-bid stratgis. A singl-bid stratgy is a uniform stratgy which puts nonzro wight on at most on non-zro vctor, i.. advrtisr randomizs btwn bidding a crtain amount b on all kywords, and not bidding at all. A singl-bid stratgy is vn asir to implmnt in practic than a two-bid stratgy. For xampl, th sarch ngins oftn allow advrtisrs to st a maximum daily budgt. In this cas, th advrtisr would simply bid b until hr budgt runs out, and th ad srving systm would rmov hr from all subsqunt auctions until th nd of th day. On could also us an ad schduling tool offrd by som sarch companis 8 to implmnt this stratgy. Th bst singl-bid stratgy can also b computd asily from th aggrgat landscap. Th optimal stratgy for a budgt U will ithr b th point x s.t. cost(x) isas larg as possibl without xcding U, or a convx combination of zro and th point y, whrcost(y) isassmallas possibl whil largr than U. 8 S py?answr=33227, for xampl. 44

6 cpc $.67 $.5 $.25 $.1 clicks cost cpc A 2 $1 $.5 B 5 $.5 $.1 C 3 $2 $.67 D 4 $1 $.25 B D Total clicks: A C wgtatlastr traffic, sinc this bid would nsur that for all q such that b q h(r ) w win as many clicks as Ω. Not that by bidding h(r ) on vry kyword, w may actually gt vn mor than r traffic, sinc for quris q whr b q is much lss than h(r )wmaywinmorclicksthanω. Howvr, all of ths xtra clicks still cost at most h(r ) pr click. Thus w s that for any r [,C Ω], if w bid h(r )on vry kyword, w rciv at last r traffic at a total spnd of at most h(r ) pr click. Not that by randomizing btwn bidding zro and bidding h(r ), w can rciv xactly r traffic at a total spnd of at most r h(r ). W summariz this discussion in th following lmma: Figur 2: Four quris and thir click-pric curv. 4. APPROIMATION ALGORITHMS In th prvious sction w proposd using uniform stratgis and gav an fficint algorithm to comput th bst such stratgy. In sction w prov that thr is always a good uniform stratgy: Thorm 3. Thr always xists a uniform bidding stratgy that is (1 1 )-optimal. Furthrmor, for any ɛ>, thr xists an instanc for which all uniform stratgis ar at most (1 1 + ɛ)-optimal. W introduc th notion of a click-pric curv, whichis cntral to our analysis. This dfinition maks it simpl to show that thr is always a singl-bid stratgy that is a approximation (and this is tight); w thn build on this to prov Thorm Click-pric curvs Considr a st of quris Q, and for ach qury q Q, lt (clicks q( ), cost q( )) b th corrsponding bid landscap. Considr an advrsarial biddr Ω with th powr to bid indpndntly on ach qury. Not that this biddr is mor powrful than an optimal biddr, which has to bid on th kywords. Suppos this stratgy bids b q for ach qury q. Thus, Ω achivs traffic C Ω = P i clicks(b i ), and incurs total spnd U Ω = P i cost(b i ). Without loss of gnrality w can assum that Ω bids so that for ach qury q, th cost pr click is qual to b q, i.. cost q(b q)/clicks q(b q)=b q. W may assum this bcaus for som qury q, ifcost q(b q)/clicks q(b q) <b q, w can always lowr b q and without changing th cost and clicks. To aid our discussion, w introduc th notion of a clickpric curv (an xampl of which is shown in Figur 2), which dscribs th cpc distribution obtaind by Ω. Formally th curv is a non-dcrasing function h :[,C Ω] R + dfind as h(r) =min{c P q:b q c clicksq(b q) r}. Anothr way to construct this curv is to sort th quris in incrasing ordr by b q =cost q(b q)/clicks q(b q), thn mak a stp function whr th qth stp has hight b q and width clicks q(b q) (s Figur 2). Not that th ara of ach stp is cost q(b q). Th following claim follows immdiatly: Claim 1. U Ω = R C Ω h(r)dr. Suppos w wantd to buy som fraction r /C Ω of th traffic that Ω is gtting. Th click-pric curv says that if w bid h(r ) on vry kyword (and thrfor vry qury), Lmma 4. For any r [,C Ω], thr xists a singl-bid stratgy that randomizs btwn bidding h(r) and bidding