Externalities among Advertisers in Sponsored Search

Externaltes among Advertsers n Sponsored Search Dmtrs Fotaks 1 Potr Krysta 2 Orests Telels 2 1 School of Electrcal and Computer Engneerng, Natonal Techncal Unversty of Athens, 157 80 Athens, Greece. Emal: fotaks@cs.ntua.gr 2 Department of Computer Scence, Unversty of Lverpool, L69 3BX Lverpool, UK. Emal: {p.krysta,o.telels}@lverpool.ac.uk Abstract. We ntroduce a novel computatonal model for sngle-keyword auctons n sponsored search, whch explctly models externalty effects among the advertsers, an aspect that has not been (fully) reflected n the exstng models, and s known to be prevalent n the behavor of real advertsers. Our model takes nto account both postve and negatve correlatons between any par of advertsers, and approprately reflects them n the way they perceve an outcome of a sponsored search aucton. In our model, the clck-through rate of an ad depends on the set of other ads appearng n the sponsored lst, on ther relatve order, and on ther dstance n the lst. In contrast to prevous modelng attempts, we avod modelng end-users behavor, but only resort to a reasonable assumpton that ther browsng focuses on any bounded scope secton of the lst. We present a comprehensve collecton of computatonal results concernng the wnner determnaton problem n our model wth an objectve to maxmze the socal welfare, showng both hardness of approxmaton results and polynomal-tme approxmaton algorthms. We also present an exact polynomal-tme algorthm for the practcally relevant cases of our model, where the length of the sponsored lst s at most logarthmc n the number of advertsers. Ths exact algorthm can be used as a truthful mechansm by employng the VCG payments, thus showng that we can fully cope wth selfsh behavor of advertsers when the mechansm s fully aware of ther correlatons. We fnally study the performance of the practcally used Generalzed Second Prce aucton mechansm n our model and show that, n presence of externaltes, pure Nash equlbra may not exst for conservatve bdders that do not outbd ther valuaton. Moreover, we exhbt nstances where pure Nash equlbra do exst, but they carry an unbounded loss n socal welfare as compared to the socally optmal soluton assgnment of slots, even for such conservatve bdders. Research partally supported by an NTUA Basc Research Grant (PEBE 2009), EPSRC grant EP/F069502/1, and DFG grant Kr 2332/1-3 wthn Emmy Noether Program.

1 Introducton Sponsored search advertsng s nowadays a predomnant and arguably most successful paradgm for advertsng products n a market, facltated by the Internet. It consttutes a major source of ncome for popular search engnes lke Google, Yahoo! or Mcrosoft Bng, who allocate up to 8 9 advertsement slots n ther stes, alongsde the organc results of keyword searches performed by end users. Each tme an end user makes a search for a keyword, slots are allocated to advertsers by means of an aucton performed automatcally; advertsers are ranked n non-ncreasng order of a score, defned as the product of ther bd wth a characterstc relevance quantty per advertser. The relevance of each advertser s nterpreted as the probablty that hs ad wll be clcked by an end user. The score corresponds then to the declared expected revenue of advertsers. Advertsers ranked hgher are matched to hgher slots. Each advertser s charged an amount dependng on the score (hence, the bd) of the one ranked below hm, when hs ad s clcked. The descrbed aucton s used n varyng flavors by search engnes. Apart from the mentoned Rank-By-Revenue rule, a planer Rank-By-Bd rule has also been used (e.g. by Yahoo!). Ths aucton, known as the Generalzed Second Prce (GSP) aucton, consttutes a generalzaton of the wellknown strategyproof Vckrey aucton [30]. The GSP aucton s not strategyproof though; t encourages strategc bddng by the advertsers nstead of elctng ther valuatons truthfully. Ths nduces a strategc game rch n Nash equlbra, and compettve behavor among advertsers that ncurs sgnfcant revenue to the search engnes. Pure Nash Equlbra of the GSP mechansm were frst studed by Edelman, Ostrovsky, Schwartz [17] and Varan [29], under what came to be known as the separable clck-through rates model. In ths model every slot s assocated wth a Clck-Through Rate (CTR),.e. the probablty that an ad dsplayed n ths slot wll be clcked. The jont probablty that an ad s clcked s gven by the product of the slot s CTR wth the relevance of the ad. Snce [17, 29], a rch lterature on keyword auctons has been publshed, concernng algorthmc and game theoretc ssues such as bddng strateges, socal effcency, revenue, see, e.g. [27] (chapter 28). Recent experments [22] show that the probablty of a dsplayed ad beng clcked (hence, the utlty of the advertser) s affected by ts relatve order and dstance to other ads on the lst. E.g., two competng ads dsplayed nearby each other may dstract a user s attenton from any of them. However, two advertsers may also proft from beng dsplayed nearby (e.g. a cars manufacturer and a spare parts suppler). We ntroduce a model for expressng such externaltes n keyword auctons. Externaltes result n complcated strategc competton among advertsers, precludng stable outcomes and socal welfare optmzaton. A way of allevatng these effects s by solvng optmally the Wnner Determnaton problem [2, 1, 23];.e., the problem of selectng wnners and ther assgnment to slots, to maxmze the Socal Welfare (the sum of the advertsers utltes and the auctoneer s revenue defned formally n Secton 2). Such a soluton, pared wth payments of the Vckrey-Clarke-Groves mechansm [14] (chapter 1), yelds a truthful mechansm. On the other hand, the study of the GSP aucton s performance n presence of externaltes [20] yelds nsghts for the current practce of sponsored search. In our model we analyze exact and approxmaton algorthms for Wnner Determnaton, and show how externaltes can harm the GSP aucton s stablty and socal effcency. Several recent works concern theoretcal and expermental study of externaltes n keyword auctons [1, 7, 19, 20, 23, 21, 22]. These works assocate the occurrence of externaltes wth a model of how end users search through the lst of ads and how ths affects the probablty of an ad beng clcked. Ths search s commonly modeled by a top-down ordered scan of the lst [7, 1, 23, 20]. The popular Cascade Model [1, 23], assocates a contnuaton probablty wth every ad,.e. the lkelhood of the user contnung hs ordered scan after vewng the ad. Usng real data from keyword auctons, Jezorsk and Segal [22] found that prevously proposed models fal to express externaltes. The Cascade Model 1

s contradcted by the fact that about half of the users do not clck on the ads sequentally,.e., they return to hgher slots after clckng on lower slots. Jezorsk and Segal arrved at a structural model of end user behavor advocatng that, after scannng all the advertsements, users focus on a subset of consecutve slots. Wthn ths focus wndow, they observed externaltes due to proxmty and relatve order of the dsplayed ads. They hghlght that an ad s CTR on the lst depends crucally on the ads dsplayed n the other slots, above and below t. Wth our model we am at quantfyng rgorously these observatons, whle avodng explct modelng of end users behavor. Contrbuton and Technques. Our man contrbuton s a novel and farly expressve model for descrbng postve or negatve nfluences between any two advertsers n keyword auctons. Our model s bult by usage of a socal context graph [6] and upon the assumpton of separable clck-through rates, whch facltates analytcal study. (see Secton 2 for precse defnton). It provdes for the descrpton of nfluences among the advertsers relevances, dependng on: () The IDs of advertsers, () Ther relatve order n the lst of sponsored lnks and () The dstance of ther ads n the lst. Motvated by results of [22], we make the assumpton that the end users attenton may be captured by any sze c wndow of consecutve slots, wthn whch externaltes take effect among advertsers. Our model s the frst to express nherently the practcally relevant possblty that socal welfare maxmzaton may occur under partal allocaton of slots (.e., where some of the slots are left empty). Under the proposed model we study the Wnner Determnaton problem wth the objectve of maxmzng the socal welfare. Usng reductons from the Longest Path problem and the Travelng Salesperson Problem wth dstances 1 and 2 [28], we show that the problem s NP-hard and APX-hard respectvely, even for wndow sze 1 and postve-only externaltes. For postve-only externaltes we study approxmaton algorthms. We develop two nterestng approxmaton-preservng reductons of the Wnner Determnaton problem to the Weghted m-set Packng problem wth sets of sze m = 3 and m = 2c + 1 respectvely. These result n a nce tradeoff between approxmablty and computatonal effcency for the Wnner Determnaton problem; for k slots and n advertsers, we obtan algorthms wth approxmaton factors: () 6c n tme O(kn 2 log n), () 4c n tme O(k 2 n 4 ) and () 2(c + 1) n polynomal tme for any constant c. These results settle almost tghtly the approxmablty of the problem for postve-only externaltes and c = O(1). On the postve sde, we buld on the color codng technque [3] to obtan an exact algorthm for Wnner Determnaton n the full generalty of our model. A derandomzaton of our color codng s possble, that yelds a determnstc algorthm. The algorthm has runnng tme 2 O(k) n 2c+1 log 2 n, whch shows that the class of practcally nterestng nstances of the problem wth c = O(1) and even k = O(log n) s n P. Ths algorthm, pared wth VCG payments [14] yelds a truthful mechansm when the advertsers valuatons are prvate nformaton, that optmzes the socal welfare n presence of externaltes. Notce that the wnner determnaton problem can be solved easly by complete enumeraton of all ( n k) tuples of ads, n O(n k ) tme. For any constant sze wndow our algorthm s sgnfcantly faster, and even for larger than constant values of k. Thus the problem s not NP-hard for k = O(poly(log n)), unless NP DTIME(n poly(log n) ). We conclude our study wth an nvestgaton of the GSP mechansm s behavor under externaltes. We fnd that pure Nash equlbra do not exst n general, for conservatve bdders that do not outbd ther valuaton. Ths s a standard assumpton when studyng non-truthful mechansms [11, 26, 13], snce overbddng results n unboundedly low socal welfare. Even when such conservatve equlbra exst, we show that ther socal welfare can be arbtrarly low compared to the socal optmum. Ths should be contrasted wth an upper bound of 1.282 shown recently by Caraganns et al. [11] for the socal neffcency of GSP equlbra. 2

