Externalities in Online Advertising

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1 Extrnalitis in Onlin Advrtising Arpita Ghosh Yahoo! Rsarch 2821 Mission Collg Blvd. Santa Clara, CA Mohammad Mahdian Yahoo! Rsarch 2821 Mission Collg Blvd. Santa Clara, CA ABSTRACT Most modls for onlin advrtising assum that an advrtisr s valu from winning an ad auction, which dpnds on th clickthrough rat or convrsion rat of th advrtismnt, is indpndnt of othr advrtismnts srvd alongsid it in th sam sssion. This ignors an important xtrnality ffct: as th advrtising audinc has a limitd attntion span, a high-quality ad on a pag can dtract attntion from othr ads on th sam pag. That is, th utility to a winnr in such an auction also dpnds on th st of othr winnrs. In this papr, w introduc th problm of modling xtrnalitis in onlin advrtising, and study th winnr dtrmination problm in ths modls. Our modls ar basd on choic modls on th audinc sid. W show that in th most gnral cas, th winnr dtrmination problm is hard vn to approximat. Howvr, w giv an approximation algorithm for this problm with an approximation factor that is logarithmic in th ratio of th maximum to th minimum bid. Furthrmor, w show that thr ar som intrsting spcial cass, such as th cas whr th audinc prfrncs ar singl pakd, whr th problm can b solvd xactly in polynomial tim. For all ths algorithms, w prov that th winnr dtrmination algorithm can b combind with VCG-styl paymnts to yild truthful mchanisms. Catgoris and Subjct Dscriptors F.2.0 [Analysis of Algorithms and Problm Complxity]: Gnral; J.4 [Social and Bhavioral Scincs]: Economics Gnral Trms Algorithms, Economics, Thory Kywords Advrtising, xtrnalitis, auctions, approximation algorithms 1. INTRODUCTION Much of th work on auctions for onlin advrtising assum that a biddr has an intrinsic valu for winning an auction: givn that a biddr is dclard a winnr of an ad Copyright is hld by th Intrnational World Wid Wb Confrnc Committ (IW3C2). Distribution of ths paprs is limitd to classroom us, and prsonal us by othrs. WWW 2008, April 21 25, 2008, Bijing, China. ACM /08/04. slot, h drivs som privat utility, which is unaffctd by th numbr or st of othr winnrs. Howvr, this is not ncssarily tru, spcially in th contxt of advrtising: advrtisrs compt for usrs attntion, and th attntion a particular winning advrtisr rcivs clarly dpnds on th st of all winnrs. In this papr, w ar intrstd in th problm of mchanism dsign in a stting with xtrnalitis, i.., whn a winnr s utility dpnds on th st of othr winnrs. Th problm of mchanism dsign with xtrnalitis can b motivatd in multipl sttings in th contxt of onlin advrtising, for instanc, in th stting of sponsord sarch (.g., Googl s Adwords or Yahoo! s Sarch Markting) or ad placmnt on contnt pags (.g., Googl s Adsns or Yahoo! s Contnt Match). In this papr w focus on a stting commonly known as onlin lad gnration, or th pay-prlad modl [11]. Th objctiv of onlin lad gnration is to sll crdibl lads (in th form of prsonal information of a potntial customr) to companis, or advrtisrs, intrstd in such lads. Th advrtisrs thn contact th potntial customr dirctly to offr quots and information about thir srvic. This modl of advrtising is currntly most popular among financial firms that offr mortgags, insuranc companis, auto dalrs intrstd in potntial buyrs of nw cars, and th distanc ducation industry 1. According to th PricwatrhousCooprs IAB Rvnu Rport for yar 2006 [12], lad gnration rvnus accountd for 8 prcnt of th 2006 full yar rvnus or $1.3 billion, up from th 6 prcnt or $753 million rportd in In addition to having bn th initial motivation of th prsnt work, focusing on th lad gnration modl has th advantag that it simplifis th discussion of xtrnalitis by abstracting away aspcts of th problm rlating to th spcific placmnt of th ads on a pag. W will brifly discuss such issus and thir intrplay with th xtrnality problm at th nd of this papr. Th main problm facd by a lad gnration company that has acquird a lad is th following tradoff: if th lad is snt to fwr advrtisrs, th valu to ach advrtisr will b highr sinc thy ar compting with fwr othr advrtisrs for th potntial customr. Furthr, th valu to an advrtisr might also dpnd spcifically on which othr advrtisrs obtain th lad, not just how many othrs: a 1 Anothr growing industry that can b studid in th framwork of lad gnration is onlin dating srvics. Howvr, sinc currntly th standard in this industry is basd on flatf subscription and not pay-for-prformanc, our modls might not b dirctly applicabl.

