Network Formation and the Structure of the Commercial World Wide Web

Transcription

1 Network Formaton and the Structure of the Commercal World Wde Web Zsolt Katona and Mklos Sarvary September 5, 2007 Zsolt Katona s a Ph.D. student and Mklos Sarvary s Professor of Marketng at INSEAD, Bd. de Constance, 77305, Fontanebleau, France. E-mal: zsolt.katona@nsead.edu, mklos.sarvary@nsead.edu. Tel: Fax:

2 Network Formaton and the Structure of the Commercal World Wde Web Abstract We model the commercal World Wde Web as a drected graph emergng as the equlbrum of a game n whch utlty maxmzng Web stes purchase (advertsng) n-lnks from each other, whle also settng the prce of these lnks. In equlbrum, hgher content stes tend to purchase more advertsng lnks (mrrorng the Dorfman-Stener rule) whle sellng less advertsng lnks themselves. As such, there seems to be specalzaton across stes n revenue models: hgh content stes tend to earn revenue from the sales of content whle low content ones from the sales of traffc (advertsng). In an extenson, we also allow stes to establsh (reference) out-lnks to each other and fnd that there s a general tendency to establsh reference lnk to stes wth hgher content. Fnally, we explore network formaton n the presence of search engnes and fnd that the hgher the proporton of people usng these, the more stes have an ncentve to specalze n certan content areas. Our results have nterestng practcal mplcatons for search-engne optmzaton, the prcng of onlne advertsng as well as the choce of Internet busness models. They also shed lght on why Google can use the Web s lnk structure to rank stes by content. Keywords: Internet advertsng, game theory, network formaton.

3 1 Introducton The Internet and ts most broadly known applcaton, the World Wde Web (WWW) s ganng tremendous mportance n our socety. It represents a new medum for dong busness that transcends natonal borders and attracts an ever larger share of socal and economc transactons. A key feature of the WWW s that, as a decentralzed network, t evolves on ts own based on ts members ncentves and actvtes. The goal of ths paper s to develop a model that helps understand what structure emerges from ths decentralzed network formaton process. The WWW ncludes an extremely broad communty of Web stes wth a vast array of motvatons and objectves. We cannot pretend to be able to capture all relevant behavors on such a dverse network. Rather, we restrct our attenton to the commercal WWW, by whch we mean the collecton of nterlnked stes whose objectve s to proft from economc exchange wth the publc and/or each other. In the followng, by WWW, we wll always refer to ths sub-network. As such, our goal s to explan the network formaton process and the resultng network structure of the commercal WWW. Understandng ths network structure s mportant for all frms partcpatng n e-commerce. The network structure has a crucal role n determnng the flow of potental consumers to each ste, whch s key for demand generaton. A prmary nterest of search engnes, for nstance, s to understand how stes contents are related to ther connectedness on the Web. In turn, Web stes need to be strategc about connectng themselves n the Web to ensure that search engnes correctly reflect or even boost ther rank under a gven search word. 1 Indeed, search-engne optmzaton has grown nto a $1.25 bllon busness wth a growth rate reachng 125% n In response to Google s regular updates of ts search algorthm, dfferent stes shuffle up and down wldly n ts search rankngs. Ths phenomenon, whch happens two or three tmes a year s called Google Dance by search professonals who gve names to these events as they do for hurrcanes (see Dancng wth Google s spders, The Economst, March 9, 2006). 1

4 Smlarly, the prmary way through whch stes can drve traffc to themselves s the purchase of advertsng lnks. 2 At the same tme, each ste also has the opton to sell the traffc reachng t by sellng such advertsng lnks to other stes. In a network where each ste s a potental advertser and a potental seller of advertsng, what determnes the tradeoff between sellng content or advertsng? In partcular, how does ths tradeoff depend on the ste s popularty or attractveness to the browsng publc? A closely related queston s how should stes prce ther advertsng lnks as a functon of ther content. Fnally, even on the commercal WWW, many of the lnks are socalled reference lnks, that stes establsh to other stes n order to boost ther own content or credblty (Mayzln and Yoganarasmhan 2006). Stes need to understand, how such lnks complement or nteract wth advertsng lnks to determne the ultmate network structure. Addressng these practcal problems requres the understandng of the forces that drve the evoluton of the network s structure and the resultng compettve dynamcs. Specfcally, we propose a network model n whch the nodes represent ratonal economc agents (stes) who make smultaneous and delberate decsons on the advertsng n-lnks they purchase from each other. Agents are heterogeneous wth respect to ther endowed content, whch may be thought of as ther nherent value n the eyes of the publc/market. Consumers are assumed to surf on the web of nodes accordng to a random process, whch s nevertheless closely lnked to the network structure. Stes generate revenue from two sources: () by sellng ther content to consumers and () by sellng lnks to other stes. We start by assumng that the prce per traffc of each lnk s an ncreasng functon of the orgnatng ste s content. Next, we show that ths s ndeed the case n an equlbrum where stes frst set ther prces for advertsng lnks and then purchase lnks at these prces n a second stage. We also extend the model to the case where beyond buyng and sellng advertsng lnks, stes can also establsh reference out- 2 In 2006, Internet advertsng has reached $10 bllon wth a yearly growth rate of over 25% (see Marketng Budgets Are Up 46% for Q2, July 5, 2006). 2