2 A Model for Externaltes n Keyword Auctons We consder a set N = [n] = {1, 2,... n} of n advertsers (or players/bdders) and a set K = {1, 2,..., k} of k n advertsement slots. Each advertser N has valuaton v for each clck and s assocated wth a probablty q that her ad receves a clck, ndependently of slot assgnment. q s often termed relevance and measures the ntrnsc qualty of ad. Each slot j s assocated wth a probablty λ j that an ad dsplayed n slot j wll be clcked; ths s called the Clck-Through Rate (CTR) of the slot. The (overall) Clck-Through Rate of an ad N occupyng slot j s λ j q (separable CTRs). We assume 1 λ 1... λ k > 0,.e. that the top slot s slot 1, the second one s slot 2 and so on. Let S [n], S k, be the set of wnnng ads and π : S K ther assgnment to slots,.e. π() K s the slot of advertser S. Every advertser S ssues to the search engne (auctoneer) a payment p per receved clck, hence receves expected utlty u (S, π) = λ π() q (v p ). The Socal Welfare s then: sw(s, π) = S λ π() q v = S u (S, π) + S λ π() q p, (1) where S λ π() q p s the expected revenue of the auctoneer. Our modelng of externaltes s bult on top of the separable CTRs model and nduces amplfed or dmnshed actual relevance Q (compared to q ) to the advertsers, dependng on ther relatve poston and dstance of ther ads on the lst. We use a drected socal context graph [6] G(N, E +, E ), defned upon the set of advertsers N. Each edge (, j) E + E s assocated wth a functon w j (0, 1). An edge (j, ) n E + or n E denotes potental postve or negatve nfluence respectvely of by j. Namely, edges n E + and E model respectvely postve and negatve externaltes among advertsers. For any edge (j, ) E + E, the potental nfluence of j to s quantfed by a functon w j : { k + 1, k + 2,..., 1, 1,..., k 2, k 1} [0, 1]. If d π (j, ) = π() π(j) s the dstance of ad from ad j n the lst, w j (d π (j, )) s the probablty that a user s nterest n a dsplayed ad j may result n attracton or dstracton of hs attenton to or from an ad respectvely, dependng on whether (j, ) E + or (j, ) E 3. It s reasonable to assume that the closer and j are n π, the stronger the nfluence of j on s. Formally, for every l, l, wth l, l 1, f l < l, then w j (l) w j (l ). Let S N be the set of wnners and π be the permutaton assgnng them to the slots. Defne the subgraph G S (S, E + S, E S ) of the context graph, nduced by S. The probablty Q (S, π) that a user s attracted by ad s expressed as product Q + (S, π) Q (S, π). Q+ (S, π) s the probablty that attracts the user s attenton ether by ts ntrnsc relevance q or by recevng postve nfluence. Q (S, π) s the probablty that the user s attenton s not dstracted from due to negatve nfluence of others. For each S defne N + (S) = {j S : (j, ) E+ S } and N (S) = {j S : (j, ) E S } to be the set of neghbors of n G S wth postve and negatve nfluence respectvely. Let us derve Q + (S, π) frst. A user s attenton s not attracted by the ad of wth probablty (1 q ) and ndependently f s not postvely nfluenced by any j N + (S). The latter occurs ether because j tself does not attract the user (wth probablty 1 q j ), or the postve nfluence of j to does not occur (wth probablty q j (1 w j (d π (j, ))). Then j does not nfluence wth probablty (1 q j ) + q j (1 w j (d π (j, ))) = (1 q j w j (d π (j, ))) and: Q + (S, π) = 1 (1 q ) j N + (S) ( 1 q j w j (d π (j, )) ) (2) 3 We note that f d π(j, ) > 0, appears below j, and f d π(j, ) < 0, j appears below n π. 3

For Q (S, π) we use smlar reasonng. A user s attenton s not dstracted by an ad of due to negatve nfluence of j N (S) f ether hs attenton s not captured by j or, f t s, j fals to nfluece negatvely. Ths event occurs wth probablty (1 q j ) + q j (1 w j (d π (j, ))) = 1 )q j w j (d π (j, ). Assumng ndependence of the events for all j N (S), we have: Q (S, π) = j N (S) ( 1 q j w j (d π (j, )) ) After dervng Q (S, π) = Q + (S, π) Q (S, π), we can restate the socal welfare as: sw(s, π, G) = S λ π() Q (S, π) v. (3) Arguably, users attenton and memory when they process the advertsements lst have a bounded scope. Therefore, we assume that there s an nteger constant c > 0, called wndow sze, so that each ad j can only affect other ads at a dstance at most c n π. Formally, f wndow sze s c, for all (j, ) E + E and all ntegers l wth l > c, w j (l) = 0. Then the relevance Q only depends on the ads n N + (S) N (S) assgned to slots at dstance at most c from. Related Work. Edelman, Ostrovsky, Schwartz [17] and Varan [29] frst modeled the game nduced by the GSP aucton mechansm under the assumpton of separable CTRs,.e. that the probablty of a specfc ad beng clcked when dsplayed n a certan slot s gven by the product of the slot s CTR and the ad s relevance. They dentfed socally optmal pure Nash equlbra of the game, n whch advertsers are ranked by non-ncreasng score computed by ther valuatons. Pror to these works, Aggarwal, Goel and Motwan [2] had already desgned a truthful mechansm for non-separable CTRs (n ths case the VCG aucton s also not applcable). For separable CTRs they proved revenue equvalence of ther mechansm to the VCG. There has been a growng nterest n modelng externaltes n sponsored search and studyng how externaltes affect the advertsers bddng strateges and the propertes of GSP equlbra. The unpublshed work of Das et al.[16] s probably the closest n sprt to ours. Das et al. consder externaltes among advertsers based on ther relatve qualty (.e., relevance), and quantfy the fact that for any sponsored lst of ads, as the relevance of an ad ncreases, ts CTR should ncrease and the CTRs of the other ads should decrease. However, the model of [16] treats advertsers as anonymous, n the sense that externaltes do not depend on ther bds, but just on ther relatve qualty. Our model makes a step further, snce n addton to the advertsers relatve qualty as measured by ther relevance, we also take nto account ther IDs, ther dependences n the context graph, and ther relatve dstance n the sponsored lst. Moreover, our model quantfes how as the relevance of an ad ncreases, ts CTR and the CTRs of postvely correlated ads appearng nearby n the lst should ncrease, and the CTRs of negatvely correlated ads appearng nearby n the lst should decrease. Athey and Ellson [7] gave one of the frst models for externaltes, where they assumed an ordered top-down scan of slots by end users. By assumng a certan cost ncurred to the end users for clckng on an ad, they derved the equlbra of the GSP aucton for ther game. Along smlar lnes, Aggarwal et al. [1] and Kempe and Mahdan [23] studed Cascade Models nvolvng Markovan end-users, where every ad s assocated wth ts ndvdual CTR and a contnuaton probablty. The latter denotes the probablty that an end-user wll contnue scannng the lst of ads (n a top-down order) after vewng that partcular ad. The authors n these works treated the algorthmc queston of wnner determnaton towards maxmzng the socal welfare. Equlbra of the GSP aucton mechansm n the cascade model were studed by Gots and Karln n [20]. Kumnov and Tennenholtz [25] study 4