2 compting advrtisr who provids a similar srvic and is vry likly to offr a bttr dal to th usr dcrass th valu of th lad much mor than a lss comptitiv advrtisr, or an advrtisr who is offring a diffrnt srvic (.g., a Toyota dalr might dcras th valu to a Honda dalr mor than a Ford dalr dos). In any cas, th utility to an advrtisr who buys a lad dpnds on othr buyrs of th sam lad. In addition, typically a lad can only b sold to a limitd numbr of advrtisrs, as spcifid in th privacy policy of th wbsit [11]. In th most gnral and abstract form, th problm w ar intrstd in is th following: thr ar n biddrs, ach with a utility function u i : 2 {1,...,n} R, whr u i(s) is th utility that biddr i drivs whn th st of winnrs is S {1,..., n}. It is rasonabl to assum that u i(s) is zro if i / S. Th problm is to dsign an incntiv compatibl mchanism to maximiz wlfar. Such a mchanism would slct th subst of biddrs that maximizs v(s) = i S u i(s). VCG paymnts can b usd to induc truthful rporting if th optimal subst can b found: vry biddr i is chargd th diffrnc btwn th valu of th optimal st whn i is rmovd from th st of biddrs and v(s) u i(s), which is th valu drivd by all rmaining winnrs in th currnt solution. Sinc spcifying a utility function of th form abov taks xponntial spac in th numbr of biddrs, w ar intrstd in invstigating modls for utility functions that ar both ralistic in th contxt of onlin advrtising, and also allow compact rprsntation. W build such a modl by looking at th choic problm from th advrtising audinc s prspctiv. Our modl assums that customrs hav (possibly intrdpndnt) valuations for diffrnt advrtisrs (ths valuations might b a function of th quot th customr rcivs from th advrtisrs in a lad gnration businss, or th prcption of th quality of th product offrd by th advrtisr) and also for an outsid option. Whn prsntd with a numbr of choics, thy pick th advrtisr whom thy hav th highst valuation for, or no advrtisr if thir valuation of th outsid option is gratr than th valuations of all advrtisrs prsntd in th st. This modl is dfind in mor dtail in th nxt sction. W will study th computational complxity of th winnr dtrmination problm in this modl, and prov that in th most gnral cas, if th distributions of th valus ar givn xplicitly, th winnr dtrmination problm is hard to approximat within a constant factor. On th othr hand, w will giv an approximation algorithm that solvs this problm within a factor that is logarithmic in th ratio of th maximum to th minimum bid. Furthrmor, in svral spcial cass, most notably in th cas that distributions ar singl-pakd, th winnr dtrmination problm can b solvd xactly in polynomial tim. W will prov that ths algorithms, combind with a VCG-styl paymnt schm, giv ris to dominantstratgy incntiv compatibl mchanisms. Finally, w discuss altrnativ modls for xtrnalitis, and dirctions for futur work. Rlatd work. Auctions with xtrnalitis hav bn studid in th conomics litratur, th arlist rlatd work bing that of Jhil, Moldovanu and Stacchtti [7], whr a losr s valu dpnds on th idntity of th winnr. Th problm of mchanism dsign with allocativ xtrnalitis is also studid by Jhil and Moldovanu [5], and Jhil, Moldovanu and Stacchtti [8]; Jhil and Moldovanu [6] study mchanism dsign with both allocativ and informational xtrnalitis. Howvr, non of ths paprs addrss computational issus arising from th mchanism dsign problm. To th bst of our knowldg, this is th first thortical work that spcifically addrsss th problm of xtrnalitis in onlin advrtising. Limitd xprimntal vidnc for th hypothsis that th click-through rat of ads dpnd on surrounding ads is providd in th work of Joachims t al. [9]. 2. THE MODEL In this sction, w dfin a modl for xtrnalitis among advrtisrs that is th focus of this papr. Our modl is basd on th intuition that ach advrtisr has a privat valu for capturing th businss of th usr (th advrtising audinc). Th usr, on th othr hand, gts xposd to a numbr of ads, and chooss at most on of ths ads basd on hr prcption of th quality of th advrtisr. For xampl, in th cas of th lad gnration businss, th usr rcivs quots from th advrtisrs who rciv th lad, and probably will choos th advrtisr who givs hr th lowst quot, or non of th advrtisrs, if sh rcivs a bttr quot through anothr mdium. Mor formally, suppos thr ar n advrtisrs numbrd 1,..., n, ach with a privat valu v i (which is th valu advrtisr i drivs whn h is chosn by a usr). Th quality of advrtisr i (from th prspctiv of th usr) is dnotd by q i. Furthrmor, lt q 0 dnot th quality of th bst outsid option. Ths quality paramtrs q i ar random variabls, drawn from a joint probability distribution Q. Intuitivly, considring q i s as random variabls (as opposd to dtrministic valus) capturs th fact that usrs do not all mak th sam choics among th advrtisrs. Also, in gnral th q i s nd not b indpndnt, sinc th choics of usrs ar oftn dictatd by th sam gnral principls. For xampl, knowing that a usr prcivs Ford autos as suprior to Toyotas incrass th liklihood that sh also prfrs Chvy to Honda. Whn a st S of advrtisrs is chosn, th usr picks th advrtisr with th largst quality q i in S, if th quality of this advrtisr is gratr than that of th outsid option. This advrtisr thn drivs a valu of v i; all othr advrtisrs driv a valu of 0. So th xpctd valu whn a st S of advrtisrs is chosn is v(s) = i S v ipr[ j S {0} : q i q j], whr th probability is ovr a random draw of (q 0, q 1,..., q n) from Q. For simplicity, w assum that q i s ar always distinct, and thrfor w do not nd to spcify how tis ar brokn. Th winnr dtrmination problm in this modl is to choos a st S of at most a givn numbr k of advrtisrs to maximiz v(s). Not that v(s) is not monoton in S: adding an advrtisr with low valu but high quality can actually caus a nt dcras in th valu of th st. Th winnr dtrmination problm can b writtn as th follow-