5 lnks to each other at a small cost. Fnally, we explore the stuaton when a substantal part of the publc uses search engnes. In ths context, we ask what happens when nodes represent multple content areas. We fnd that n equlbrum, hgher content stes tend to buy more advertsng lnks, mrrorng the Dorfman-Stener rule well-known for tradtonal meda but, so far, not explored for a network medum. Smlarly, reference lnks tend to pont from low content stes to hgh content ones. As such, n equlbrum, the number of all n-lnks s closely correlated wth the ste s content. Ths explans why search engnes have so much success usng algorthms based prmarly on n-lnks (e.g. Google s Page Rank) for orderng pages n terms of content n the context of a search word. The model also has a number of practcal mplcatons for the prcng of Internet advertsng. We fnd for nstance, that stes wth hgher content should set a hgher prceper-clck for ther advertsng lnks. Ths, combned wth our result on the purchase of advertsng lnks ndcates that there s a tendency for specalzaton of commercal stes busness models. Hgher content stes emphasze product sales drvng traffc to the ste, whle lower content ones emphasze the sales of traffc by manly sellng advertsng lnks. A tendency for specalzaton also exsts n content areas. Specfcally, f we allow stes to cover multple content areas, we can show that, the more consumers use search engnes, the more stes have an ncentve to specalze n terms of content areas. Fnally, we can show that the above equlbrum patterns are generally consstent wth the emprcal realty of the commercal WWW. In partcular, we fnd that n-lnks follow a smlar degree dstrbuton as out-lnks as t s emprcally observed on the WWW, but not predcted by exstng models of network formaton. The paper s organzed as follows. The next secton revews the relevant lterature. Secton 3 presents the basc model, whch consders advertsng lnks and exogenous prces. Secton 4 extends ths model to a two-stage game where stes prce advertsng lnks n the frst stage and then, purchase n-lnks from each other. Secton 5 explores two further extensons: () the 3

6 ntroducton of reference out-lnks and () the exstence of search engnes n a context where content s mult-dmensonal. The paper ends wth a general dscusson and concludng remarks. To mprove readablty, most proofs have been delegated to the Appendx. 2 Relevant Lterature Whle the marketng lterature related to the Internet has grown consderably n recent years, there s vrtually no research explorng the lnk-structure of ths new medum or the lkely forces that drve ts evoluton. Ths s not to say that socal scences and economcs n partcular have not examned the endogenous formaton of networks. In an nfluental paper, Bala and Goyal (2000), for nstance, develop a model of non-cooperatve network formaton where ndvduals ncur a cost of formng and mantanng lnks wth other agents n return for access to benefts avalable to these agents. Recent extensons of the model (Bramoulle et al. 2004) also consder the choce of behavor n an (ant-)coordnaton game wth network partners beyond the choce of these partners. 3 These models have several features, whch do not really apply to the WWW. Frst, they concentrate on the cost of lnk formaton, whch s shown to be crtcal for the outcome. More mportantly, the above papers consder that ndvduals n the network are dentcal. For example, n Bala and Goyal (2000), lnkng to a well-connected person costs the same as connectng to an dle one. Ths s clearly not the case on the WWW, where large dfferences exst between the stes contents and ther connectedness. Also, on the WWW the cost of establshng a lnk largely depends on where ths lnk orgnates from. Fnally, the equlbrum networks emergng from the above models clearly do not comply wth the structure of the WWW. Bala and Goyal (2000), for nstance, fnd two possble equlb- 3 See also Jackson and Wolnsky (1996) for an early paper concerned wth the relatonshp between socal network stablty and effcency and Jackson (2003) for a recent summary of ths lterature. 4

7 rum network archtectures, the wheel and the star or ther respectve generalzatons. Our work also relates to the vast lterature on advertsng (see Bagwell (2005) for a good recent revew). 4 Of partcular nterest for us are studes dealng wth advertsng frms choces of advertsng quanttes and the prcng of advertsng by meda frms. Advertsng quanttes have been known to be determned by the advertsers product margns (Dorfman and Stener 1954) and, of course, by the effectveness of advertsng. Advertsng expendtures have also been shown to be affected by product qualty n a varety of context. Nelson (1974) and Schmalensee (1978) develop a theory of advertsng as a sgnal of qualty. Vllas-Boas (2004) studes advertsng effort n the context of dscrmnaton between hgh and low qualty products and Agrawal (1996) computes equlbrum advertsng levels n the presence of dfferental brand loyalty. Our model does not map nto these stuatons but our results lnkng advertsng quanttes to stes content relate to the varety of outcomes dentfed n these papers. On the supply sde, recent papers n marketng (see Dukes and Gal-Or 2003) have shown that advertser- and meda-competton also have a sgnfcant effect on advertsng quanttes. Advertsng prces have also been shown to be nfluenced by the above market features but recently, two addtonal factors have been revealed to be of further nterest: () the dsutlty of advertsng (Masson et al. 1990) and () the compettve prcng of meda content (Godes et al. 2006). Our paper bulds on ths lterature but s markedly dfferent from t n many respects. Frst, our model studes advertsng va lnks of a network,.e. advertsng effectveness s endogenous as t depends on the network s structure. Also, advertsng s used to ncrease traffc, not to nform, nor to sgnal qualty or affect brand loyalty. More mportantly, n our model, advertsers and the meda are not separate enttes. Each ste s 4 See Zeff and Aronson (1999) for an early summary of advertsng on the Internet and Hoffman and Novak (2000) for a qualtatve descrpton of onlne advertsng prcng models. See also Iyer and Padmanabhan (2006) on Internet referral servces. 5