equlbra of the GSP and VCG (cf. [27, Ch. 9]) auctons n a smlar model. The cascade model was assessed expermentally by Craswell et al. [15]. Gomes, Immorlca and Markaks [21] were the frst to document emprcally externaltes under a verson of the cascade model, usng real data from keyword auctons. Externaltes among the bdders have also been consdered n other smlar settngs. For nstance, Ghosh and Mahdan [19] present a model for externaltes n onlne lead generaton, study the complexty of the wnner determnaton problem, and present computatonally effcent ncentve compatble mechansms for several cases. Chen and Kempe [12] consder postve and negatve socal dependences among the bdders for sngle tem auctons, employng an dea smlar to socal context graph, and study the propertes of equlbra for the frst and second prce auctons. Wnner Determnaton Problems. We denote by MSW-E the problem of Maxmum Socal Welfare wth Externaltes,.e. selectng a subset of wnnng players S and a permutaton π of S that maxmze sw(s, π, G) n our model. The complexty and performance of our algorthms are parameterzed by the wndow sze c and we wrte MSW-E(c). An nterestng specal case of MSW-E(c) occurs for E =. We refer to ths postve-only externaltes verson by MSW-PE(c). Motvated by the Cascade Model, we also consder the case of forward-only postve-only externaltes denoted by MSW-FPE(c) where an ad j may only nfluence an ad ff π(j) < π(). For dervng our hardness results we consder a smplfcaton of the model where the probablty w j that ad j affects ad only depends on the wndow sze n π, and not on ther ds,.e. w j = w for all (j, ) E + E. In presence of negatve externaltes, t may be proftable to select less than k ads and arrange them n a broken lst, namely a lst wth empty slots appearng between slots occuped by negatvely correlated ads. In practce however, ths may not be feasble. Therefore, we restrct feasble solutons to so-called unbroken lsts, namely lsts where the selected ads occupy consecutve slots n the ad lst startng from the frst one 4. Hence a feasble soluton can select less than k adds, but t s not allowed to use empty slots and separate negatvely correlated ads (.e. the empty slots, f any, occupy the last k S slots n π). The analyss of our algorthms assumes the case of unbroken lsts. However, t s not dffcult to generalze our algorthmc results to the case of broken lsts. 3 Computatonal Hardness and the GSP Mechansm In ths secton, we restrct our attenton to the smplest specal case of MSW-FPE(1), where there s a sngle functon w, and poston multplers λ j, valuatons v, and qualtes q are unform. We prove that even n ths very restrcted settng, MSW-FPE(1) s APX-hard. We note that a smple and elegant transformaton from the Longest Path problem can be used to prove NP-hardness of MSW-FPE(1); ths proof s descrbed n the Appendx, Secton A.1. Below we descrbe a PTAS reducton for APX-hardness. Theorem 1. MSW-FPE(1) s APX-hard even n the specal case of unform poston multples, valuatons, and qualtes. Proof sketch. The proof s by a PTAS reducton from the Travelng Salesperson Problem wth dstances 1 and 2, aka TSP(1, 2), whch s known to be APX-hard [28]. An nstance of TSP(1, 2) s defned by an undrected graph G(V, E). Each edge {u, u } E has length 1 and each non-edge {u, u } E has length 2. The goal s to fnd a tour that ncludes all vertces and has a mnmum 4 A reasonable assumpton that may justfy the restrcton above s that there exst an adequate number of neutral ads (k 1 of them suffce) that do not negatvely affect any other ad. 5

total length, or equvalently contans a mnmum number of non-edges. TSP(1, 2) s napproxmable wthn 741/740 unless P = NP [18], and approxmable wthn 8/7 [9]. Gven an undrected graph G(V, E), V = n, we construct an nstance I of MSW-FPE(1) wth n advertsers, all of them wth valuaton 1 and qualty q (0, 1), context graph G(V, E, ), a sngle functon w wth w(1) = β (0, 1) and w(l) = 0 for all l 1, n slots avalable n the sponsored lst, and unform poston multplers equal to 1. In the Appendx, Secton A.2, we show that gven a (1 + ε)-approxmate soluton to I, we can effcently construct a (1 + O(ε))-approxmate soluton to the nstance of TSP(1, 2) defned by G(V, E). 4 An Exact Algorthm Based on Color Codng In ths secton we develop and analyze an exact algorthm for MSW-E(c). We employ color codng [3] and dynamc programmng, to prove the followng result: Theorem 2. MSW-E(c) can be solved optmally n 2 O(k) n 2c+1 log 2 n tme. For smplcty, we assume that the optmal soluton conssts of k ads. We can remove ths assumpton by runnng the algorthm for every sponsored lst sze up to k, and keep the best soluton. Snce the runnng tme s exponental n k, ths does not change the asymptotcs of the runnng tme. To apply the technque of color codng, we consder a fxed colorng h : N [k] of the ads wth k colors. A lst (S, π) of k ads s colorful f all ads n S are assgned dfferent colors by h. In the followng, we formulate a dynamc programmng algorthm that computes the best colorful lst. For each 2c-tuple of ads ( 1,..., 2c ) N 2c, wth all ads assgned dfferent colors by h, and each color set C [k], C k 2c, that does not nclude any of the colors of 1,..., 2c, we compute sw( 1,..., 2c, C), namely the maxmum socal welfare f the last C postons n the lst are colored accordng to C, and on top of them, there are ads 1,..., 2c n ths order from top to bottom. More precsely, the soluton correspondng to sw( 1,..., 2c, C) assgns ad p to slot k ( C + 2c) + p, p = 1,..., 2c, and consders the best choce of ads colored accordng to C for the last C slots. Clearly, there are at most n 2c 2 k 2c dfferent sw values to compute, and the maxmum sw value for all colorful tuples ( 1,..., 2c, C), wth C = k 2c, corresponds to the best colorful lst of k ads. The proof of Theorem 2 follows: Proof. For the bass of our dynamc programmng, let C = and for all 2c-tuples ( 1,..., 2c ) N 2c wth all ads assgned dfferent colors by h, we have: sw( 1,..., 2c, ) = c λ k 2c+p Q p ( 1,..., p,..., p+c ) v p p=1 + 2c p=c+1 λ k 2c+p Q p ( p c,..., p,..., 2c ) v p, where Q p ( 1,..., p,..., c+p ) (resp. Q p ( p c,..., p,..., 2c )) s the CTR of ad p gven that the (only) ads n the lst at dstance at most c from p are 1,..., c+p (resp. p c,..., 2c ) arranged n ths order from top to bottom. 6

Gven the values of sw for all 2c-tuples of ads and all color sets of cardnalty s < k 2c, we compute the values of sw for all 2c-tuples ( 1,..., 2c ) and all color sets C of cardnalty s + 1: sw( 1,..., 2c, C) = max {sw( 2,..., 2c,, C {h()}) + :h() C + λ k ( C +2c)+1 Q 1 ( 1,..., c+1 ) v 1 + c + λ k ( C +2c)+p [Q p ( 1, 2,..., p,..., p+c ) Q p ( 2,..., p,..., p+c )] v p + p=2 + λ k C c+1 [Q c+1 ( 1, 2,..., c+1,..., 2c, ) Q p ( 2,..., c+1,..., 2c, )] v c+1 } In the recurson above, the second term accounts for the addtonal socal welfare due to 1, and the thrd and the fourth term account for the dfference n the socal welfare due to ads 2,..., c+1, whose CTRs Q 2,..., Q c+1 are affected by 1. Ads c+2,..., 2c, are used to calculate the dfference n the CTRs Q 2,..., Q c+1. The CTRs of ads c+2,..., 2c, and of the ads at the bottom of the lst wth colors n C {h()} are not affected by 1, snce ther dstance to 1 s greater than c. Therefore, for any fxed colorng h, the best colorful lst of k ads can be computed n tme O(n 2c+1 2 k ). If we select a random colorng h, the probablty that the optmal soluton s colorful under h s k!/k k > e k. If we run the algorthm for e k ln n randomly and ndependently chosen colorngs and keep the best soluton, the probablty that we fal to fnd the optmal soluton s at most 1/n. The approach can be derandomzed usng a k-perfect famly of hash functons of sze 2 O(k) log 2 n (see [3, Secton 4] for the detals). Hence we obtan the followng result for practcally nterestng szes of the problem: Corollary 1. For k = O(log n) and c = O(1), MSW-E(c) s n P. Moreover, unless NP DTIME(n poly(log n) ), MSW-E(c) s not NP-hard for c = O(1) and k = O(poly(log n)). In a mechansm desgn settng, suppose that advertsers clck valuatons are ther prvate nformaton. Havng an exact algorthm for MSW-E(c), we can drectly use t together wth the VCG payments (see e.g. [27, Ch. 9]) to obtan a truthful mechansm for MSW-E. Computaton of payments ncurs an addtonal factor O(k) to the runnng tme of the algorthm gven by Theorem 2. 5 O(c)-Approxmaton Algorthms for Postve Externaltes In ths secton, we show how to use polynomal-tme approxmaton algorthms for the Weghted m-set Packng problem and approxmate MSW-PE(c) wthn a factor of O(c) n polynomal tme. In Weghted m-set Packng, we are gven a collecton of sets, each wth at most m elements and a postve weght, and seek a collecton of dsjont sets of maxmum total weght. The greedy algorthm for Weghted m-set Packng acheves an approxmaton rato of m, the algorthm of [10] acheves an approxmaton rato of (2/3)m n tme quadratc n the number of sets, and the algorthm of [8] acheves an approxmaton rato of (m + 1)/2 n polynomal tme for any constant m. Theorem 3. Gven an α-approxmaton T (ν)-tme algorthm for Weghted 3-Set Packng wth ν sets, we obtan a 2αc-approxmaton T (kn 2 )-tme algorthm for MSW-PE(c) wth n ads and k slots. Proof. We transform any nstance of MSW-PE(c) to an nstance of Weghted 3-Set Packng wth kn 2 /4 sets so that any α-approxmaton to the optmal set packng gves a 2αc-approxmaton to the 7