3 ing mathmatical program: maximiz xi s.t. P n Pi=1 n i=1 vixipr[ j = 0,..., n : qi qjxj] xi k, x i {0, 1}, i = 1,..., n, x 0 = 1. (1) Bfor w can start talking about th computational complxity of th winnr dtrmination problm, w nd to spcify how th input is rprsntd. In particular, thr ar many ways th distribution Q can b rprsntd. In this papr, w mainly considr an xplicit rprsntation of Q, i.., whn th distribution has a finit support and all lmnts of th support of this distribution ar listd as part of th input (as xplaind blow). This is prhaps th simplst way to rprsnt th distribution, and our hardnss rsult in th nxt sction (showing that th winnr dtrmination problm is hard to approximat for this rprsntation) clarly carris ovr to strongr rprsntations, such as modls whr Q is givn by an oracl. Obsrv that in our modl, th actual valus of th quality paramtrs q i do not mattr; all that mattrs is th rlativ ordring of th qualitis. Spcifically, th only ral information w us from th distribution is th probability of ach ranking of th biddrs qualitis and th quality of th outsid option. In othr words, w can assum that thr ar a finit numbr of diffrnt usr typs, and ach usr typ j is givn by a ral numbr which indicats th probability that a random usr is of typ j, and a prmutation of th n + 1 options {0,..., n} (with 0 rprsnting th outsid option and 1,..., n rprsnting th advrtisrs). Not that th ordring of advrtisrs that occur aftr th outsid option in a prmutation is irrlvant, and thrfor can b omittd. To summariz, th xtrnality problm with an xplicitly givn distribution can b formulatd as follows. Winnr Dtrmination Problm with Extrnalitis: Input: - Intgrs n, m, and k, A subst S of {1,..., n} with S k that max- Output: imizs v(s) := - a valu v i for ach i = 1,..., n, - a probability p j for ach j = 1,..., m with P m j=1 pj = 1, and - a prmutation π j of {0,..., n} for ach j = 1,..., m. m p j v ii( l S {0} : π j(i) π j(l)), j=1 i S whr I( l S {0} : π j(i) π j(l)) dnots th indicator variabl for th vnt that i is th highst lmnt in prmutation π j of all mmbrs of S 0. In th nxt sction, w will show a strong hardnss rsult for th abov problm. In Sction 4 w will giv an approximation algorithm, and in Sction 5 w prov that th problm is solvabl in polynomial tim if prfrncs π j ar singl-pakd, and also in anothr spcial cas of th problm with implicitly givn distributions. 3. HARDNESS OF THE WINNER DETER- MINATION PROBLEM W show that th winnr dtrmination problm dfind in th prvious sction, vn with no cardinality constraints (i.., k = n), is NP-hard, and hard to approximat. Th proof is basd on a rduction from th indpndnt st problm in graphs. Rcall that in th indpndnt st problm, w ar givn a graph G and th objctiv is to find a maximum-cardinality subst S of vrtics with no dg btwn any two vrtics of S. Håstad [14] provd that this problm cannot b approximatd in polynomial tim to within a factor of n 1 ɛ for any ɛ > 0, unlss NP = ZPP. Thorm 1. Th winnr dtrmination problm with xtrnalitis is hard to approximat in polynomial tim within a factor n 1 ɛ for any ɛ > 0, unlss NP = ZPP. Proof. Givn an instanc G of th indpndnt st problm, w map it to an instanc of th winnr dtrmination problm with xtrnalitis as follows. Lt n = m = k = V (G), and assum th nods in th graph G ar numbrd 1,..., n. Corrsponding to ach nod i, w crat an advrtisr i with valu v i = L i, whr L is a sufficintly larg numbr (as w will s, it is nough to tak L = n 2 ). Also, for ach nod i, w crat a usr typ i with probability p P i = c/l i, whr c is a normalizing constant that nsurs n i=1 pi = 1. Th prmutation π i corrsponding to usr typ i is constructd as follows. Lt N i := {j : j < i and ij E(G)} dnot th st of nighbors of i in G that hav an indx lss than i. Th prmutation π i ranks th lmnts of N i in an arbitrary ordr at th top, followd by i, followd by th outsid option 0 (rcall that th ordring of lmnts aftr th outsid option is not important). This complts th dfinition of th instanc of th winnr dtrmination problm. Now, w show that if th siz of th maximum indpndnt st in th graph G is t, thn th valu of th solution of th abov instanc of th winnr dtrmination problm is btwn ct and ct + cn. Lt I dnot th maximum indpndnt st of G ( I = t). First, w prov that th valu of L th solution to th winnr dtrmination problm is at last ct. To show this, it is nough to tak S = I. Sinc I is an indpndnt st, for vry i S, th first lmnt of π i that is in S is i. Thrfor, for vry such i, usrs of typ i contribut a total valu of p i v i = c to th objctiv function. Hnc, th valu of th st S is prcisly v(s) = ct. Nxt, w prov that no st S in th instanc of th winnr dtrmination problm has valu mor than ct + cn. To L show this, tak th optimal st S in th winnr dtrmination problm, and dfin an indpndnt st I in th graph as follows: start with I =, and procss vrtics of S in incrasing ordr of thir indx. For vry vrtx i S, if no nighbor of i is addd to I so far, add i to I. Clarly, at th nd of this procdur, w obtain an indpndnt st I of G. W show that th valu of th st S is at most c I + cn. To L s this, not that for vry lmnt i that is in S but not in I, th vrtx i must hav a nighbor j S with j < i. Considr such a j with th smallst indx. By dfinition, j appars bfor i in π i. Thrfor, th contribution of ach such usr typ i to v(s) is at most p iv j c/l. For vry i I, th contribution of i to v(s) is at most p iv i = c. Summing up ths contributions, w obtain v(s) c I + cn. L Sinc I is an indpndnt st, w gt v(s) ct + cn. L