8 a buyer as well as a seller of advertsng. A central queston s: whch one of these actvtes domnates and how does ths decson depend on the ste s content. Fnally, our work s also related to recent papers modelng consumers browsng process on the WWW. Our demand structure s based on the classc model by Brn and Page (1998) to provde a consstent descrpton of how consumers flow on a complex network of stes. We use some of the recent mathematcal results related to ths framework, n partcular Langvlle and Meyer (2004). We extend our model usng the concept of a reference-lnk, as n Mayzln and Yoganarasmhan (2006), to desgnate out-lnks that stes establsh to other stes n order to mprove ther own perceved value by consumers. Wth these elements, we develop a model that s more consstent wth the realty of the WWW than those of the exstng network formaton lterature. Ths model s presented next. 3 The Model We descrbe Web stes and the lnks between them as a drected graph, G. The nodes of the graph correspond to the stes and the drected edges to the lnks between the stes. Let j denote f there s a lnk from node to node j and j f there s no lnk between them. The number of lnks gong out from a ste s the out-degree of the ste, denoted by, and the n-degree s the number of ts ncomng lnks, denoted by d n. It s mportant to note that we consder as the unt of analyss a sngle Web ste, whch may possbly nclude multple pages. Techncally, on the WWW, the nodes correspond to the Web pages. However, most of the tme, a Web ste offerng a sngle product conssts of several pages havng almost all lnks establshed between them. The ncomng lnks of the ste usually go to one of the man pages and the outgong lnks can go from any page. We argue that n a model of network formaton, these pages should be consdered 6

9 as one sngle node representng the Web ste. All the lnks gong out and comng nto a ste s sub-pages should be assgned to ths one node. 5 Beyond structural reasons, consderng stes as the unt of analyss also makes sense because they represent a sngle decson maker. In what follows, we wll descrbe consumers browsng behavor on such a graph, followed by the descrpton of the network formaton game played by the stes. In dong so, we need to stay at a relatvely hgh level of abstracton. In partcular, we wll consder a homogeneous group of consumers and a reduced form proft functon for stes. 3.1 Consumer browsng process The prmary task n modelng the WWW s to descrbe the process through whch users browse the Web,.e. how they move from one ste to the other. We wll consder these users as potental consumers, who may buy the content (product) sold at a partcular ste. We normalze ther total number to 1. Furthermore, we wll neglect consumer heterogenety and smply assume that a consumer reachng a ste may consume the content of that ste or purchase t wth probablty ρ, that we can assume to be 1, wthout loss of generalty. Our goal s to establsh the number of vstors at a ste (n a gven unt of tme). To do ths consstently s not a trval task because the weght (ncomng traffc) of ncomng lnks depends on how much traffc reaches ther orgnatng stes,.e. how many n-lnks the ncomng lnks themselves have. Obvously, two ncomng lnks have very dfferent effect on a ste s traffc f they orgnate from dfferent locatons. In other words, we need to descrbe the flow of consumers consstently across all nodes of the network. We wll use the smple but very powerful soluton proposed to ths prob- 5 Ths perspectve s shared by search professonals. When Google calculates the rank of a page n ts search functon for nstance, t calculates t for the whole ste and not for sngle pages wthn a ste. A possble way to do ths s to consder all the pages that are n the sub-drectores under the same doman name of a ste. For example any page wth an address s consdered as part of the Amazon ste. 7

10 lem by Brn and Page (1998), whch became one of the basc prncples for Page Rank, the algorthm that Google s search engne uses to order Web pages. Assume n stes and magne that the total mass of consumers (1 unt) s ntally dstrbuted equally between these n stes. A consumer follows a random browsng behavor n every step. Startng from ste, wth probablty δ, s/he randomly follows a lnk gong out from that ste or stays there, choosng each of these + 1 optons wth equal probablty. 6 Wth probablty 1 δ, s/he jumps to a random ste on the Web, agan choosng each ste wth equal probablty. The number of steps whle the user follows the lnks wthout jumpng then follows a geometrc dstrbuton, wth expectaton 1 1 δ. δ s called the dampng factor and n practce t s often set to δ = 0.85, whch corresponds to an expected surfng dstance of around 6.67, that s, almost seven lnks. It can be shown that the teraton of the above process results n a lmt dstrbuton of consumers between Web stes. Ths lmt dstrbuton s called Page Rank (PR). 7 It can be thought of as the number of vstors at a Web ste per unt tme. By defnton, PR has to satsfy the followng equaton: ( ) + δ r, (1) r = 1 δ n r r r k k + 1 where r s the Page Rank of ste (.e. the proporton of vstors reachng t), 1, 2,..., k are the stes lnkng to ste and j denotes the number of lnks gong out from ste j, that s the j-th ste lnkng to ste (wthout countng the loops). Descrbng the process over tme for all stes, let r (t) denote the row vector resultng from the teraton after step t. Wth ths notaton r (0) denotes the ntal vector of the teraton whch, we set wthout loss of generalty to r (0) = ( 1, 1,..., 1 ),.e. we dstrbute browsers unformly across all nodes. n n n 6 The event when a consumer stays at the webste can be formally represented by drawng a loop around the node. 7 Although Page Rank usually refers to the score that Web stes receve from Google, we use Page Rank to descrbe the scores that are calculated of ths smple verson of the algorthm. 8

11 The teraton s defned through the M transton probablty matrx, whose cells are: [M] j = { 1, f ( j), +1 0 otherwse. Notce, that the -th row of the matrx represents node and the number n cell j represents the probablty of movng to node j from node. Usng M, the teraton descrbed above reads: r (t+1) = δ r (t) M + (1 δ)r (0). (2) If the seres r (t) s convergent as t and t converges to r, then r provdes the PR values of the nodes n the network. These can be thought of as the steady number of vstors at a Web ste per unt tme. It can be shown usng Markov-chan theory that the teraton s ndeed convergent f the graph satsfes some propertes (see Langvlle and Meyer (2004) for detals). We only use the followng lemma. Lemma 1 (Langvlle and Meyer 2004) If r (t) s a probablty dstrbuton for every t, then the seres s convergent as t. Obvously, n the ntal step, r (0) s a probablty dstrbuton, but r (t+1) does not satsfy ths unless each row of the matrx M contans at least one non-zero element, that s, every node n the graph has at least one out-lnk. The loops added to the nodes ensure that ths holds. Usng the matrx form of defnton (1), f teraton (2) s convergent and t converges to r, then t has to satsfy: r = δ rm + (1 δ)r (0). (3) Notce that f r s a probablty dstrbuton, then for any matrx [U] j = 1 n, ru = ( 1, 1,..., 1 ). Hence (3) can be wrtten as n n n r = δ rm + (1 δ)ru = r(δm + (1 δ)u). (4) 9