optmal socal welfare for the orgnal MSW-PE(c) nstance. To smplfy the presentaton, we assume that k s even. Our proof can be easly extended to the case where k s odd. Gven an nstance of MSW-PE(c) wth n ads and k slots, we partton the lst nto k/2 blocks of 2 consecutve slots each. The set packng nstance conssts of ( n 2) 3-element sets for each block. Namely, for every block p = 1, 3, 5,..., k 1 and every subset { 1, 2 } of 2 ads, there s a set { 1, 2, p} n the set packng nstance 5. The weght W ( 1, 2, p) of each set { 1, 2, p} s the maxmum socal welfare f ads 1 and 2 are assgned to slots p and p + 1, and there s no nfluence on 1 and 2 from any other ad n the lst. Formally, W ( 1, 2, p) = max{λ p Q 1 ( 1, 2 )v 1 + λ p+1 Q 2 ( 1, 2 )v 2, λ p Q 2 ( 2, 1 )v 2 + λ p+1 Q 1 ( 2, 1 )v 1 } where Q 1 ( 1, 2 ) = 1 (1 q 1 )(1 q 2 w 2 1 (1)) (resp. Q 2 ( 1, 2 ) = 1 (1 q 2 )(1 q 1 w 1 2 ( 1))) denotes the relevance of ad 1 (resp. 2 ) gven that the only ad n the lst wth an nfluence on 1 (resp. 2 ) s 2 (resp. 1 ) located just above (resp. below) 1 (resp. 2 ) n the lst. Gven an nstance of MSW-PE(c) wth n advertsers and k slots, the correspondng nstance of Weghted 3-Set Packng can be computed n O(kn 2 ) tme. To show that the transformaton above s approxmaton preservng, we prove that () the optmal set packng has weght at least 1/(2c) of the maxmum socal welfare, and that () gven a set packng of weght W, we can effcently compute a soluton for the orgnal nstance of MSW-PE(c) wth a socal welfare of at least W. To prove (), we assume (by renumberng the ads approprately f needed) that the optmal lst for the MSW-PE(c) nstance s (1,..., k). We let Q be the relevance of ad n (1,..., k), and let W = k =1 λ Q v be the optmal socal welfare. We construct a collecton of 2c feasble set packngs of total weght at least W. Thus, at least one of them has a weght of at least W /(2c). The constructon s based on the followng clam, whch can be proven by nducton on c (see Appendx A.3). Clam 1 Let c be any postve nteger. Gven a lst (1,..., k), there s a collecton of 2c feasble 3-set packngs such that for each par 1, 2 of ads n (1,..., k) wth 1 2 c, the unon of these packngs contans a set { 1, 2, p} wth p mn{ 1, 2 }. Intutvely, for each par 1, 2 of ads located n (1,..., k) wthn a dstance no more than the wndow sze c, and thus possbly havng a postve nfluence on each other, the collecton of set packngs constructed n the proof of Clam 1 ncludes a set { 1, 2, p} whose weght accounts for the ncrease n 1 s and 2 s socal welfare due to 2 s and 1 s postve nfluence, respectvely. Summng up the weghts of all those sets, we account for the postve nfluence between all pars of ads n (1,..., k), and thus end up wth a total weght of at least W. Formally, let W (j) be the total weght of the j-th set packng constructed n the proof of Clam 1, and let 1, 2, wth 1 < 2, be any par of ads n (1,..., k) ncluded n the same set { 1, 2, p} of the j-th packng. Snce each ad appears n each set packng at most once, we let Q (j) 1 = Q 1 ( 1, 2 ) and Q (j) 2 = Q 2 ( 1, 2 ) be the relevance of 1 and 2 n the calculaton of W ( 1, 2, p). Snce Clam 1 ensures that p 1, and snce slot CTRs are non-ncreasng, λ 1 (resp. λ 2 ) s no greater than λ p (resp. λ p+1 ). Therefore, W ( 1, 2, p) λ 1 Q (j) 1 v 1 + λ 2 Q (j) 2 v 2. Settng Q (j) = 0 for all ads n (1,..., k) whch do not appear n any set of the j-th packng, we obtan that W (j) k =1 λ Q (j) v. We show that 2c j=1 W (j) k 2c λ v =1 j=1 Q (j) k λ Q v = W 5 Throughout the proof, we mplctly adopt the smplfyng (and easly mplementable) assumpton that the range of block descrptors 1, 3, 5,..., k 1 and the range of ad descrptors 1,..., n are dsjont. j=1 8

The frst nequalty follows from the dscusson above and by changng the order of the summaton. To establsh the second nequalty, we show that for every ad n (1,..., k), 2c j=1 Q(j) Q. To smplfy the presentaton, we focus on an ad wth c < k c. Ads 1,..., c and k c + 1,..., k can be treated smlarly. We recall that Q = 1 (1 q ) +c j= c,j P (j), where for each ad j n the sublst ( c,..., 1, + 1,..., + c), P (j) = (1 q j w j (j )) [0, 1] accounts for j s postve nfluence on s relevance. Clam 1 ensures that for each j n ( c,..., 1, + 1,..., + c), ads and j are ncluded n the same set of some set packng. Snce ad appears n each set packng at most once, for smplcty, we can renumber the set packngs of Clam 1, and say that and j are ncluded n the same set of the j-th set packng. Then, f j <, Q (j) = 1 (1 q )(1 q j w j ( 1)) 1 (1 q )P (j), because w j s non-ncreasng wth the dstance of j and n the lst, and thus w j ( 1) w j (j ). The same holds f j >. Therefore, Q (j) 1 (1 q )P (j), for any j. To conclude the proof of (), we observe that: +c j= c,j (1 (1 q )P (j)) 1 (1 q ) +c j= c,j P (j) = Q (4) To establsh (4), we repeatedly apply that for every x, y, z [0, 1], (1 xy) + (1 xz) 1 xyz. We proceed to establsh clam (), namely that gven a set packng of weght W, we can effcently construct a sponsored lst of socal welfare at least W for the orgnal nstance of MSW-PE(c). By constructon, we can restrct our attenton to set packngs of the form {{ p, p+1, p}} p=1,3,...,k 1, where the weght of the packng s W = p W ( p, p+1, p), and where ads p and p+1 are ndexed accordng to ther best order, wth respect to whch W ( p, p+1, p) s calculated. Snce we consder postve externaltes, the sponsored lst ( 1, 2,..., k 1, k ) has a socal welfare of at least W. Combnng Theorem 3 wth the greedy 3-approxmaton algorthm for Weghted 3-Set Packng and wth the algorthm of [10] we obtan respectvely: Corollary 2. For n ads and k slots, the MSW-PE(c) problem can be approxmated wthn factor 6c n O(kn 2 log n) tme and wthn factor 4c n O(k 2 n 4 ) tme. A smlar approxmaton preservng reducton from MSW-PE(c) to Weghted (2c + 1)-Set Packng yelds: Theorem 4. An f(m)-approxmaton T (ν, m)-tme algorthm for Weghted m-set Packng wth ν sets yelds a 2f(2c + 1)-approxmaton O(ckn 2c + T (kn 2c, 2c + 1))-tme algorthm for MSW-PE(c) wth n ads and k slots. The full proof appears n the Appendx, Secton A.4; the lst s parttoned nto k/(2c) blocks of 2c consecutve slots each, and the set packng nstance conssts of ( n 2c) sets of 2c + 1 elements each for each block. Then we exhbt a par of feasble set packngs whose total weght s no less than the optmal socal welfare. Combnng Theorem 4 wth the approxmaton algorthm of [8], we obtan a polynomal tme 2(c + 1)-approxmaton algorthm MSW-PE(c), for any constant c. 9