4 Thrfor, if w tak L > n 2, th valu of th solution of th winnr dtrmination problm dividd by c is within (1 ± o(1)) of th siz th maximum indpndnt st of G. Thus, by th hardnss of th indpndnt st problm [14], th winnr dtrmination problm cannot b approximatd within a factor of n 1 ɛ for any ɛ > 0, unlss NP = ZPP. Th abov thorm ruls out th possibility of finding an algorithm for th winnr dtrmination problm with any approximation factor that is a rasonabl function of n. Howvr, not that th instancs constructd in th abov hardnss rsult contain advrtisrs whos valus diffr by a larg factor (n 2n ). This raiss th qustion of whthr on can approximat th winnr dtrmination problm within a factor that dpnds on th sprad btwn th largst and th smallst valus. Th answr to this qustion is indd positiv, as w will s in th nxt sction. 4. APPROIMATION ALGORITHM W now prsnt an approximation algorithm for th winnr dtrmination problm which can b usd to dsign an incntiv compatibl mchanism for th auction problm with xtrnalitis. Th hardnss rsult in th prvious sction ruls out any approximation that is bttr than a linar factor in th numbr of advrtisrs; our approximation is thrfor in trms of th sprad of th advrtisr valus: w show that th winnr dtrmination problm can b approximatd to within a factor of 2 ( ln R + 1), whr R is an uppr bound on th ratio btwn th highst valu and th lowst valu an advrtisr could hav. W bgin with th following lmma, which shows that th problm can b solvd approximatly whn all valus ar clos to ach othr. Lmma 1. Th winnr dtrmination problm can b approximatd to within a factor R, whr R is an uppr bound on th ratio btwn th highst valu and th lowst valu an advrtisr could hav. Proof. First suppos that v 1 =... = v n, i.., all advrtisr valus ar qual. Thn th problm of winnr dtrmination with xtrnalitis is xactly a wightd vrsion of th classical max k-covrag problm [15]: th lmnts ar th m usr typs with wights p j (j = 1,..., m), th sts ar th advrtisrs i = 1,..., n, and an advrtisr i covrs a usr typ j if i appars in π j bfor 0, i.., if usr typ j prfrs i ovr th outsid option. Th grdy approximation algorithm for th maximum k-covrag problm can b asily gnralizd to solv th wightd vrsion: th algorithm procds in itrations, in ach itration picking an advrtisr that covrs a st of prviously uncovrd usr typs of maximum total wight. It is not hard to s that this algorithm achivs an approximation factor of /() for th wightd maximum k-covrag problm [15]. Th winnr dtrmination problm can b approximatd to within th sam factor whn all valus ar qual. If advrtisrs hav unqual valus, w can obtain a factor R by simply ignoring th valus and solving th wightd max-covrag problm. W will now build on this obsrvation to obtain a randomizd algorithm with a ratio that is logarithmic in R. Thn w will show how this algorithm can b turnd into a monoton algorithm (a proprty that is ndd in ordr to achiv incntiv compatibility), and how it can b drandomizd. For a givn subst S {1,..., n}, dnot by OP T k (S) th valu of th optimal solution to th wightd max-k-covrag problm rstrictd to th subst of advrtisrs S, P k (S) th valu of th solution rturnd by th grdy algorithm to th wightd max k-covrag problm, and dnot by Sk(S) th subst of advrtisrs from S chosn by th grdy algorithm. Assum v min is a known lowr bound, and v minr is a known uppr bound on th valu of an advrtisr. W will show latr that our drandomizd algorithm works vn if w do not know th valus of v min and R. Divid advrtisrs into buckts 1 through L, whr L = ln R and buckt B l consists of advrtisrs with valus v i [ l 1 v min, l v min). Algorithm A 1: Randomly choos on of th buckts B 1,..., B L, say B l. Solv th wightd max k-covrag problm with input B l ; lt S k(b l ) b th st of advrtisrs rturnd by th grdy algorithm. Rturn S k(b l ) as th st of winnrs. Thorm 2. Algorithm A 1 approximats th winnr dtrmination problm to within a factor of ln R. 2 Proof. Lt SOL 1 dnot th xpctd wlfar from th solution chosn by th abov algorithm: SOL 1 L l=1 1 L vminl 1 P k (B l ), (2) whr P k (B l ) is th wight of th solution chosn by th grdy algorithm for max k-covrag, with input rstrictd to th st of advrtisrs with valus in [ l 1 v min, l v min], as dfind bfor. Th wlfar from th optimal st of advrtisrs for th winnr dtrmination problm S OP T is OP T = v ip (i, S OP T ) i S OP T = L l=1 L l=1 i S OP T B l v ip (i, S OP T ) v min l i S OP T B l P (i, S OP T ), (3) whr P (i, S OP T ) = P {j: i S OP T {0}, π j (i) π j (i )} pj is th total wight of th prmutations whr i is rankd abov all othr lmnts in S OP T and th outsid option. Sinc P k (B l ) is th solution rturnd by th grdy algorithm for th maximum k-covrag problm rstrictd to advrtisrs in B l, and S OP T B l is a solution for this problm of valu at last P i S OP T B l P (i, S OP T ), w hav i S OP T B l P (i, S OP T ) Combining (2) and (3) with this, w gt OP T 2 L 1 SOL, i.., an approximation factor of 1 P k(b l ). 2 ln R.