12 Ths formula helps nterpret the meanng of Page Rank by descrbng t as the weghted average of two matrces (M and U) each representng a dfferent random process. M contans the transton probabltes across lnked stes,.e. t moves browsers along the lnks of the network. Thus, t encapsulates the structure of the Web. In contrast U represents a process that scatters browsers randomly around to any of the stes. The weghts gven to these two processes are defned by δ, the dampng factor. 8 Thus, Page Rank and the underlyng process s a consstent descrpton of how traffc s dstrbuted across stes for any gven lnk structure of the network. 3.2 Network formaton Assume that there are n nodes (stes) wth gven constants c 1,..., c n, representng ther contents. These content parameters can be thought of as some measure of the Web stes value for the publc n a partcular content doman. For nstance, the ste may sell a product and c may represent consumers wllngness to pay for ths product. Then, the varaton n c may be thought of as heterogenety across stes n terms of product qualty. In ths sprt, we assume that the ste s net revenue from a consumer s proportonal to ths parameter: the hgher the publc values the ste, the hgher the ncome from a consumer vstng t. The ste s net revenue wll also be proportonal to the total number of consumers beng at the ste, as measured by r,.e. ste s total ncome from ts consumers s: r c. The cost of each ste has a fxed and a varable component. The fxed component can be set to 0 wthout loss of generalty. We assume that the varable component (e.g. a shppng cost) that s proportonal to the number of vstors s dentcal across stes. Let C denote ths per-vstor cost. Then, the 8 It s also nterestng to note that r s the egenvector of the matrx δm + (1 δ)u wth ts prncpal egenvalue, 1. 10

13 total cost of a ste s: r C. We assume that there s a market for lnks between stes. Every node, offers lnks for a fxed prce-per-clck, q, whch vares across nodes as wll be clarfed below. Ths s consstent wth general meda (or Internet) practce where ad rates are typcally quoted as rates per clck-through. The number of clcks on a partcular lnk can be calculated from the consumer flow model. If ste has traffc r and out-lnks, then the number of vstors clckng on a partcular out-lnk wll be δr /( + 1). Then, the total prce of an advertsng lnk from ste wll be p = δr q /( + 1). If another node purchases a lnk then ths lnk wll be created and pontng from the seller to the buyer. Gven prces, nodes makes smultaneous decsons about ther ncomng lnks, that s, whch other nodes they buys lnks from. Each node s allowed to buy one lnk from every other node. Essentally, ths market can be thought of as the advertsng market. If a node buys a lnk, t pays for an advertsement to be placed on the seller s page. In our baselne model, the per-clck prces for lnks are exogenous but we wll relax ths assumpton n Secton 4.2. Specfcally, n ths secton we wll assume that q = q(c ) s an ncreasng functon of content c and that prces are not too hgh (see (16) n the Appendx). In Secton 4.2, we show that n a two-stage game where prces are set frst, followed by the purchase of lnks, equlbrum prces are ndeed set ths way. Nevertheless, even ths exogenous prcng structure as reflected by the choce of q(c) s qute ntutve. Prceper-clck ncreasng n content allows us to capture the basc tradeoff between keepng a consumer or handng t over to another ste. The hgher the gan from a consumer (.e. the hgher c), the hgher the ste wants to charge for potentally lettng hm/her to surf to another ste. In other words, ths prce functon captures the tradeoff between stes two revenue streams. 9 9 Notce, that n our model, stes control ther sold advertsng lnks only through ther 11

14 Wth these elements, a ste s proft, for a gven network structure conssts of ts ncome from ts consumers plus the advertsng ncome (from sold lnks) mnus the advertsng costs (of bought lnks). Formally: u = r (c C) + p j p j. (5) 3.3 Equlbrum analyss Our objectve s to determne the Nash-equlbra of a game where players objectve functon s gven by (5) and ther strateges consst of buyng lnks from one another n a smultaneous decson. These equlbra represent a network or a graph (a set of lnks between the nodes) and our man nterest s n understandng the structure of ths graph. The followng proposton descrbes the general structure of these equlbra. Proposton 1 At least one Nash-equlbrum always exsts and all the equlbra have the followng propertes. () The out-degree s a weakly decreasng functon of content n the followng sense. If, for a gven par of nodes c k < c l, then k l. () If all the content parameters are dfferent, then n-degree and Page Rank are ncreasng functons of content. Proof (Sketch): Here we gve the man logc of the proof whle the detaled proof s provded n the Appendx. In the frst step, we show that n equlbrum all the nodes buy lnks from the nodes wth the lowest q s. Ths does not mean that they wll buy from the nodes chargng the lowest prce for lnks, but rather from those, whch sell ther traffc at the lowest per-clck prcng. Ths may not entrely capture the strategc nteracton between stes. For example, a ste may not allow advertsng by a strong rval even at a hgh prce. We wll dscuss ths ssue n detal at the end of the paper and would lke to thank the revew team for pontng t out. 12