6 On the GSP Mechansm wth Externaltes In examnng the behavor of the GSP aucton mechansm we make the reasonable assumpton that only a snapshot of the players ntrnsc relevances q, N s avalable to the mechansm. In practce, estmates of the players relevances are deduced by software machnery of the sponsored search platform; t s therefore concevable that the mechansm wll eventually extract nformaton ndcatve of externaltes. By the tme ths occurs however, assocatons among advertsers may also be updated. Ths justfes the mechansm s unawareness of externaltes. On the other hand, each advertser s aware of the socal context assocatons that may harm hm or boost hs relevance. Gven a socal context graph G(N, E +, E ), wth functons w j for every (j, ) E + E and wndow sze c, we assume that every advertser N s aware of: E + = {(, ) E + N }, E = {(, ) E N } and of c and w j for every j E + E. In a keyword aucton each advertser bds b for recevng a slot n the lst. The GSP mechansm n ts most common flavors uses the Rank-By-Revenue (RBR) rule to assgn advertsers to slots. Under RBR, advertsers are ranked n order of non-ncreasng score q b and hgher scores are assgned hgher CTR slots. The score of a bdder s hs declared expected revenue for a clck. The planer RBB rule s obtaned by takng q = 1 for all N. Gven a bd vector b, let φ b : K N denote the rankng of bdders n order of non-ncreasng expected revenue,.e. φ b (j) s the bdder assgned to slot j. Accordng to the prevously used defnton of π, we use φ as an extenson of π 1. Every slot wnnng player φ b (j), for j = 1,..., k pays per clck a prce equal to (q φb (j+1) b φb (j+1))/q φb (j);.e., the score of the bdder occupyng the next poston under b, dvded by the ntrnsc relevance of φ b (j). Usng hs knowledge of externaltes that nfluence hm, player φ b (j) experences a relevance Q φb (j)(b) and estmates hs expected proft (utlty) as: ( u φb (j)(b) = λ j Q φb (j)(b) v φb (j) q ) φ b (j+1) b q φb (j+1) (5) φb (j) We assume a complete nformaton settng, as advertsers typcally employ machne learnng technques to estmate how much they should outbd a compettor. Such technques reveal the rankng nformaton used by the GSP mechansm. Thus, n computng hs best response under a bd vector b, every advertser N s assumed to know only q and b for each N \ {}, and not the actual relevance Q perceved by. In studyng pure Nash equlbra of the GSP mechansm we make the standard assumpton of conservatve (or ex-post ndvdually ratonal) bdders. It states that n fear of recevng negatve utlty, no bdder ever outbds hs valuaton,.e. b v holds for all N. Ths allows to prove meanngful bounds for the socal welfare relatve to the socal optmum [26, 13] snce, otherwse, the worst-case socal welfare of equlbra may be arbtrarly low. For conservatve bdders and under our defntons of the mechansm s and the bdders awareness of externaltes we show: Proposton 1. The strategc game nduced by the Generalzed Second Prce Aucton mechansm under the RBR rule and determnstc te-breakng does not generally have pure Nash equlbra n presence of forward postve externaltes, even for 3 conservatve players and 2 slots. Proof. Consder 3 players, and 2 slots. The te-breakng rule pcks player 3 frst f he ssues the same bd wth any of the other two players. Among the other 2 players tes may be resolved n any arbtrarly chosen fxed way. All players have ntrnsc relevance q and valuatons v 1 = v 2 = V > 2 q 1 q v, where v = v 3. The slots have CTRs λ 1 > λ 2, such that γ = λ 2 /λ 1 > V/[(2 q) (V v)]. Notce that V = (2 q) V (1 q) V < (2 q) (V v), because V > 2 q 1 q v. Thus γ can be feasbly chosen wthn a non-empty range (V/[(2 q) (V v)], 1]. 10

Set the wndow sze to be c = 1 and consder a drected cycle {(1, 2), (2, 1)} as the context graph. Let w j (1) = 1. When both players 1,2 are slot wnners, f player s ranked below, hs relevance s amplfed to Q = q(2 q). We clam that none of the orderngs 1, 2 and 2, 1 has a bd vector that s a pure Nash equlbrum. For every bd vector b that makes player 3 a looser (wthout tes), we show that player φ b (1) has ncentve to am for slot 2. b wll denote the bd vector after such a devaton of player φ b (1), gven b φb (1). Then: u (1) (b) = λ 1 q (v φb (1) b φb (2)) = γ 1 λ 2 q (v φb (1) b φb (2)) < γ 1 λ 2 q V (2 q) (V v) < V λ 2 q V = λ 2 Q (V v) λ 2 Q (v φb (1) b 3 ) = u φb (1)(b ) When at least one of players 1, 2 bds equally to b 3, at least one of 1,2 looses (.e. does not wn a slot), because of the te-breakng rule. Under such a confguraton the loosng player has ncentve (and the capacty - because V > v b 3 ) to outbd b 3 and get a slot. Ths proves that for partcular determnstc te-breakng rules, conservatve pure Nash equlbra do not exst for the GSP under the RBR rule. The proof of the followng proposton can be found n Appendx A.5. Proposton 2. The strategc game nduced by the Generalzed Second Prce Aucton under the RBR rule and determnstc te-breakng does not generally have pure Nash equlbra n presence of bdrectonal postve externaltes, even for 3 conservatve players and 3 slots. In provng these propostons we used equal ntrnsc relevance factors among players. Thus, even the relevance-ndependent RBB rule cannot yeld stablty aganst the players perceptons about ther expected payoff. Even when pure Nash equlbra exst, externaltes may cause the optmum socal welfare to be arbtrarly larger than the best welfare achevable n equlbrum,.e. an unbounded Prce of Stablty [4]: Proposton 3. There s an nfnte famly of nstances of the stategc game nduced by the Generalzed Second Prce aucton mechansm wth unbounded Prce of Stablty, even wth conservatve bdders. Proof. Consder frst an nstance wth 2 slots of equal CTRs, λ 1 = λ 2 = 1, and 5 players. Take q 1 = q 2 = q 5 = δ, v 1 = v 2 = v 5 = 1, and q 3 = q 4 = 1, v 3 = v 4 = δ such that δ s always very much smaller than δ. Let the socal context graph have edge set E + = {(3, 1), (4, 2)} and E =. Set the wndow sze to c = 1 and let w(1) = 1. The bd vector b where b = v s a pure Nash equlbrum: under any te-breakng rule, two players out of 1, 2, 5 receve a slot and have zero utlty. Then, sw(b) = 2δ. For the socal optmum, assgn the two slots to any of the pars {3, 1} or {4, 2}. The optmum socal welfare s then OP T = 1 + δ, n any of these pars one player has hs relevance amplfed to 1, due to postve nfluence by the other. OP T/sw(b) = (1 + δ )/(2δ), whch becomes arbtrarly large as δ 0. Notce that, because players are conservatve, any bd confguraton n whch a player {3, 4} receves a slot, cannot be an equlbrum. Ths s because at least two of players 1,2,5 would not receve a slot and any of them can devate by outbddng. Thus all equlbra of the game have equal socal welfare. One can straghtforwardly generalze ths nstance for any number of slots k, usng 2k + 1 players. 11