5 It is a wll-known thorm (s, for xampl, Archr and Tardos [1]) that in a stting with on-dimnsional typs, in ordr to dsign an incntiv compatibl mchanism, on nds an allocation algorithm that is monoton: th probability of winning should not dcras if an advrtisr s valu incrass. Th abov algorithm dos not hav this proprty: for xampl, if advrtisr i is th only advrtisr in an intrval B l, sh wins with probability 1/L indpndnt of hr position in th prmutations, whras sh may not b chosn as a winnr in hr nw buckt if hr valu incrass. Th following modification nsurs that th allocation algorithm is monoton. Dfin th (ovrlapping) buckts B l = {i : v i v min l 1 }, i.., B l = B l... B L is th union of th buckts B j from l to L. Dnot by P k (B l ) th valu of th grdy solution of th wightd max k-covrag problm rstrictd to advrtisrs in B l. Algorithm A 2: Randomly choos on of th buckts B 1,..., B L, say B l. Solv th wightd max-k-covrag on B l using th grdy algorithm; lt Sk(B l ) b th st of advrtisrs rturnd by th grdy algorithm. Rturn S k(b l ) as th st of winnrs. Thorm 3. Th algorithm A 2 is monoton, and approximats th winnr dtrmination problm to within a factor of ln R. 2 Proof. Th proof of th approximation ratio follows simply from Thorm 2 and noting that P k (B l ) P k (B l ), sinc B l B l. To prov th monotonicity, not that for any buckt B l and any advrtisr i, if this advrtisr is in B l, aftr incrasing th valu v i, sh still rmains in this buckt. Thrfor, conditiond on any valu of l, incrasing v i (whil holding vrything ls constant) cannot rmov i from B l and thrfor cannot dcras i s chanc for bing includd in th solution. Finally, w show that our algorithm can b drandomizd whil maintaining th monotonicity proprty. Th ida of drandomization is asy: instad of picking a random buckt, chck all buckts and pick on that givs th highst valu. Howvr, on nds to b carful as this transformation can somtims turn a monoton algorithm into a non-monoton on. In our cas, w can us proprtis of th grdy maximum k-covrag algorithm to prov that th algorithm rmains monoton. In addition to dcrasing uncrtainty, on advantag of th drandomizd algorithm is that it can b implmntd without knowldg of th valus of v min or vn R. 2 In ordr 2 Not that for th purpos of solving th algorithmic qustion, having a prior knowldg of v min and R is not important, sinc ths valus can b computd from th input. Howvr, doing so can violat th monotonicity proprty; for xampl, if th advrtisr that has th minimum v i incrass hr valu, this changs th structur of th buckts and could rsult in dcrasing hr chanc of gtting slctd. to do this, for vry intgr l, w dfin th buckt B l as th st of advrtisrs of valu at last l (B l = {i : v i l }). By th dfinition of R, thr ar at most ln R + 1 nonmpty distinct buckts. W can now dfin th dtrministic algorithm as follows: Algorithm A 3: Run th grdy algorithm for th max-k-covrag problm on th buckts B l for all l. Choos buckt B l such that l = arg max B l v 12 l 1 P k (B l ). Rturn S k(b l ) as th st of winnrs. Thorm 4. Th algorithm A 3 is monoton, and approximats th winnr dtrmination problm to within a factor of 2 ( ln R + 1), whr R is an uppr bound on th ratio btwn th highst valu and th lowst valu an advrtisr could hav. Proof. Th approximation ratio follows from Thorm 3 and th fact that th maximum of a st of numbrs is not lss than thir avrag. W now prov that th algorithm A 3 is monton. Suppos that biddr i is a winnr, and has valu v i [ l, l+1 ), i.., i is in buckt B l and vry buckt bfor that. Sinc sh is a winnr, j P k (B j) is maximizd for som j = l l. Suppos v + i > v i is such that v + i [ l+, l+ +1 ). W nd to show that aftr incrasing i s valu to v + i, i is still a winnr. Not that P (B l ) is unchangd for all l l, sinc B l is unchangd for ths sts, and th valus of biddrs in B l do not affct th valu of P (B l ), or th st of biddrs corrsponding to P (B l ). For th sts B l with l < l l +, th only addition is th biddr i. Sinc th algorithm usd to comput P (B l ) is th dtrministic grdy algorithm for wightd max-k covrag [15], P (B l ) changs only if th algorithm chooss i in th winning st. Thus if P (B l ) P (B l ), i is chosn as a winnr in B l. Thrfor, th allocation algorithm is monoton. Th abov thorm, togthr with th thorm of Archr and Tardos [1] implis that thr is a dominant-stratgy incntiv-compatibl mchanism for ad auctions with xtrnalitis that can approximat th social wlfar to within a factor of 2 ( ln R + 1) of th optimum. 5. ALGORITHMS FOR SPECIAL CLASSES OF USER PREFERENCES Th rsult in th prvious sction says that th problm of choosing th optimal st of advrtisrs cannot vn b approximatd wll, in th most gnral cas. Howvr, as w will show in this sction, th problm can b solvd xactly in crtain spcial cass. 5.1 Singl-pakd prfrncs Singl-pakd prfrncs form an important domain of prfrncs, and ar wll-studid in th contxts of majority voting and Arrovian social wlfar functions, stratgyproof voting ruls, and fair division, among othrs (s, for xampl, [3, 13]).