15 prce. Based on the ncreasng prce structure, these must be the stes wth lowest content parameters, hence out-degree s a decreasng functon of the content parameter. Then, we show that nodes wth hgher content can buy more lnks, hence n-degree s an ncreasng functon of the content. Due to the specal structure of the network ths yelds that the Page Rank s also an ncreasng functon of content. Fgure 1 shows a possble equlbrum network structure. Once the nodes are arranged accordng to ther content (top left graph), the network structure reveals the smple tendency whereby most lnks orgnate from low content pages (small dots) and are drected towards hgh ones (large dots). The lower part of the fgure shows how n- an-lnks depend on content, where nodes are arranged n ncreasng order of content. Of course, f we suppose that all the content parameters are dfferent, then () s equvalent to sayng that the out-degree s a decreasng functon of the content parameter. If there are dentcal content values, the nodes can stll be ordered (as s done on the fgure) such that both the contents are ncreasng and the out-degrees are decreasng. Ths general equlbrum structure of the model, that advertsng lnks tend to go from lower content stes to hgher content ones, s qute nterestng. Essentally, t means that hgh content stes are the most mportant buyers of advertsng. Ths result s smlar to the Dorfman-Stener advertsng rule well-known n tradtonal meda. 10 It s partcularly nterestng that ths result contnues to hold even n a network context where sellers of advertsng are competng for traffc to sell ther own content. The result also seems to have face valdty as the bggest advertsng stes tend to be large well-known brands. Surveyng the last decade n onlne advertsng, DoubleClck, for example, documents that by 2005, Fortune 500 companes share of all onlne advertsng reached 30% and has steadly ncreased over tme. Smlar, trends 10 We would lke to thank the Area Edtor for pontng out ths smlarty. 13

16 out lnks (sold advertsng) n lnks (bought advertsng) low content hgh content low content hgh content Fgure 1: The top two fgures depct the same network, a possble equlbrum network, where larger nodes denote hgher content. The bottom graphs represent the number of out- and n-lnks for each node, where nodes are arranged n ncreasng order of content. 14

17 emerge for Europe as well. 11 The result s also nterestng, because t suggests that stes have a tendency to specalze n ther busness model. Certan stes, the ones wth low content specalze n sellng lnks (.e. traffc), whle stes wth hgh content tend to buy lnks (advertse) n order to beneft from content (product) sales. However, there are also stes that do both, whch s specfc to the Web. To summarze, the network s formaton s characterzed by two features: () pages tend to buy lnks from other stes wth lower contents and () the hgher the content of a ste the more lnks t wll buy from other stes. Ths results n a network where the number of n-lnks correlates wth the value of the correspondng ste. 4 Endogenous prces and nfntely many stes After analyzng network formaton wth per-clck prces as parameters, we now study a game where prces and lnks are both decson varables. partcular, a key drver of our results so far was the assumpton that q s ncreasng n content. Our goal s to show that ths s true even wth endogenous prces and that the network formaton results hold. Specfcally, we analyze a two-stage game where n the frst stage, stes set per-clck prces for advertsng lnks and n the second stage, they establsh lnks between each other, gven prces. The second stage game, as t was descrbed n Secton 3.2, would be too complex to solve for any fxed set of q parameters. However, the sze of the Web suggests that we should consder the case when the number of players s large enough so that a sngle ste s decson does not have a sgnfcant effect on the other stes. To capture ths dea, we suppose that there are nfntely many stes or a contnuum of stes. We descrbe such a model next. 11 See, The Decade n Onlne Advertsng and The Onlne Advertsng Landscape n Europe, DoubleClck, Aprl/September 2005 as well as Zeff and Aronson (1999) p.7. In 15

18 4.1 Network formaton In the nfnte verson of the orgnal network formaton game, suppose that the set of players s the nterval I = [0, 1] and each player corresponds to a node of the nfnte drected graph. Defnton 1 A drected graph on the set I s defned as a subset G I I, where an element (x, y) G corresponds to a drected lnk from x I to y I. The defnton of the degrees of the graph requres measure theory. We wll call the subsets of I measurable f they are measurable wth respect to the Lebesgue-measure on the nterval I, denoted Λ. Defnton 2 The out-degree of x I n the graph G, s the measure of those nodes to whch lnks from x exst, that s (x) = Λ{y I (x, y) G} f the set s measurable, otherwse the out-degree does not exst. Smlarly, the n-degree of y I s defned as d n (y) = Λ{x I (x, y) G} f the set s measurable. We wll restrct ourselves to graphs where all the degrees exst, that s, the correspondng sets are measurable. We wll show that any equlbrum graph has to be such. Drectly generalzng the game, we assume that the measurable functon c() provdes the content of ste I and the measurable functon q() represents the per-clck prces. We can assume wthout loss of generalty that c() s ncreasng,.e. stes are ordered by content on I. The Page Rank functon s also drectly generalzable. However, n the nfnte case, we have to deal wth the problem of zero out-degrees. If the set of nodes that buy lnks from node, s a zero measure set, then () = 0. In the fnte case, the soluton s to establsh a loop around node, but that would also be a zero-measure set n the nfnte case. Hence, we ntroduce the varable s > 0, accountng for the vstors who stay at ste. Then, the proporton of vstors who stay at the ste s 16 s. Therefore, the equaton s+ ()

19 defnng Page Rank wll be s r() = (1 δ) + δ () + s r() + δ x r(x) dx. (6) (x) + s It can be nterpreted as a densty functon descrbng the margnal probablty of vstors beng at dfferent stes. A (1 δ) proporton of vstors s jumpng to random pages and the rest of them are followng the lnks. To make sure that players are not ndfferent between dfferent choces, we assume that Λ(q 1 (x)) = 0 for every x, that s, not many stes have the exact same prce. The total prce for a lnk at ste s p() = δr()q()/( () + s). Then, ste has the followng utlty functon. u = r()(c() C) p() () j p(j)dj. (7) For ths nfnte game, the man results that were vald for the dscrete case stll hold. If q(.) s an ncreasng functon of content and satsfes (16), there always exsts an equlbrum and n ths equlbrum, n-degree s ncreasng an-degree s decreasng n content (and n ). Proposton 2 formally states ths result. Proposton 2 If q() s ncreasng satsfyng (16), and the functons c and q are contnuous, at least one pure-strategy Nash-equlbrum exsts and n any equlbrum d n () s ncreasng and () s decreasng. Proof: See the Appendx. Snce the number of players s nfnte, a sngle player does not have a sgnfcant mpact on the game. Let us capture ths by the followng defnton. Defnton 3 Two measurable functons q and q : [0, 1] R are equal almost everywhere (q = q a.e.) f Λ{x q(x) q (x)} = 0, that s, f they only dffer n a small set. 17