zro, and this stratgy rcivs xactly r traffic with total spnd at most r h(r). Lmma 4 dscribs a landscap as a continuous function. For our lowr bounds, w will nd to show that givn any continuous function, thr xists a discrt landscap that approximats it arbitrarily wll. Lmma 5. For any C, U > and non-dcrasing function f : [,C] R + such that R C f(r)dr = U, and any small ɛ >, thr xists an instanc of Budgt Optimization with budgt U +ɛ, whr th optimal solution achivs C clicks at cost U + ɛ, and all uniform bidding stratgis ar convx combinations of singl-bid stratgis that achiv xactly r clicks at cost xactly rf(r) by bidding f(r) on all kywords. Proof. Construct an instanc as follows. Lt ɛ>ba small numbr that w will latr dfin in trms of ɛ. Dfin r =,r 1,r 2,...,r m = C such that r i 1 <r i r i 1 + ɛ, f(r i 1) f(r i) f(r i 1)+ɛ, andm (C +f(c))/ɛ. (This is possibl by choosing r i s spacd by min(ɛ, f(r i) f(r i 1))) Now mak a qury q i for all i [m] with biddrs bidding h(r i),h(r i+1),...,h(r m), and ctrs α[1] = α[2] = = α[m i+1] = r i r i 1. Th graph is a matching with on kyword pr qury, and so w can imagin th optimal solution as bidding on quris. Th optimal solution will always bid xactly h(r i)onquryq i, and if it did so on all quris, it would spnd U := P m i=1 (ri ri 1)h(ri). Dfin ɛ small nough so that U = U + ɛ, which is always possibl sinc Z C m f(r)dr + (r i r i 1)(h(r i) h(r i 1)) U i=1 U + ɛ 2 m U + ɛ(c + f(c)). Not that th only possibl bids (i.., all othrs hav th sam rsults as on of ths) ar f(r ),...,f(r m), and bidding uniformly with f(r P i i)rsultsin j=1 ri ri 1 = ri clicks at cost h(r i)r i. 4.2 A 1 -approximation algorithm 2 Using Lmma 4 w can now show that thr is a uniform singl-bid stratgy that is 1 -optimal. In addition to bing an 2 intrsting rsult in its own right, it also srvs as a warm-up for our main rsult. Thorm 6. Thr always xists a uniform singl-bid stratgy that is 1 -optimal. Furthrmor, for any ɛ > thr 2 xists an instanc for which all singl-bid stratgis ar at most ( 1 + ɛ)-optimal. 2 45

7 Proof. Applying Lmma 4 with r = C Ω/2, w s that thr is a stratgy that achivs traffic C Ω/2 withspnd C Ω/2 h(c Ω/2). Now, using th fact that h is a non-dcrasing function combind with Claim 1, w hav (C Ω/2)h(C Ω/2) C Ω /2 h(r)dr h(r)dr = U Ω, (4) which shows that w spnd at most U Ω. W conclud that thr is a 1 -optimal singl-bid stratgy randomizing btwn 2 bidding C Ω/2 andzro. For th scond part of th thorm, w construct a tight xampl using two quris Q = {x, y}, twokywordsk = {u, v}, anddgse = {(u, x), (v, y)}. Fix som α whr <α 1, and fix som vry small ɛ>. Qury x has two positions, with bids of b x 1 =1/α and b x 2 = ɛ, and with idntical click-through rats α x [1] = α x [2] = α. Quryy has on position, with a bid of b y 1 =1/α andaclick-throughratofα y [1] = α. Th budgt is U = 1+ɛα. Th optimal solution is to bid ɛ on u (and thrfor x) and bid 1/α on v (and thrfor y), both with probability 1. This achivs a total of 2α clicks and spnds th budgt xactly. Th only usful bids ar, ɛ and 1/α, sincfor both quris all othr bids ar idntical in trms of cost and clicks to on of thos thr. Any singl-bid solution that uss ɛ as its non-zro bid gts at most α clicks. Bidding 1/α on both kywords rsults in 2α clicks and total cost 2. Thus, sinc th budgt is U =1+ɛα < 2, a singl-bid solution using 1/α can put wight at most (1+ɛα)/2 onth1/α bid. This rsults in at most α(1 + ɛα) clicks. This can b mad arbitrarily clos to α by lowring ɛ. 4.3 A (1 1 )-approximation algorithm Th ky to th proof of Thorm 3 is to show that thr is a distribution ovr singl-bid stratgis from Lmma 4 that obtains at last (1 1 )CΩ clicks. In ordr to figur out th bst distribution, w wrot a linar program that modls th bhavior of a playr who is trying to maximiz clicks and an advrsary who is trying to crat an input that is hard for th