Ths result demonstrates how externaltes may harm the performance of the GSP mechansm arbtrarly, n contrast to the recent upper bound of 1.282 shown recently [11] on the mechansm s Prce of Anarchy [24] (the rato of the optmum welfare over the worst equlbrum welfare), wthout externaltes. Gots and Karln showed n [20] a lower bound of k n the Cascade Model of externaltes wth conservatve bdders. 7 Concluson and Further Drectons We presented a comprehensve lst of computatonal results for allevatng the mpact of externaltes n sponsored search, under a novel farly expressve model. We gave an exact algorthm for optmzng the socal welfare under postve and/or negatve externaltes, whch can be pared wth VCG payments, to yeld a truthful polynomal tme mechansm for practcally relevant nput szes. We settled APX-completeness of the assocated Wnner Determnaton problem wth postve-only externaltes and constant-sze wndow of ther scope, by provng hardness and desgnng constant factor approxmaton algorthms. These algorthms do not yeld truthful mechansms n domnant strateges, because they are not monotone 6. The dependence of the advertsers CTRs depend on the CTRs of others caused even smple approxmaton algorthms that we tred not to be monotone. On the other hand we showed that the non-truthful GSP mechansm may not have pure Nash equlbra n presence of externaltes; when t does, these may have unboundedly low socal welfare. As our negatve results rely on the generalty of our model, many drectons emerge regardng the desgn and performance analyss of mechansms (ncludng the GSP), for specal practcally relevant cases; these nclude e.g. smple cases of socal context graphs and bounded nfluence among advertsers. Acknowledgments: Dmtrs Fotaks wshes to thank Vasleos Syrgkans for many helpful dscussons on the model of externaltes among advertsers presented n ths work. We would also lke to thank Moshe Tennenholtz for many nterestng dscussons and suggestons concernng ths project. References 1. G. Aggarwal, J. Feldman, S. Muthukrshnan, and M. Pal. Sponsored Search Auctons wth Markovan Users. In C. H. Papadmtrou and S. Zhang, edtors, WINE 2008, volume 5385 of LNCS, pages 621 628. Sprnger, Hedelberg, 2008. 2. G. Aggarwal, A. Goel, and R. Motwan. Truthful auctons for prcng search keywords. In Proc. of 7th ACM Conference on Electronc Commerce (EC), pages 1 7. ACM, 2006. 3. N. Alon, R. Yuster, and U. Zwck. Color Codng. Journal of the ACM, 42:844 856, 1995. 4. E. Anshelevch, A. Dasgupta, J. M. Klenberg, E. Tardos, T. Wexler, and T. Roughgarden. The Prce of Stablty for Network Desgn wth Far Cost Allocaton. SIAM Journal on Computng, 38(4):1602 1623, 2008. 5. A. Archer and É. Tardos. Truthful mechansms for one-parameter agents. In Proc. of 42nd IEEE Symposum on Foundatons of Computer Scence (FOCS), pages 482 491. IEEE, 2001. 6. I. Ashlag, P. Krysta, and M. Tennenholtz. Socal Context Games. In C. H. Papadmtrou and S. Zhang, edtors, WINE 2008, volume 5385 of LNCS, pages 675 683, 2008. 7. S. Athey and G. Ellson. Poston Auctons wth Consumer Search. Quartetly Journal of Economs, page (to appear), 2011. 8. P. Berman. A d/2 Approxmaton for Maxmum Weght Independent Set n d-claw Free Graphs. In M. M. Halldórsson, edtor, SWAT 2000, volume 1851 of LNCS, pages 214 219, 2000. 9. P. Berman and M. Karpnsk. 8/7-approxmaton algorthm for (1, 2)-TSP. In Proc. of the 17th Annual ACM Symposum on Dscrete Algorthms (SODA), pages 641 648. ACM, 2006. 10. P. Berman and P. Krysta. Optmzng msdrecton. In Proceedngs of the ACM-SIAM Symposum on Dscrete Algorthms (SODA), pages 192 201, 2003. 6 A certan monotoncty property s requred for an algorthm to mply a truthful mechansm n sngle parameter settngs as s ours, see, e.g., [5]. 12

11. I. Caraganns, C. Kaklamans, P. Kanellopoulos, and M. Kyropoulou. On the Effcency of Equlbra n Generalzed Second Prce Auctons. In Proceedngs of the ACM Conference on Electronc Commerce (EC), to appear. ACM, 2011. 12. P. Chen and D. Kempe. Bayesan Auctons wth Frends and Foes. In M. Mavroncolas and V. G. Papadopoulou, edtors, SAGT 2009, volume 5814 of LNCS, pages 335 346. Sprnger, Hedelberg, 2009. 13. G. Chrstodoulou, A. Kovács, and M. Schapra. Bayesan Combnatoral Auctons. In L. Aceto, I. Damgaard, L. A. Goldberg, M. M. Halldórsson, A. Ingólfsdóttr, and I. Walukewcz, edtors, ICALP 2008 (1), volume 5125 of LNCS, pages 820 832. Sprnger, Hedelberg, 2008. 14. P. Cramton, Y. Shoham, and R. Stenberg. Combnatoral Auctons. MIT Press, 2006. 15. N. Craswell, O. Zoeter, M. Taylor, and B. Ramsey. An expermental comparson of clck poston-bas models. In WSDM 08: Proceedngs of the nternatonal conference on Web search and web data mnng, pages 87 94. ACM, 2008. 16. A. Das, I. Gots, A.R. Karln, and C. Matheu. On the Effects of Competng Advertsements n Keyword Auctons. In Unpublshed manuscrpt, 2008. 17. B. Edelman, M. Ostrovsky, and M. Schwarz. Internet advertsng and the generalzed second prce aucton: Sellng bllons of dollars worth of keywords. Amercan Economc Revew, 97(1):242 259, 2007. 18. L. Engebretsen and M. Karpnsk. TSP wth bounded metrcs. Journal of Comp. and Sys. Scences, 72(4):509 546, 2006. 19. A. Ghosh and M. Mahdan. Externaltes n Onlne Advertsng. In Proc. of the 17th Internatonal Conference on World Wde Web Conference (WWW), pages 161 168. ACM, 2008. 20. I. Gots and A. R. Karln. On the equlbra and effcency of the GSP mechansm n keyword auctons wth externaltes. In C. H. Papadmtrou and S. Zhang, edtors, WINE 2008, volume 5385 of LNCS, pages 629 638. Sprnger, Hedelberg, 2008. 21. R. Gomes, N. Immorlca, and E. Markaks. Externaltes n Keyword Auctons: an emprcal and theoretcal assessment. In S. Leonard, edtor, WINE 2009, LNCS, pages 172 183. Sprnger, Hedelberg, 2009. 22. P. Jezorsk and I. Segal. What makes them clck: Emprcal analyss of consumer demand for search advertsng. Workng paper, Department of Economcs, Stanford Unversty, 2009. 23. D. Kempe and M. Mahdan. A Cascade Model for Externaltes n Sponsored Search. In C. H. Papadmtrou and S. Zhang, edtors, WINE 2008, volume 5385 of LNCS, pages 585 596. Sprnger, Hedelberg, 2008. 24. E. Koutsoupas and C. H. Papadmtrou. Worst-case equlbra. Computer Scence Revew, 3(2):65 69, 2009. 25. D. Kumnov and M. Tennenholtz. User Modelng n Poston Auctons: Re-Consderng the GSP and VCG Mechansms. In Proc. of 8th Int. Conf. on Autonomous Agents and Multagent Systems (AAMAS), pages 273 280. ACM, 2009. 26. R. Paes Leme and E. Tardos. Pure and Bayes-Nash Prce of Anarchy for Generalzed Second Prce Aucton. In Proc. of the IEEE Symposum on Foundatons of Computer Scence (FOCS), pages 735 744. IEEE, 2010. 27. N. Nsan, T. Roughgarden, É. Tardos, and V. Vazran. Algorthmc Game Theory. Cambrdge Unversty Press, 2007. 28. C.H. Papadmtrou and M. Yannakaks. The Travellng Salesman Problem wth Dstances One and Two. Mathematcs of Operatons Research, 18(1):1 11, 1993. 29. H. R. Varan. Poston auctons. Internatonal Journal of Industral Organzaton, 25(6):1163 1178, 2007. 30. W. Vckrey. Counterspeculaton, Auctons and Compettve Sealed Tenders. Journal of Fnance, pages 8 37, 1961. 13

A Appendx A.1 NP-hardness of MSW-FPE(1) Proposton 4. MSW-FPE(1) s NP-hard even n the specal case of unform poston multplers, valuatons, and qualtes. Proof. We use transformaton from the Longest Path problem. Gven a drected graph G(V, E) and an nteger k 2, we construct an nstance I of MSW-FPE(1). In I, there are V advertsers, all of them wth valuaton 1 and qualty 1/2, there are k slots n the sponsored lst, all poston multplers are equal 1, the advertsers context graph s G(V, E, ), and there s a sngle externalty functon wth w(1) = 1/2 and w(l) = 0 for all l 1. The ad at slot 1 contrbutes 1/2 to the socal welfare. The ad at each slot l, l {2,..., k}, contrbutes 5/8 to the socal welfare f the ad on slot l 1 s correlated to t, and 1/2 otherwse. It s mmedate that G has a smple path of length k 1 ff I admts a soluton of socal welfare at least (5k 1)/8. A.2 The Proof of Theorem 1 We show that for any constant ε > 0, an (1 + ε)-approxmaton algorthm for MSW-FPE(1) mples a (1 + O(ε))-approxmaton algorthm for TSP(1, 2). Gven an nstance G(V, E), V = n, of TSP(1, 2), we construct an nstance I of MSW-FPE(1). In I, there are n advertsers, all of them wth valuaton 1 and qualty q (0, 1), the context graph s G(V, E, ), there s a sngle functon w wth w(1) = β (0, 1) and w(l) = 0 for all l 1, the number of slots s n (.e. all ads ft n the sponsored lst), and all poston multplers are equal to 1. A soluton to I s a permutaton π of the n advertsers correspondng to the vertces of G. The ad at slot 1 contrbutes q to the socal welfare. The ad at each slot l, l {2,..., n}, contrbutes q(1 + β(1 q)) to the socal welfare f the ad at slot l 1 s correlated to t (.e. there s an edge n G between π 1 (l 1) and π 1 (l)), and q otherwse. To smplfy the notaton, we let ζ = β(1 q). Therefore any soluton to I that contans µ non-edges of G,.e. assgns µ pars of ads not connected by an edge n G to consecutve slots on the ad lst, has a socal welfare of q(n(1 + ζ) (µ + 1)ζ). Let t = (u 1,..., u n ) be an optmal tour for the TSP(1, 2) nstance, and let α be the number of non-edges n t, and n+α be t s total length. For smplcty, we assume that f α > 0, {u n, u 1 } E. The soluton to I that arranges the ads n the same order as n t, namely (u 1,..., u n ), has a socal welfare of at least q(n(1 + ζ) ζ) f α = 0, and q(n(1 + ζ) αζ) otherwse. For any constant ε > 0, we consder a (1 + ε)-approxmate soluton π to I. Let π contan α non-edges. If α > 0, the socal welfare of π s: whch mples that q(n(1 + ζ) (α + 1)ζ) α + 1 If α = 0, we work smlarly and obtan that α + 1 q(n(1 + ζ) αζ) 1 + ε ε ζ+1 1+ε ζ n + 1 1+ε α ε ζ+1 1+ε ζ n + 1 1+ε Arrangng the vertces of G accordng to π gves a tour wth at most α +1 non-edges and a length no greater than n + α + 1. If α > 0, we obtan that ( ) n + α + 1 (n + α) 1 + ε ζ+1 ζ 14,