20 Lemma 2 If q = q a.e., then the set of equlbra of the games correspondng to the two functons are equal a.e., that s, for any equlbrum functon d n () for q, there exsts an equlbrum for q wth a d n () = d n () a.e. Proof: Let X denote the set { q() q ()}. The payoffs and the optmal decsons do not change for the stes that are not n X. For those, who are n X, the optmal decsons may be dfferent, but these players are n a null set. Now that we have characterzed the equlbra n the second stage (network formaton) game, we wll show that q() s ncreasng n any equlbrum of the two-stage game. 4.2 Prce settng In the frst stage, every ste selects ts q() smultaneously, only knowng the content functon. In the second stage, stes establsh lnks. Snce the two-stage game may have several sub-game perfect Nash-equlbra, even unreasonable ones, we wll rule out some of them based on Lemma 2. Defnton 4 A sub-game perfect equlbrum (q, E(q)) of the two-stage game s a refned sub-game perfect Nash-equlbrum, f () E(q) s a pure-strategy Nash-equlbrum of the second stage and () If q = q a.e., then E(p) = E(p ) a.e. Ths defnton makes sure, that to any refned SPNE corresponds an SPNE, and any SPNE wth the property that an nfntesmal perturbaton n prces (q q ) leads to a qualtatvely dfferent network n the second stage s not a refned SPNE. Therefore, stes have an expectaton about the second stage s network structure n the frst stage, and ths expectaton does not change f only a few stes change ther prces. Ths approach gnores certan drect 18

21 strategc effects of the prcng decson. Specfcally, we assume that stes react to the dstrbuton of prces across all other stes. Wth nfntely many stes, ths dstrbuton does not change f a sngle ste alone changes ts prce. Ths assumpton s realstc n the context of the WWW where there are over 10 bllon pages and no ste domnates the traffc on the entre network. Usng ths equlbrum concept, our man result s the followng. Proposton 3 For any refned SPNE of the two-stage game, the frst stage s q(.) functon has to be ncreasng. Proof: See the Appendx. The sgnfcance of Proposton 3 s that t supports our assumpton that n the network formaton stage of the game, the per-clck prces of advertsng lnks ncrease wth respect to the stes content. Among other fndngs, ths renforces our prevous result that stes tend to be specalzed n terms of ther revenue models. Stes wth low content tend to sell traffc to hgher content stes by sellng advertsng lnks for relatvely low prces. Hgh-content stes on the other hand beneft more from the sales of ther content to the publc. They prce ther advertsng lnks hgh and, as a result, sell few advertsng lnks. 12 The ntuton behnd the result s that stes wth a hgher content have a hgher potental of makng profts on ther vstors. Hence they set hgher prces to be able to sell fewer lnks. Ths way a hgher proporton of ther vstors become ther customers, resultng n a hgher average margn per vstor. In the second stage these stes purchase more advertsng, snce they can more effectvely leverage the traffc they buy. 5 Extensons In what follows, we explore two extensons to the model. Frst, we allow stes to create reference lnks. These are out-lnks that stes may establsh to 12 Hot, well-targeted content stes have [..] been able to command very hgh prces. Zeff and Aronson (1999), Chapter 7, p

22 boost ther effectve content. Second, we explore the mpact of search engnes allowng stes to have multple content areas. 5.1 Reference lnks So far, we have focused on a specfc type of lnks: advertsng lnks. These lnks are establshed for a fee to drect consumers to the Web ste of the advertser. Here, we ntroduce another type of lnk that s commonly used n the non-commercal Web: reference lnks. 13 These lnks also have an mportant role n formng the structure of the commercal Web. Reference lnks are used to ncrease the referrng stes content wth the help of the referred pages (Mayzln and Yoganarasmhan 2006). The number of reference lnks gong out from (comng n) a ste s denoted by R (d n R ). Every node s allowed to establsh one reference lnk from tself to every other node at mantenance cost κ. Each ste s allowed to establsh an (outgong) reference lnk to every other ste. The advertsng lnks are stll ncluded n the model, as they were n the orgnal verson, that s, each ste s allowed to buy one (ncomng) advertsng lnk from every other ste. Let R j denote f there s a reference lnk from to j and A j f there s an advertsng lnk between them, whereas the number of ncomng (outgong) advertsng lnks s denoted by d n A ( A ). Thus, the strategy of player can be descrbed by two vectors, each consstng of 0 s and 1 s. The frst vector x R determnes to whch nodes player establshes reference lnks to (x R(j) = 1 f s/he forms a reference lnk to node j and 0 f not). The second vector x A descrbes whch nodes s/he buys advertsng lnks from (x A(j) = 1 f s/he buys a lnk from node j and 0 f not). In the case when decdes to refer to j and j decdes to buy an advertsng lnk from, we assume that both lnks are establshed and ths s the only case when two lnks pontng n the same drecton are allowed between two nodes. Also, n order to get around the problem that 13 We are ndebted to one of the revewers for suggestng ths extenson. 20