6 W start by dfining th notion of singl-pakd prfrncs in th contxt of th xtrnality problm. Rcall that th prfrnc of ach usr typ j is givn as a prmutation π j of {0, 1,..., n}, whr 0 is th outsid option and 1,..., n rprsnt advrtisrs. W say that th usr prfrncs ar singl-pakd (with rspct to th ordring 1,..., n of th advrtisrs), if for vry usr typ j, thr is a valu a j {1,..., n}, such that for vry 1 x < y a j, advrtisr y is prfrrd to advrtisr x according to π j, and for vry a j x < y n, advrtisr x is prfrrd to advrtisr y according to π j. In othr words, ach usr typ j has an idal advrtisr a j, and advrtisrs bfor a j ar rankd according to thir distanc to a j, and similarly for advrtisrs aftr a j. No rstriction is placd on how j ranks two advrtisrs, on bfor a j and th othr aftr a j, or how sh ranks any advrtisr in comparison to th outsid option. Th following thorm givs an algorithm for th winnr dtrmination problm with xtrnalitis, whn prfrncs ar singl-pakd, with rspct to a known ordring 1,..., n. Thorm 5. Th winnr dtrmination problm with xtrnalitis can b solvd in polynomial tim if usr prfrncs ar singl-pakd. Proof. W giv a dynamic programming algorithm for this problm. Th main stp is to dfin an appropriat subproblm, which can b solvd rcursivly. Th subproblm w us is paramtrizd by two paramtrs i and r, with 1 i n and 1 r k, and is dfind as follows: Considr an instanc of th winnr dtrmination problm with xtrnalitis whr th st of usr typs is rstrictd to {j : a j i} (not that w do not chang th probabilitis p j s, so in this rstriction th probabilitis can add up to lss than 1. Howvr, th problm is still wll-dfind in this cas). Lt SOL i,r b th maximum valu v(s) of a st S satisfying th following: i S, S {i,..., n}, and S r. W show how SOL i,r can b computd rcursivly. Th ida is to focus on th first advrtisr aftr i that will b includd in th st S. If i is th indx of this advrtisr, th valu drivd from usr typs j with a j i can b writtn as SOL i,r 1, sinc by th singl-pakd proprty of th prfrncs, non of ths usrs prfrs any advrtisr bfor i to i. Usrs j with i a j < i will choos on of th advrtisrs i and i or th outsid option (again, by th singl-pakd proprty). Thrfor, th total valu of th solution in this cas can b writtn as SOL i,r 1 + p jv i + p jv i, j S i,i j S i,i whr S i,i := {j : min(i, i ) a j < max(i, i ) and π j(i) < min(π j(i ), π j(0))} is th st of usrs whos idal advrtisr is btwn i and i and prfr i to i and also to th outsid option. To comput SOL i,r, w nd to tak th maximum ovr all i of th abov xprssion, and also of th cas whr thr is no such i, i.., i is th last advrtisr that is includd in th st. In this cas, th valu of th subproblm is P j S i,n+1 p jv i, whr S i,n+1 := {j : a j i and π j(i) < π j(0)} is th st of usrs whos idal advrtisr is aftr i and who prfr i to th outsid option. To summariz, SOL i,r can b computd using th following rcursiv formula for vry r 2. SOL i,r = max{ p jv i, max{sol i i,r 1 (4) >i j S i,n+1 + p jv i + p jv i }}. (5) j S i,i j S i,i For r = 1, w hav SOL i,r = P j S i,n+1 p jv i. Using th abov rcursions, on can comput all th SOL i,r s in tim O(n 2 km). Using th abov argumnt, it is asy to s that th solution of th problm can b computd in trms of ths valus as follows: max i=1,...,n 8 9 < = p jv i + SOL i,r : ;, j S 0,i whr S 0,i = {j : a j < i and π j(i) < π j(0)} is th st of usrs whos idal advrtisr is bfor i and who prfr i to th outsid option. This givs an O(n 2 km)-tim algorithm to solv th winnr dtrmination problm with xtrnalitis whn usr prfrncs ar singl-pakd. Rmark 1. Not that th siz of th dynamic programming tabl in th abov algorithm is O(nk), which is indpndnt of th numbr of diffrnt usr typs. Thrfor, our algorithm can b adaptd to th cass whr thr ar xponntially many usr typs, and thy ar givn ithr by an oracl, or with an implicit rprsntation. Th only ingrdint ndd is an algorithm that computs summations lik th ons in Equation (4). 5.2 Prturbd singl ranking W now considr anothr spcial cas of th winnr dtrmination problm, whr usr prfrncs ar givn implicitly using th following distribution of advrtisr qualitis: Advrtisr i has quality q i = x i with probability p i (whr x i s ar a givn distinct valus), and 1 with probability 1 p i; th quality is indpndnt of all othr advrtisrs qualitis. Th quality paramtr for th outsid option is fixd at q 0 = 0. This givs ris to xponntially many prmutations which ar all substs of a singl undrlying prmutation dfind by th x i s, but ach advrtisr i is droppd from th prmutation (indpndntly) with probability 1 p i. An intrprtation of this modl is that thr is an undrlying tru quality rating amongst all advrtisrs (as givn by th ordring of th x i s), and a usr who is informd about two advrtisrs i and j ranks thm in this ordr. Howvr, with probability 1 p i, th usr has not hard of advrtisr i, in which cas sh will not choos this advrtisr. In othr words, usrs rank th advrtisrs according to indpndnt prturbations of th sam ranking, undr a particular modl of prturbation. As w will point out in Rmark 2, th ida can b gnralizd to mor gnral local prturbation modls. Th valu of a st of advrtisrs S in this modl can b writtn as v(s) = i S v ip i Y j S,x j >x i (1 p j). (6) Thorm 6. Th winnr dtrmination problm in th abov modl can b solvd xactly in polynomial tim.