playr. Thn using linar programming duality, w wr abl to driv both an optimal stratgy and a tight instanc. Aftr solving th LP numrically, w wr also abl to s that thr is a uniform stratgy for th playr that always obtains (1 1 )CΩ clicks; and thn from th solution wr asily abl to guss th optimal distribution. This mthodology is similar to that usd in work on factor-rvaling LPs [8, 1] An LP for th worst-cas click-pric curv. Considr th advrsary s problm of finding a click-pric curv for which no uniform bidding stratgy can achiv αc Ω clicks. Rcall that by Lmma 1 w can assum that a uniform stratgy randomizs btwn two bids u and v. W also assum that th uniform stratgy uss a convx combination of stratgis from Lmma 4, which w can assum by Lmma 5. Thus, to achiv αc Ω clicks, a uniform stratgy must randomiz btwn bids h(u) andh(v) whru αc Ω and v αc Ω. Call th st of such stratgis S. Givn a (u, v) S, th ncssary probabilitis in ordr to achiv αc Ω clicks ar asily dtrmind, and w dnot thm by p 1(u, v) andp 2(u, v) rspctivly. Not furthr that th advrtisr is trying to figur out which of ths stratgis to us, and ultimatly wants to comput a distribution ovr uniform stratgis. In th LP, sh is actually going to comput a distribution ovr pairs of stratgis in S, whichw will thn intrprt as a distribution ovr stratgis. Using this st of uniform stratgis as constraints, w can charactriz a st of worst-cas click-pric curvs by th constraints h(r)dr U (u, v) S p 1(u, v)uh(u)+p 2(u, v)vh(v) U Acurvh that satisfis ths constraints has th proprty that all uniform stratgis that obtain αc Ω clicks spnd mor than U. Discrtizing this st of inqualitis, and pushing th first constraint into th objctiv function, w gt th following LP ovr variabls h r rprsnting th curv: min ɛ h r s.t. r {,ɛ,2ɛ,...,c Ω } (u, v) S, p 1(u, v)uh u + p 2(u, v)vh v U In this LP, S is dfind in th discrt domain as S = {(u, v) {,ɛ,2ɛ,...,c Ω} 2 : u αc Ω v C Ω}. Solving this LP for a particular α, if w gt an objctiv lss than U, w know (up to som discrtization) that an instanc of Budgt Optimization xists that cannot b approximatd bttr than α. (Th instanc is constructd as in th proof of Lmma 5.) A binary sarch yilds th smallst such α whr th objctiv is xactly U. To obtain a stratgy for th advrtisr, w look at th dual, constraining th objctiv to b qual to U in ordr to gt th polytop of optimum solutions: w u,v =1 (u, v) S, (u,v) S v :(u,v ) S u :(u,v) S p 1(u, v ) u w u,v ɛ and p 2(u,v) v w u,v ɛ. It is straightforward to show that th scond st of constraints is quivalnt to th following: (u,v) S h R C Ω/ɛ : r ɛh r = U, w u,v(p 1(u, v) u h u + p 2(u, v) v h v) U. Hr th variabls can b intrprtd as wights on stratgis in S. A point in this polytop rprsnts a convx combination ovr stratgis in S, with th proprty that for any click-pric curv h, th cost of th mixd stratgy is at most U. Sinc all stratgis in S gt at last αc Ω clicks, w hav a stratgy that achivs an α-approximation. Intrstingly, th quivalnc btwn this polytop and th LP dual abov shows that thr is a mixtur ovr valus r [,C] that achivs an α-approximation for any curv h. Aftr a sarch for th appropriat α (which turnd out to b 1 1 ), w solvd ths two LPs and cam up with th plots in Figur 3, which rval not only th right approximation ratio, but also a pictur of th worst-cas distribution and th approximation-achiving stratgy. 9 From th pic- 9 Th paramtrs U and C Ω can b st arbitrarily using scaling argumnts. 46