23 players mght be ndfferent between two or more possble choces of lnks, we wll assume that f a player s ndfferent s/he establshes as many lnks as possble. The ncentve to create reference lnks s to ncrease a ste s content by referrng to other stes. Therefore, we generalze the payoff functon by usng the accumulated or effectve content term, whch conssts of two elements: () the ste s resdent content, c, () the sum of the content of stes lnked to through reference lnks multpled by a scalng constant 0 β < 1. Therefore, the total payoff of node s defned as follows: ( u = r c + β ) c j C R j κ R + p A p j. (8) j A Introducng the reference lnks makes the problem much more complex, snce a ste cannot control ts traffc by buyng the approprate number of advertsng lnks, the traffc s also affected by the ncomng reference lnks. In order to solve the game we use the followng smplfcaton. Instead of usng the stochastc model, to descrbe the flow of consumers, we use a traffc functon wth the followng propertes. Let r = f(d n R, d n A ) be the traffc or demand that reaches the ste. f s a functon of the ste s n-degrees and we assume that t s ncreasng and strctly concave n both advertsng lnks (d n A ) and reference lnks (d n R ). Ths assumpton s strongly supported by practce and s one of the basc prncples behnd search engne desgn. Descrbng Google s search engne, The Economst clams for example, that [t]he most powerful determnant of a Web page s mportance s the number of ncomng referral lnks, whch s regarded as a gauge of a ste s popularty. 14 We also make the natural assumpton that f has ncreasng dfferences n d n R and d n A. That s, f(x+h 1, y+h 2 ) f(x, y+h 2 ) f(x+h 1, y) f(x, y) for any x, y 0 and h 1, h 2 0,.e. the two knds of n-degrees are weakly 14 Ibd. See also How Google works, The Economsts Technology Quarterly, September 18,

24 complements. Then, the utlty functon becomes: ( u = f(d n A, d n R ) c + β ) c j C κ R + p A p j. (9) R j Wth ths generalzaton we can show the followng. j A Proposton 4 If p = p(c ) s ncreasng, then the game has an equlbrum, and n any equlbrum, f c > c j then d n R and R R j. Proof: See the Appendx. d n R j, A A j, d n A d n A j Keepng the assumpton that prces are ncreasng n content, we can show that the structure of the network formed by the advertsng lnks s qualtatvely the same as wthout reference lnks. The network formed by the reference lnks has a smlar structure but wth the opposte order of outdegrees. For both networks, the n-degrees are ncreasng n content, whereas the out-degrees are decreasng n content for advertsng lnks and ncreasng for reference lnks. The ntuton for the dstrbuton of reference lnks s qute smple. Clearly, each ste wll try to establsh reference lnks to the hghest content stes, whch beneft more from these n-lnks as they have a hgher margn on the addtonal traffc generated by these n-lnks. Therefore, hgh content stes can afford to establsh more reference out-lnks ncreasng ther margn even more. The presence of advertsng lnks ntensfes ths effect snce outgong reference lnks and ncomng advertsng lnks are complements. The more reference lnks a ste establshes the more advertsng lnks t has an ncentve to buy. Thus, the ncreased traffc from these advertsng lnks results (ndrectly) n extra proft from outgong reference lnks. The general feature of the equlbrum network, that hgher content results n more reference n-lnks s very nterestng. It provdes, for nstance, an explanaton for why the famous search engne, Google had so much success 22

25 ntroducng the quantty Page Rank for search. Google s objectve s not only to fnd all the pages contanng the search expresson, but also to rank them accordng to ther content. Snce measurng content drectly s dffcult, t can use Page Rank as an ndrect measure because, accordng to our model, n equlbrum, hgh Page Rank should be correlated wth hgh content. 5.2 Search engnes and multple content areas Search engnes (SE) play an mportant role n the formaton of the network. If some consumers use SEs, then the number of vstors at a Web ste does not only depend on the structure of the network but also on how search engnes dsplay the ste n the result of a gven search. Today s SEs use a twofold method to determne whch pages and n what order to dsplay the result of a search. On the one hand, they measure content drectly, on the other hand, they measure content ndrectly through the structure of the network, usng methods such as Page Rank. To examne the effect of SEs we wll assume a sngle SE that flters the s hghest content stes for ts users, where s s a fxed nteger. We also assume that traffc s dstrbuted across these s stes proportonal to each ste s Page Rank. Note that we do not consder the SE as a strategc player. As wll become clear later, when consderng SEs, we need to generalze our model n another respect, lettng content have multple dmensons. Specfcally, we assume that content s a D-dmensonal vector c = (c 1, c 2,..., c D ). These dmensons can be seen as content areas (e.g. entertanment or e-commerce n varous domans, etc.). We assume that a partcular consumer vstng the ste s only nterested n one dmenson of the ste. 15 The proporton of consumers nterested n the dfferent dmensons s represented by the weght vector w. Ths vector can also be nterpreted as the probablty dstrbuton on content dmensons descrbng the nterest of 15 Ths assumpton can be relaxed. If a consumer s nterested n several dmensons we assgn a probablty dstrbuton to hs/her nterest. 23

26 a randomly selected consumer. Thus, the expected consumer-specfc content at ste s the scalar product w c, whch can also be called the (weghted) average content of a page. Then, n the generalzaton of our model (5), the ncome of a Web ste from sellng ts content changes from r c to r w c. Thus, stll wthout the presence of SEs, the total utlty of node s u = r (w c C) + p j p j, (10) where we assume that p = δq r /( + 1) and q = q(w c ) s an ncreasng functon of average content. It s easy to see that ths generalzed model results n the same equlbrum as the one descrbed n Proposton 1. The only dfference s that we need to replace content wth the weghted average content n the Proposton. Ths shows that wthout ntroducng the SEs n the model, mult-dmensonal content does not make much dfference. In partcular, f stes had the possblty to change the allocaton (dstrbuton) of ther total content across specfc content areas, they would not have an ncentve to do so, snce only (weghted) average content matters. 16 What happens f we ncorporate SEs n the model? Let us assume that only a b proporton of consumers s browsng accordng to the process descrbed n Secton 3.1. The remanng (1 b) consumers use a SE n every step of browsng, whch drects them to a Web ste n the followng way. As we mentoned before, a consumer s only nterested n one dmenson of content, hence s/he runs a search n that dmenson. Through the result of the search, the SE drects the consumer randomly to one of the top content stes n that dmenson. More precsely, the SE selects the pages wth the s hghest content parameters n every dmenson and drects consumers to one of these wth probablty proportonal to ther Page Rank. 17 Let S d denote the set of 16 Notce that the cost of content assocated wth a certan area s proportonal to the consumer nterest n that dmenson. 17 Ths s consstent wth practce. For example, there are very few consumers who go 24