7 Proof. Again, w us dynamic programming to giv a Θ(nk)-tim algorithm for th winnr dtrmination problm. Numbr advrtisrs in dcrasing ordr of x i, so that x 1 > x 2 > > x n. W considr th following subproblm: Lt SOL i,r dnot th optimal valu whn no mor than r advrtisrs can b slctd from th subst i,..., n of advrtisrs. In othr words, SOL i,r := max S {i,...,n}, S r {v(s)}. Not that v(s {i}) for i < min(s) is p iv i + (1 p i)v(s). Using this, it is asy to prov th following rcursion. SOL i,r = max(sol i+1,r, v ip i + (1 p i)sol i+1,r 1). Starting from i = n (SOL n,r = p nv n for all r), w populat a tabl with nk ntris; computing ach ntry in th tabl taks Θ(1) tim. Thrfor, th solution of th winnr dtrmination problm, which is givn by SOL 1,k, can b computd in tim O(nk). Rmark 2. Not that th only thing our algorithm rlis on is that if i is bfor all lmnts of S, v(s {i}) can b computd from v(s). For many othr modls of prturbation that prturb th prmutation locally and indpndntly, a similar approach works, prhaps by kping a limitd amount of information about th st S. As an xampl, considr prturbations of th following form: swap th ordr of th advrtisrs 2i 1 and 2i in th undrlying prmutation with probability p i, indpndntly for all i; thn from th rsulting prmutation, drop lmnt j with probability p j, indpndntly for all j. If i is bfor all lmnts of S, v(s {i}) can b computd givn v(s) and th lowst-indx lmnt of S. Thrfor, th winnr dtrmination problm can b solvd in this prturbation modl by a dynamic programming algorithm with an n n k tabl. 6. ALTERNATIVE MODELS In this sction w discuss two othr modls that ar not basd on a modl of usrs choics, but ar simplr and thrfor might b mor applicabl in practic. Pairwis multiplicativ modl. As bfor, thr ar n advrtisrs with privat valus v 1,..., v n. Th modl is dfind in trms of paramtrs a ij [0, 1] for 1 i, j n. Th valu of a ij rprsnts th factor by which advrtisr j dcrass th valu to advrtisr i, if both ar chosn togthr in th winning st. Th valu of a st S is v(s) = Y v i a ij. i S j S\{i} In trms of th computational complxity of th winnr dtrmination problm, this problm is still hard to approximat, sinc by taking a ij {0, 1}, on can s asily that th indpndnt st problm is a spcial cas of this problm. Uniform discount modl. This modl assums that diffrnt advrtisrs xprinc th sam dgr of discount, which dpnds only on total wight of th st of winnrs. Mor prcisly, w assum ach advrtisr i has a privat valu v i and a wight w i (th wight could corrspond to th convrsion rat or anothr masur of th importanc of th advrtisr), and w ar givn a non-incrasing function f : R [0, 1], which spcifis th discount as a function of th total wight of th advrtisrs in th st. Th valu of a st S is givn by v(s) = i S v if( j S w j). It is not hard to s that th winnr dtrmination problm in this modl can b solvd using th algorithm for th knapsack problm: for vry lvl W of th total wight, w can comput th st S W of advrtisrs of maximum total valu whos wight add up to at most W. This can b computd using a knapsack algorithm. Th solution of th winnr dtrmination problm is th maximum, ovr all choics of W, of th valu of S W. With an appropriat discrtization of valus of W, this can b mad into a polynomial tim approximation schm for th problm. Whn wights ar small intgrs, this approach givs a polynomial tim xact algorithm for th problm. Turning th approximation schm for th winnr dtrmination problm into a truthful mchanism rmains an opn qustion. On approach to tackl this problm would b to us th tchniqus dvlopd in th contxt of multi-unit auctions [10]. 7. DISCUSSION In this papr, w discussd modls for th xtrnality problm in onlin advrtising. W provd that for th most gnral modl, th winnr dtrmination problm is computationally hard, and gav algorithms for this problm in som spcial cass. W bliv that th xtrnality problm is a major issu in th study of onlin advrtising, and so far has not rcivd nough attntion from th rsarch community. In th following, w list a fw dirctions for futur rsarch. Location-dpndnt xtrnalitis. Th modls studid in this papr, motivatd by th lad gnration businss, was dvlopd for a stting whr th only dcision mad by th winnr dtrmination algorithm is th st of advrtisrs who gt an advrtising opportunity (by rciving a lad), without any particular ordr. Howvr, in th cass whr th auction dcids which advrtismnts should b displayd on a pag (which is th mor common cas), th winnr dtrmination algorithm should not only spcify which ads ar displayd, but also in which slot ach ad is displayd. Furthrmor, in this stting th xtrnalitis can b location dpndnt: for xampl, a sponsord sarch ad displayd in th 10th slot might impos no xtrnality