8 C/ C C/ Figur 3: Th worst-cas click-pric curv and (1 1/)-approximat uniform bidding stratgy, as found by linar programming. turs, w wr abl to quickly guss th optimal stratgy and worst cas xampl Proof of Thorm 3 By Lmma 4, w know that for ach r U Ω,thrisa stratgy that can obtain traffic r at cost r h(r). By mixing stratgis for multipl valus of r, w construct a uniform stratgy that is guarantd to achiv at last 1 1 =.63 fraction of Ω s traffic and rmain within budgt. Not that th final rsulting bid distribution will hav som wight on th zro bid, sinc th singl-bid stratgis from Lmma 4 put som wight on bidding zro. Considr th following probability dnsity function ovr such stratgis (also dpictd in Figur 3): j for r<cω/, g(r) = 1/r for r C Ω/. Not that R C Ω g(r) dr = R C Ω 1 dr = 1, i.. g is a probability dnsity function. Th traffic achivd by our stratgy C Ω / r is qual to 1 traffic = g(r) rdr = C Ω / r rdr = Th xpctd total spnd of this stratgy is at most spnd = = C Ω / g(r) rh(r) dr h(r) dr C 1 1 «C Ω. h(r) dr = U Ω. Thus w hav shown that thr xists a uniform bidding stratgy that is (1 1 )-optimal. W now show that no uniform stratgy can do bttr. W will prov that for all ɛ> thr xists an instanc for which all uniform stratgis ar at most (1 1 + ɛ)-optimal. First w dfin th following click-pric curv ovr th domain [, 1]: 8 < h(r) = : 1 2 for r< 1 1 «for r 1 r Not that h is non-dcrasing and non-ngativ. Sinc th curv is ovr th domain [, 1] it corrsponds to an instanc whr C Ω =1. Notalsothat R 1 R 1 1 h(r) dr = 2 1/ 1 dr = 1. Thus, this curv corrsponds to an instanc whr r U Ω = 1. Using Lmma 5, w construct an actual instanc whr th bst uniform stratgis ar convx combinations of stratgis that bid h(u) and achiv u clicks and u h(u) cost. Suppos for th sak of contradiction that thr xists a uniform bidding stratgy that achivs α>1 1 traffic on this instanc. By Lmma 1 thr is always a two-bid optimal uniform bidding stratgy and so w may assum that th stratgy achiving α clicks randomizs ovr two bids. To achiv α clicks, th two bids must b on valus h(u) and h(v) with probabilitis p u and p v such that p u + p v =1, u α v and p uu + p vv = α. To calculat th spnd of this stratgy considr two cass: if u = thn w ar bidding h(v) with probability p v = α/v. Th spnd in this cas is: α α/v spnd = p v v h(v) =αh(v) =. 2 Using v α and thn α>1 1 w gt spnd α 1 2 > (1 1/) 1 2 =1, contradicting th assumption. W turn to th cas u >. Hr w hav p u = v α v u and p v = α u v u Not that for r (, 1] w hav h(r) 1 ). Thus 2 r spnd p u uh(u)+p v vh(v) = (v α)(u 1) + (α u)(v 1) (v u)( 2) = α 1 2 > 1. Th final inquality follows from α>1 1. Thus in both cass th spnd of our stratgy is ovr th budgt of Exprimntal Rsults W ran simulations using th data availabl at Googl which w brifly summariz hr. W took a larg advrtising campaign, and, using th st of kywords in th campaign, computd thr diffrnt curvs (s Figur 4) for thr diffrnt bidding stratgis. Th x-axis is th budgt (units rmovd), and th y-axis is th numbr of clicks obtaind (again without units) by th optimal bid(s) undr ach rspctiv stratgy. Qury bidding rprsnts our (unachivabl) uppr bound Ω, bidding on ach qury indpndntly. Th uniform bidding curvs rprsnt th rsults of applying our algorithm: dtrministic uss a singl bid lvl, whil randomizd uss a distribution. For rfrnc, w includ th lowr bound of a ( 1)/ fraction of th top curv. Th data clarly dmonstrat that th bst singl uniform bid obtains almost all th possibl clicks in practic. Of cours in a mor ralistic nvironmnt without full knowldg, it is not always possibl to find th bst such bid, so furthr invstigation is rquird to mak this approach usful. Howvr, just knowing that thr is such a bid availabl should mak th on-lin vrsions of th problm simplr. 5. HARDNESS RESULTS By a rduction from vrtx covr w can show th following (proof omittd): Thorm 7. Budgt Optimization is strongly NP-hard. 47