If α = 0, we obtan that ( ) ( n + α + 1 n 1 + ε ζ+1 1+ε ζ + 1 n(1+ε) n 1 + ε ζ + 1 ), ζ ζ ε 2 (1+ζ). where the last nequalty holds for all n Therefore, for any constant ε > 0, any (1 + ε)-approxmate soluton to MSW-FPE(1) nstance I can be translated nto a (1 + ε ζ+1 ζ )-approxmate soluton to the orgnal nstance of TSP(1, 2). We recall that ζ = β(1 q). Thus ζ+1 ζ can be arbtrarly close to 2 f q s selected suffcently close to 0 and β s selected suffcently close to 1. A.3 The Proof of Clam 1 n Theorem 3 The proof s by nducton on c. For c = 1, the frst set packng s {{p, p + 1, p}} p=1,3,5,...,k 1 and the second set packng s {{p + 1, p + 2, p}} 7 p=1,3,5,...,k 3. It s straghtforward to verfy that for each ad n (1,..., k 1), there s a set {, + 1, p} wth p = 1 f s even, and wth p = f s odd (see also the par of rows correspondng to c = 1 n Fg. 1). For some nteger c 2, we nductvely assume that the clam holds for a wndow sze up to c 1, and we show that the clam holds for a wndow sze of c. Thus, by nductve hypothess, we already have a collecton of 2(c 1) feasble set packngs such that for each par 1, 2 of ads wth 1 2 c 1, the unon of these packngs contans a set { 1, 2, p} wth p mn{ 1, 2 } (see e.g. the eght rows correspondng to c = 1, 2, 3, 4 n Fg. 1). To complete the constructon, we add two new set packngs such that for each ad k c, they nclude a set {, + c, p}, wth p. If c s odd, the frst new set packng s {{p, p + c, p}} p=1,3,5,...,k c and the second new set packng s {{p + 1, p + c + 1, p}} p=1,3,5,...,k c 2 (see the par of rows correspondng to c = 5 n Fg. 1). Both set packngs are feasble. In partcular, none of them contans the same ad twce, snce each set contans an even and an odd ad, and the ad ds ncrease by 2 from each set to the next. Moreover, for each ad k c, there s a set {, + c, p} wth p = f s odd, and wth p = 1 f s even. If c s even, we use a more complcated perodc constructon, whch s requred for the feasblty of the two set packngs. The frst new set packng s: {{2cl + p, 2cl + p + c, 2cl + p}} p=1,3,5,...,c 1,l=0,1,..., (k 3c)/(2c) {{2cl + p + c + 1, 2cl + p + 2c + 1, 2cl + p + c}} p=1,3,5,...,c 1,l=0,1,..., (k 3c)/(2c) The second new set packng s: {{2cl + p + 1, 2cl + p + c + 1, 2cl + p}} p=1,3,5,...,c 1,l=0,1,..., (k 3c)/(2c) {{2cl + p + c, 2cl + p + 2c, 2cl + p + c}} p=1,3,5,...,c 1,l=0,1,..., (k 3c)/(2c) wth the understandng that we exclude any sets wth an ad of d greater than k (see the par of rows correspondng to c = 6 n Fg. 1). In the frst (resp. the second) set packng, the sets n the frst group nclude each ad wth odd (resp. even) d once, and the sets n the second group nclude each ad wth even (resp. odd) d at most once. Thus we ensure that both set packngs are feasble. In both cases, the 7 We recall that the thrd element of each set s the so-called block descrptor, and refers to the lst postons accordng to whch the weght of the set s calculated n the proof of Theorem 3. Even though we convenently dentfy both ads and block descrptors wth postve ntegers, we hghlght that ads and block descrptors are dfferent enttes. Thus, e.g. {p, p + 1, p} should be regarded as a set wth three dfferent elements, and {p, p + 1, p 1} and {p + 2, p + 3, p + 1} should be regarded as dsjont sets, because the contan dfferent ads and have dfferent block descrptors. 15

p = 1 p = 3 p = 5 p = 7 p = 9 p = 11 p = 13 p = 15 p = 17 p = 19 p = 21 p = 23 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 c = 1 1 3 4 6 5 7 8 10 9 11 12 14 13 15 16 18 17 19 20 22 21 23 2 4 3 5 6 8 7 9 10 12 11 13 14 16 15 17 18 20 19 21 22 24 1 4 3 6 5 8 7 10 9 12 11 14 13 16 15 18 17 20 19 22 21 24 2 5 4 7 6 9 8 11 10 13 12 15 14 17 16 19 18 21 20 23 c = 2 c = 3 1 5 3 7 6 10 8 12 9 13 11 15 14 18 16 20 17 21 19 23 2 6 4 8 5 9 7 11 10 14 12 16 13 17 15 19 18 22 20 24 1 6 3 8 5 10 7 12 9 14 11 16 13 18 15 20 17 22 19 24 2 7 4 9 6 11 8 13 10 15 12 17 14 19 16 21 18 23 c = 4 c = 5 1 7 3 9 5 11 8 14 10 16 12 18 13 19 15 21 17 23 2 8 4 10 6 12 7 13 9 15 11 17 14 20 16 22 18 24 c = 6 Fg. 1. The set packngs constructed n the proof of Clam 1 for k = 24 and c = 1,..., 6. For each value of c, we have two rows, one for each of the addtonal set packngs n the nductve step. Each cell contanng two ntegers 1, 2 corresponds to a set wth ads 1 and 2. The block descrptor of each set s gven by the value of p n the correspondng column. Cells / sets n the same row belong to the same set packng, and thus each add appears n each row at most once, whle cells /set n the same column have the same block descrptor. For even values of c, we mark, by swtchng from grey to whte color, the groups of sets obtaned for each dfferent value of l. ads are pared so that the ds of two ads n the same set dffer by c. Moreover, the frst group of the frst (resp. second) packng and the second group of the second (resp. frst) packng are complmentary, n the sense ther unon ncludes all sets { 1, 2, p} where 1 and 2 are odd (resp. even) and have 1 2 = c. Therefore, for each ad k c, there s a set {, + c, p} wth p = f s odd, and wth p = 1 f s even. Hence, for every wndow sze c and any lst (1,..., k), we can construct a collecton of 2c feasble 3-set packngs such that for each par 1, 2 of ads wth 1 2 c, there s a set { 1, 2, l} wth l mn{ 1, 2 }. A.4 The Proof of Theorem 4 We transform any nstance of MSW-PE(c) to an nstance of Weghted (2c + 1)-Set Packng wth at most kn 2c sets so that any α-approxmaton to the optmal set packng gves a 2α-approxmaton to the optmal soluton of the MSW-PE(c) nstance. To smplfy the presentaton, we assume that the optmal soluton to the orgnal nstance conssts of exactly k ads, and that k s a multple of 2c. Our proof can be easly extended to handle the remanng cases as well. Gven an nstance of MSW-PE(c) wth n advertsers and k slots, we partton the lst nto k/(2c) blocks of 2c consecutve slots each. The set packng nstance conssts of ( n 2c) sets of 2c + 1 elements each for each block. Namely, for every p = 1, 2c+1, 4c+1,..., k 2c+1, and every set { 1,..., 2c } of 2c ads, there s a set { 1,..., 2c, p} n the set packng nstance 8. Thus, we create at most kn 2c sets. To compute the weght of each set { 1,..., 2c, p}, we consder all possble permutatons π of ads 1,..., 2c. For each permutaton π, let W ({ 1,..., 2c, p}, π) denote the socal welfare f ads 8 As n the proof of Theorem 3, we mplctly adopt the smplfyng assumpton that the range of block descrptors 1, 2c + 1, 4c + 1,..., k 2c + 1 and the range of ad descrptors 1,..., n are dsjont. 16