on th ad displayd on th top slot. Thrfor, in ordr to b applicabl to this stting, our modl for xtrnalitis has to b modifid to tak th locations into account. A simpl way to incorporat th location componnt into our modl is as follows: assum th slots ar numbrd 1, 2,..., K, from top to bottom. W assum a random usr only looks at th ads in th top slots, whr is a random variabl with a givn distribution. Th usr thn dcids which of th ads sh has lookd at to click on, according to on of our modls. Clarly, our hardnss rsult for th winnr dtrmination problm works for this mor gnral modl as wll. It would b intrsting to find intrsting spcial cass of this problm that can b solvd in polynomial tim. Th long-trm xtrnality ffct. In this papr, our focus was on th immdiat xtrnality that advrtisrs impos on ach othr, in trms of lowring th convrsion rat or th click-through rat of othr advrtisrs in th sam

8 sssion. Thr is also a long-trm xtrnality ffct: if a usr finds th ads displayd on a wbsit (.g., sponsord sarch ads on Googl) hlpful, h or sh is mor likly to click on ads in th futur, and convrsly, if th ads ar found to b not rlvant, th usr will pay lss attntion to ads in th futur. This xtrnality ffct is wll undrstood in th contxt of traditional advrtising, and is somtims rfrrd to as th rottn-appl thory of advrtising [4]. Th implication of this ffct in th contxt of traditional advrtising mdia is in th domain of public policy: it is usd to justify adopting rgulations against fals advrtising. In th contxt of onlin advrtising, howvr, thr is much mor a publishr can do to masur this typ of xtrnality and rward or punish advrtisrs basd on whthr thy crat positiv or ngativ xtrnality. Publishrs such as Googl or Yahoo oftn can track whn a usr rvisits thir wbsit and clicks on an ad, and also whthr an ad lads to a convrsion or som action on th advrtisr s sit. Dsigning mchanisms to xtract th rlvant information from this walth of data and us it to ovrcom th xtrnality problm and maximiz th fficincy of th systm in th long run is an intrsting rsarch dirction. Larning xtrnalitis. Throughout this papr, w studid th winnr dtrmination problm with xtrnalitis, assuming that th paramtrs of th modl usr prfrncs in our main modl ar known to th algorithm. Larning ths paramtrs givn th history of choics mad by th usrs, or dsigning xprimnts in ordr to larn ths paramtrs rmains opn. Divrsity problm. It is known that having divrsity among th st of wb sarch rsults or among th st of products displayd on an lctronic stor front is valuabl. Thr has bn som ffort on dsigning algorithms in ths applications that giv som wight to th divrsity of th solution st (s, for xampl, [16]). Howvr, it is not clar what is th right way to incorporat th divrsity componnt in th objctiv. Not that in our modl, optimizing for th valu of solution in prsnc of xtrnalitis can automatically rsult in a divrs solution st. This is bcaus advrtisrs that ar similar to ach othr impos gratr ngativ xtrnality on ach othr (.g., prsumably an advrtisr who slls Appl computrs imposs littl xtrnality on on that slls th fruit). This suggsts that our framwork might b a good starting point for dfining a divrsity optimization problm that is basd on conomic principls. Dating problm. As mntiond arlir, onlin dating srvics can also b considrd in th framwork of lad gnration. Similar xtrnalitis xist in th onlin dating industry: snding a woman w on a dat with a man m imposs a ngativ xtrnality on all othr mn, sinc it dcrass thir chancs with w. Currntly, onlin dating srvics tak on of th two xtrms of ithr allowing unrstrictd sarch (i.., a subscribr has accss to th profils of anyon who mts his or hr critria, and can contact thm, which is th modl adaptd by Yahoo! Prsonals or match.com), or matching popl on pair at a tim (this is th modl adaptd by Harmony). Chn t al. [2] initiatd th study of th Harmony modl from a stochastic optimization point of viw. An intrsting dirction for futur rsarch is to study this problm in a modl with xtrnalitis in ordr to strik th right balanc btwn ths two xtrms. 8. REFERENCES [1] Aaron Archr and Eva Tardos. Truthful mchanisms for on-paramtr agnts. In IEEE Symposium on Foundations of Computr Scinc, pags , [2] Ning Chn, Nicol Immorlica, Anna Karlin, Mohammad Mahdian, and Atri Rudra. Approximating matchs mad in havn. in prparation, [3] W. Gartnr. Domain rstrictions. In K.J. Arrow, A.K. Sn, and K. Suzumura, ditors, Handbook of Social Choic and Wlfar, volum I, chaptr 3, pags Elsvir Scinc, [4] Z. K. Hansn and M. Law. Th political conomy of truth-in-advrtising rgulation in th progrssiv ra. Forthcoming at th Journal of Law and Economics. 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