9 1 Qury Bidding Uniform Bidding (randomizd) Uniform Bidding (dtrministic) Lowr bound dsigning a hirarchical kyword st (.g., shos, highhl shos, athltic shos ). W call a solution dtrministic if it consists of on bid vctor, rathr than a gnral distribution ovr bid vctors. Th following lmma will b usful for giving a structur to th optimal solution, and will allow dynamic programming. Clicks.5 Lmma 1. For kywords i, j K, ifq i Q j thn thr xists an optimal dtrministic solution to th Budgt Optimization problm with a i a j..5 Budgt Figur 4: An xampl with ral data. Now suppos w introduc wights on th quris that indicat th rlativ valu of a click from th various sarch usrs. Formally, w hav wights w q for all q Q and our goal is maximiz th total wightd traffic givn a budgt. Call this th Wightd Kyword Bidding problm. With this additional gnralization w can show hardnss of approximation via a simpl rduction from th Maximum Covrag problm, which is known to b (1 1/)-hard [6] (proof omittd). Thorm 8. Th Wightd Kyword Bidding problm is hard to approximat to within (1 1/). 6. EACT ALGORITHMS FOR LAMINAR GRAPHS If a graph has spcial structur, w can somtims solv th budgt optimization problm xactly. Not that th knapsack algorithm in Sction 2 solvs th problm for th cas whn th graph is a simpl matching. Hr w gnraliz this to th cas whn th graph has a laminar structur, which will allow us to impos a (partial) ordring on th possibl bid valus, and thrby giv a psudopolynomial algorithm via dynamic programming. W first show that to solv th Budgt Optimization problm (for gnral graphs) optimally in psudopolynomial tim, it suffics to provid an algorithm that solvs th dtrministic cas. Th proof (omittd) uss idas similar to Obsrvation 1 and Lmma 1. Lmma 9. Lt I b an input to th Budgt Optimization problm and suppos that w find th optimal dtrministic solution for vry possibl budgt U U. Thnw can find th optimal solution in tim O(U log U). A collction S of n sts S 1,...,S 2 is laminar if, for any two sts S i and S j,ifs i S j thn ithr S i S j or S j S i. Givn a kyword intraction graph G, w associat a st of nighboring quris Q k = {q :(k, q) E} with ach kyword k. If this collction of sts if laminar, w say that th graph has th laminar proprty. Not that a laminar intraction graph would naturally fall out as a consqunc of 1 W can viw th laminar ordr as a tr with kyword j as a parnt of kyword i if Q j is th minimal st containing Q i. In this cas w say that j is a child of i. Givn a kyword j with c childrn i 1,...,i c, w now nd to numrat ovr all ways to allocat th budgt among th childrn and also ovr all possibl minimum bids for th childrn. A complication is that a nod may hav many childrn and thus a trm of U c would not vn b psudopolynomial. W can solv this problm by showing that givn any laminar ordring, thr is an quivalnt on in which ach kyword has at most 2 childrn. Lmma 11. Lt G b a graph with th laminar proprty. Thr xists anothr graph G with th sam optimal solution to th Budgt Optimization problm, whr ach nod has at most two childrn in th laminar ordring. Furthrmor, G has at most twic as many nods as G. Givn a graph with at most two childrn pr nod, w dfin F [i, b, U] to b th maximum numbr of clicks achivabl by bidding at last b on ach of kywords j s.t. Q j Q i (and xactly b on kyword i) whil spnding at most U. For this dfinition, w us Z(b, U) to dnot st of allowabl bids and budgts ovr childrn: Z(b, U) = {b, b,u,u : b b, U U, b b, U U, U + U U} Givn a kyword i and a bid a i, comput an incrmntal spnd and traffic associatd with bidding a i on kyword i, that is ˆt(i, a i) = clicks q(a i), and q Q i \Q i 1 ŝ(i, a i) = cost q(a i). q Q i \Q i 1 Now w dfin F [i, b, U] as max b, b,u,u Z(b,U) j F [j,b,u ]+F [j,b,u ]+ˆt(i, b) if (ŝ(i, b) U U U and i>), and F [i, b, U] = othrwis. Lmma 12. If th graph G has th laminar proprty, thn, aftr applying Lmma 11, th dynamic programming rcurrnc in (5) finds an optimal dtrministic solution to th Budgt Optimization problm xactly in O(B 3 U 3 n) tim. In addition, w can apply Lmma 9 to comput th optimal (randomizd) solution. Obsrv that in th dynamic program, w hav alrady solvd th instanc for vry budgt U U, so w can find th randomizd solution with no additional asymptotic ovrhad. ff (5) 48