1,..., 2c alone are assgned to the slots p, p + 1,..., p + 2c 1 accordng to π. Formally, W ({ 1,..., 2c, p}, π) = + 2c j=c+1 c λ p+j 1 Q π 1 (j)(π 1 (1),..., π 1 (j),..., π 1 (j + c)) v π 1 (j) + j=1 λ p+j 1 Q π 1 (j)(π 1 (j c),..., π 1 (j),..., π 1 (2c)) v π 1 (j) (6) where π 1 (j) s the ad assgned to slot p+j 1 by π, and Q π 1 (j)(π 1 (1),..., π 1 (j),..., π 1 (j + c)) (resp. Q π 1 (j)(π 1 (j c),..., π 1 (j),..., π 1 (2c))) denotes the relevance of ad π 1 (j) gven that the only ads n the lst at dstance at most c from π 1 (j) are π 1 (1),..., π 1 (j),..., π 1 (j + c) (resp. π 1 (j c),..., π 1 (j),..., π 1 (2c)) arranged n ths order from top to bottom. The weght W ({ 1,..., 2c, p}) of each set { 1,..., 2c, p} s the maxmum weght for all permutatons π of 1,..., 2c. Snce the weght of each set can be calculated n O((2c)!c 2 ) tme, the Weghted (2c + 1)-Set Packng nstance can be computed n O(ckn 2c ) tme. To show that the transformaton above s approxmaton preservng, we prove that () the optmal set packng has weght at least half of the maxmum socal welfare, and that () gven a set packng of weght W, we can effcently compute a soluton to the nstance of MSW-PE(c) of socal welfare at least W. To prove (), we let (1,..., k) be the optmal lst for the MSW-PE(c) nstance, let Q be the relevance of ad n (1,..., k), and let W = k =1 λ Q v be the optmal socal welfare. Next, we construct a par of feasble packngs of total weght at least W. For the frst set packng, we partton (1,..., k) nto k/(2c) sets each contanng 2c ads n consecutve slots and the correspondng block descrptor. In partcular, the frst set packng contans a set = {p, p+1,..., p+2c 1, p} for each block descrptor p = 1, 2c+1, 4c+1,..., k 2c+1. Let π p (1) be the permutaton that assgns the ads n s (1) p to ther slots n (1,..., k), and for each s (1) p, let Q (1) be the relevance wth whch ad contrbutes to W (s (1) p, π p (1) ) n (6). Snce W (s (1) p ) s the s (1) p maxmum weght of s (1) p over all permutatons of ts ads, the total weght of the frst packng s W (1) j=k j=1 λ j Q (1) v. For the second packng, we partton (c + 1,..., k, 1,..., c) nto k/(2c) sets smlarly. In partcular, there s a set s (2) p wth the ads n slots p, p + 1..., p 2c + 1 n (c + 1,..., k, 1,..., c) for each block descrptor p = 1, 2c + 1, 4c + 1,..., k 2c + 1. As before, we let π p (2) be the permutaton that assgns the ads n s (2) p to ther slots n (c + 1,..., k, 1,..., c), and for each s (2) p, we let Q (2) be the relevance wth whch ad contrbutes to W (s (2) p, π p (2) ) n (6). Snce W (s (2) p ) s the maxmum weght of s (2) p over all permutatons, and snce slot CTRs are non-ncreasng, the total weght of the second packng s W (2) j=k j=c+1 λ j Q (2) v. We clam that: j=c W (1) + W (2) λ j Q (1) j v j + j=1 j=k j=c+1 λ j (Q (1) j v j + Q (2) ) W The frst nequalty follows from the dscusson above. To establsh the second nequalty, we frst observe that for all = 1,..., c, Q = Q(1), snce all ads that affect the relevance of n (1,..., k) are ncluded n s (1) 1, and thus are taken nto account n the calculaton of Q(1). For the remanng ads = c + 1,..., k, we work as n the proof of Theorem 3 and show that Q. We frst consder an ad assgned to the frst c slots of block s(1) p by π p (1), for Q (1) + Q (2) 17 j

some p = 2c + 1, 4c + 1,..., k 2c + 1. Then ad s assgned to the last c slots of block s (2) p 2c by π (2) p 2c. Therefore, Q(1) takes nto account externaltes due to ads +1,..., +c after and Q (2) takes nto account externaltes due to ads c,..., 1 before n (1,..., k). Snce we consder postve externaltes, Q (1) Q (, +1,..., +c) and Q (2) Q ( c,..., 1, ). By the defnton of ads relevance for postve externaltes, there are P +, P [0, 1], whch correspond to the contrbuton of ads + 1,..., + c and c,..., 1 to the product n (2), such that: Q (, + 1,..., + c) = 1 (1 q )P +, Q ( c,..., 1, ) = 1 (1 q )P, and Q = Q ( c,..., j,..., + c) = 1 (1 q )P + P. As n the proof of Theorem 3, 1 (1 q )P + + 1 (1 q )P 1 (1 q )P + P + (1 q )(1 P + )(1 P ) 1 (1 q )P + P, whch mples that Q (1) + Q (2) Q, for all ads = c + 1,..., k. The case where s assgned to the last c slots of some block s (1) p by π p (1) s symmetrc, and can be handled smlarly, wth the roles of Q (1) and Q (2) swtched. Ths concludes the proof of clam (). We proceed to establsh that gven a set packng of weght W, we can effcently construct a sponsored lst of socal welfare at least W for the orgnal nstance. Let s 1,..., s k/(2c) be a packng of total weght W = W (s 1 ) + + W (s k/(2c) ), and for each p = 1, 2c + 1, 4c + 1,..., k 2c + 1, let π p be the best permutaton of ads n s p. Snce the unon of s 1,..., s k/(2c) contans k ads, snce W (s p, π p ) = W (s p ), and snce we consder postve externaltes, the sponsored lst where each ad n set s p s assgned to slot p + π p () 1 of the sponsored lst has a socal welfare of at least W. A.5 Proof of Proposton 2 Take 3 players wth v 1 = v 2 = V, v 3 = v wth equal ntrnsc relevances q < 1 and choose V, v so that V > 2 q 2 q 1 v. We consder the socal context graph wth E + = {(3, 1), (3, 2)}, E =, wndow sze c = 1, and w 31 (1) = w 32 = 1. The CTRs of slots are such that γ 2 = λ 2 /λ 1 > (2 q)(v v), γ 2 < 1 v V and γ 3 = λ 3 /λ 2 < 1 v V. Notce that γ 2 can be feasbly chosen wthn a non-empty V range ( (2 q)(v v), 1 v V ), because V > 2 q 2 q 1 v. Also, observe that, nfluence to any of players 1, 2 by player 3, boosts ther relevance to Q = q(2 q). By condtonng on the slot assgnment of player 3, we show that one of the other advertsers has ncentve to devate. We wll frst assume dstnct bds that produce the correspondng assgnments and comment on falure of a determnstc te-breakng rule afterwards. CASE I. In any assgnment where φ 1 b (3) = 3, should player φ b(1) am for slot 2, hs utlty would become: V λ 2 Q (v φb (1) b 3 ) λ 2 Q (V v) {because b 3 v} q(2 q) (V v) = V λ 2 V {because Q = q(2 q)} > γ2 1 V λ 2 q (V b φb (2)) {because γ 2 > (2 q)(v v) } = λ 1 q (v φb (1) b φb (2)) = u φb (1)(b) {because γ 2 = λ 2 /λ 1 } 18

CASE II. In any assgnment wth φ 1 b (3) = 2, should player φ b(3) am for slot 2, hs utlty would become: λ 2 Q (v φb (3) b 3 ) λ 2 Q (V v) {because b 3 v} = ( 1 v ) V λ2 Q V > γ 3 λ 2 Q V {because γ 3 < 1 v V } = λ 3 Q v φb (3) = u φb (3)(b) {because γ 3 = λ 3 /λ 2 } CASE III. In any assgnment wth φ 1 b (3) = 1, should player φ b(2) am for slot 1, hs utlty would become: λ 1 Q (v φb (2) b 3 ) λ 1 Q (V v) {because b 3 v} = ( 1 v ) V λ1 Q V > γ 2 λ 1 Q (V b φb (3)) {because γ 2 < 1 v V } = λ 2 Q (v φb (2) b φb (3)) = u φb (2)(b) {because γ 2 = λ 2 /λ 1 } Havng examned all possble assgnments, we turn our attenton to the possblty of tes. We consder a determnstc te-breakng rule that ranks player 3 hgher than players 1 and 2 n case that he tes wth them. An arbtrary determnstc rule s chosen for resolvng a te among 1 and 2. Then t s easy to see that any possble te leads to one of the cases I., II., III. above. 19