27 the s hghest content pages n dmenson d and I d denote the ndcator of the event ( S d ), that s, whether the content of ste n dmenson d s among the top s contents. Then, the probablty that a consumer from a SE gets to a gven page n dmenson d s ether 0, f t s not one of the top content stes n the search dmenson, or r /R d, where R d = l S d r l s a normalzng constant n dmenson d. Thus, the ncome from consumers n dmenson d at ste s: br c d + (1 b)r c d I d R d = r c d (b + (1 b)i d /R d ). Usng notaton C = (C 1, C 2,..., C D ), where C d = c d I d /R d, the expected ncome from sellng content at page s: r (bw c + (1 b)w C ). It s mportant to see the dfference between c and C, the latter beng the content vector truncated by the search engne by elmnatng (settng to 0) the dmensons that do not make t n the top s ranks. The term (1 b)w C can then be nterpreted as the expected reward from the search engne for beng a top ste n one of the content dmensons.e. a sort of specalzaton reward. Let E denote the modfed average content bw c + (1 b)w C. Then, the total utlty of ste s u = r (E C) + p j p j, (11) where p = δq r /( + 1) and q = q(.) s an ncreasng functon of the modfed average content, E, as defned before. Clearly, wth a sngle content area, the exstence of a search engne does not matter qualtatvely. It smply makes the dvde between low and hgh content pages more pronounced. Assumng multple content areas, the equlbra can be descrbed by the followng proposton. Proposton 5 At least one pure strategy Nash-equlbrum always exsts and all the equlbra have the followng propertes. beyond the second page of Google s search results. 25

28 () The out-degree s a weakly decreasng functon of the modfed average content n the followng sense. If, for a gven par of nodes E k < E l, then k l. () If we suppose that all the modfed average contents are dfferent, then the n-degree and the Page Rank are ncreasng functons of the modfed average content. Proof: The proof follows from that of Proposton 1, replacng c wth E. The above propertes of the equlbrum graph show that the stes wth the hghest E wll have the hghest n-degree and Page Rank. Snce E k s the lnear combnaton of () the average content of ste k and, () the expected reward from the SE for offerng leadng content n partcular dmensons, the proposton mples that n the presence of a search engne the allocaton of content between dmenson really matters. Specfcally, there s an ncentve to specalze n a certan content area n order to be one of the top stes of a partcular dmenson and, n ths way maxmze the specalzaton reward. On the other hand, ths ncentve to specalze decreases as the average content of a ste s hgher, snce a hgh average content ste does not have to allocate all ts resources to one dmenson, t can afford to dversfy ts content. Thus, we would expect stes wth low total content to specalze, whle those wth hgh general content to dversfy. However, as more and more people use search engnes the advantage from hgh average content dsappears and ultmately all stes compete for hgher content n a specfc area. 6 Dscusson and concluson We proposed to model the commercal WWW based on the dea that proft maxmzng Web stes purchase (advertsng) n-lnks from each other to drect traffc to themselves n order to sell ther content. A key feature of the model s that stes are heterogeneous n terms of ther content. Homogeneous 26

29 consumers are assumed to browse the Web n a random process drected by the network s lnk structure. Frst, we supposed exogenous per-clck prces for n-lnks that ncrease n content. Later, we showed that wth endogenous prces ths pattern s confrmed n equlbrum. In two extensons, we ntroduced the presence of search engnes and the possblty for stes to establsh reference out-lnks to each other. In each case, we were nterested n the equlbrum network structure as well as stes dfferng ncentves as a functon of ther content. Overall, we found that n all equlbra, both advertsng and reference lnks pont to hgher content stes. Ths result strongly supports the broadly accepted search heurstc, whch heavly reles on the number of n-lnks to rank stes wth respect to content. Ths can explan, for nstance, why Google s Page Rank algorthm works so well n practce, by showng that n equlbrum, the number of n-lnks s postvely related to a ste s content. In contrast to n-lnks, the pattern of out-lnks s markedly dfferent for advertsng and reference lnks. Stes tend to purchase advertsng lnks from lower content stes,.e. the number of advertsng out-lnks s negatvely related to the content of a gven ste. In the case of reference lnks however, t s hgher content stes that tend to establsh more out-lnks. We also show that, n the presence of search engnes, ths structure becomes more pronounced. These results provde useful gudelnes for marketng managers on how to manage ther frms ste(s) n terms of ther connectedness n the Web. Frst, competton seems to provde strong ncentves for stes to specalze n terms of ther busness models. Low content stes beneft more from the sales of traffc (advertsng) even though they can only prce such traffc at modest rates. Hgh content stes on the other hand, beneft more from revenues earned from content sales to consumers. These stes should charge hgh prces for advertsng lnks and, as a result, sell few of these. Instead, they are better off attractng traffc by purchasng advertsng lnks. Because of ths ncreased traffc, hgh content stes also beneft more from reference lnks and should therefore, establsh more such lnks. Fnally, f we consder 27