10 Lmma 13. If th graph G has th laminar proprty, thn, by applying Lmma 11, th dynamic programming rcurrnc in (5), and Lmma 9, w can find an optimal solution to th Budgt Optimization problm in O(B 3 U 3 n) tim. Th bounds in this sction mak pssimistic assumptions about having to try vry budgt and vry lvl. For many problms, you only nd to choos from a discrt st of bid lvls (.g., multipls of on cnt). Doing so yilds th obvious improvmnt in th bounds. 7. BID OPTIMIZATION UNDER VCG Th GSP auction is not th only possibl auction on could us for sponsord sarch. Indd th VCG auction and variants [14, 4, 7, 1] offr altrnativs with complling gam-thortic proprtis. In this sction w argu that th budgt optimization problm is asy undr th VCG auction. For a full dfinition of VCG and its application to sponsord sarch w rfr th radr to [1, 2, 5]. For th sak of th budgt optimization problm w can dfin VCG by just rdfining cost q(b) (rplacing Equation (2)): p 1 cost q(b) = (α[j] α[j +1]) b[j] j=i whr i =pos(b). Obsrvation 1 still holds, and w can construct a landscap as bfor, whr ach landscap point corrsponds to a particular bid b[i]. W claim that in th VCG auction, th landscaps ar convx. To s this, considr two conscutiv positions i,i + 1. Th slop of lin sgmnt btwn th points corrsponding to thos two positions is cost(b[i]) cost(b[i +1]) (α[i] α[i +1]) b[i] = = b[i]. clicks(b[i]) clicks(b[i +1]) α[i] α[i +1] Sinc b[i] b[i + 1], th slops of th pics of th landscap dcras, and w gt that th curv is convx. Now considr running th algorithm dscribd in Sction for finding th optimal bids for a st of quris. In this algorithm w took all th pics from th landscap curvs, sortd thm by incrmntal cpc, thn took a prfix of thos pics, giving us bids for ach of th quris. But, th quation abov shows that ach pic has its incrmntal cpc qual to th bid that achivs it; thus in th cas of VCG th pics ar also sortd by bid. Hnc w can obtain any prfix of th pics via a uniform bid on all th kywords. W conclud that th bst uniform bid is an optimal solution to th budgt optimization problm. 8. CONCLUDING REMARKS Our algorithmic rsult prsnts an intriguing huristic in practic: bid a singl valu b on all kywords; at th nd of th day, if th budgt is undr-spnt, adjust b to b highr; if budgt is ovrspnt, adjust b to b lowr; ls, maintain b. If th scnario dos not chang from day to day, this simpl stratgy will hav th sam thortical proprtis as our on-bid stratgy, and in practic, is likly to b much bttr. Of cours th scnario dos chang, howvr, and so coming up with a stochastic bidding stratgy rmains an important opn dirction, xplord somwhat by [11, 13]. Anothr intrsting gnralization is to considr wights on th clicks, which is a way to modl convrsions. (Aconvrsion corrsponds to an action on th part of th usr who clickd through to th advrtisr sit;.g., a sal or an account sign-up.) Finally, w hav lookd at this systm as a black box rturning clicks as a function of bid, whras in rality it is a complx rpatd gam involving multipl advrtisrs. In [3], it was shown that whn a st of advrtisrs us a stratgy similar to th on w suggst hr, undr a slightly modifid first-pric auction, th prics approach a wll-undrstood markt quilibrium. Acknowldgmnts W thank Rohit Rao, Zoya Svitkina and Adam Wildavsky for hlpful discussions. 9. REFERENCES [1] G. Aggarwal, A. Gol and R. Motwani. Truthful auctions for pricing sarch kywords. ACM Confrnc on Elctronic Commrc, 1-7, 26. [2] G. Aggarwal, J. Fldman and S. Muthukrishnan Bidding to th Top: VCG and Equilibria of Position-Basd Auctions Proc. WAOA, 26. [3] C. Borgs, J. Chays, O. Etsami, N. Immorlica, K. Jain, and M. Mahdian. Dynamics of bid optimization in onlin advrtismnt auctions. Proc. WWW 27. [4] E. Clark. 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