Consumer Tracking and E cient Matching in Online Advertising Markets

Transcription

1 Cosumer Trackig ad E ciet Matchig i Olie Advertisig Markets Susa C. Athey, Emilio Calvao ad Joshua S. Gas December 3, 0 Oe of the advaces i our uderstadig of two-sided markets or platforms is the otio of a competitive bottleeck. Thisarisesithecotextof competig platforms whe a group o oe side of the market always multihomes; that is, they pay to access each platform. More strogly, they pay to access ad use ay oe platform idepedetly of what they are doig o other platforms. This has a sigificat impact o the ature of competitio betwee platforms. If oe side always multi-homes the it is, i some sese, captured by each platform. The platform ca act like a moopolist with respect to those customers. However, this does ot elimiate competitio as, by the very ature of two-sided markets, the level of activity o the other side of the market impacts o the quality of the product served to the other. I this situatio, platforms may compete for icreased use o oe side of the market i order to icrease the supply ad quality of the product they ca provide to captured customers. This model is pervasive i two-sided markets ivolvig advertisers. The caoical model of media ecoomics (Aderso ad Coate, 005) has, as its baselie, a model where each advertiser wats to commuicate to each broadcast viewer. Cosequetly, competig broadcasters ca charge advertisers a moopoly price for access to ay give viewer. The broadcasters

2 the compete over the umber of viewers o their chael. A importat implicatio of this is that if broadcasters were to merge, there would be o chage i prices or welfare i the advertisig-side of the market although there would be a reductio i competitio for viewers. This type of aalysis has played a sigificat role i ati-trust aalysis ivolvig advertisig markets ad mergers. For this reaso, it is importat to uderstad the behavior of advertisers with respect to sigle or multi-homig or somethig i betwee. The baselie view is that advertisers wat to sed messages to cosumers ad, ideed, place a value o impressig each ad every oe of them. Thus, if a platform happes to have attracted a cosumer, the a advertiser must deal with the platform to sed a message to that cosumer. Thus, multi-homig behavior is a direct implicatio of a desire to sed a message to all cosumers. Of course, this assumptio might be relaxed if, for istace, advertisers are ot iterested i such complete coverage. For istace, a advertiser may have alimitedquatityofproductstosellormayhaveaotherlimitotheir marketig budget. This will costrai their behavior i sedig messages to all cosumers. Aother factor is that the platforms may ot capture a cosumer etirely. As modelled by Ambrus, Calvao ad Reisiger (0), some cosumers may cosume both platforms weakeig platforms claim to capture those cosumers ad leadig to direct competitio for them. Competitio for captured cosumers would ot arise. Or alteratively, as modelled by Athey, Calvao ad Gas (0), some share of the cosumer market may allocate attetio across platforms while others may cocetrate it o platforms but it may ot be possible ex ate to idetify particular cosumers by their sigle ad multi-homig behavior. I this case, all cosumers are potetially cotestable ad this will impact o advertiser behavior i terms of sigle or multi-homig. I each case, a merger betwee platforms may reduce the competitio for advertisers. Here we preset a alterative driver of advertiser behavior that we be-

3 lieve is of relevace for olie advertisig markets. The markets we have i mid are ewly emergig ad etworks that o er display advertisig for may web-pages ad outlets. Ulike traditioal media (such as ewspapers), at a base case, we ca assume that cosumers locatio o a platform is ot kow ex ate; that is, there are o captured cosumers as ad etworks caot be assumed to drive a cosumer s choice of cotet. I this sese, from the perspective of a ad etwork, all cosumers are multi-homers. By abstractig away from the stadard, assumed behavior of cosumers, we are able to idetify a ew driver of advertiser behavior: the techology by which messages are commuicated to cosumers. Put simply, the ature of the commuicatig techology impacts o the e ciecy of choices advertisers face i allocatig messages across platforms. I the process, we demostrate that the type of techology available or adopted will impact o coclusios regardig advertiser multi-homig. We model commuicatio as a stochastic process i which a seder (for example a advertiser) trasmits a umber of (costly) messages (for example ads) to a set of receivers (the cosumers) through two di eret chaels (the outlets). The process determies who (ad hece how may) get at least oe message (ad hece are iformed) if the seder trasmits a certai umber of messages through give outlet. Such stochastic process is what we call a commuicatio techology. Di eret outlets supply oe such techology. It ca be e ciet, i which case each additioal message hits a uiformed receiver. Or it ca be dumb, i which case each additioal message hits a radom receiver, possibly a already iformed oe. Or it ca be aythig i betwee. Commuicatio is costly i the sese that messages are uit-priced. The paper proceeds as follows. I Sectio, we preset our model framework embedded withi a simple theory of commuicatio. I Sectio 3, we cosider our baselie case with idetical outlets both of which share the same techology ad the same set of receivers. This allows us to characterize the impact of trackig techology o advertiser multi-homer behavior. I Sec- 3

4 tio 4, we relax the latter assumptio ad cosider how trackig techology iteracts with the share of overlappig receivers outlets have. I Sectio 5, we cosider asymmetric outlets specifically, a case where oe outlet s cosumers are a subset of the other s. This allows us to cosider trade-o s betwee reach ad readership ad the impact of this o advertiser behavior. We the tur to cosider the welfare implicatios of advertiser behavior; particularly, o receivers. This is doe by recogizig that wasted messages are likely to cost receivers attetio with o other potetial beefits. We provide some speculative ad illustrative discussio o these welfare e ects. Afialsectiococludes. A simple theory of commuicatio Cosider a set (measure oe) of seders, a set (measure oe) of receivers ad two outlets. Outlets are idetical but for their set of receivers deoted respectively R ad R ad assumed to be of equal measure. The measure of R [R is. Let D s ad Di l deote the measure of R \R ad R i \(R \R ) respectively (s stads for switcher adl for loyal). With a abuse of otatio we deote with the operator that maps a set to its measure. Seders wish to iform receivers. The uit price of a message o either outlet is p i > 0. If p i is equal to p j the we use p to deote the commo prce. Let v deote the expected value of iformig a receiver. Type v s payo whe choosig to purchase (, )messagesisv times the expected fractio of the populatio iformed mius commuicatio costs p + p. Acommuicatio(ortrackig)techologyisafuctio i : R +! [0, ] that maps the umber of messages per receiver set through a give outlet i to the probability of iformig a give receiver at least oce. If there are D receivers i total the i (/D) isalsotheexpectedfractioofthereceivers reached by at least oe message out of the messages set through outlet i. Weassume i strictly icreasig o [0, ) (if i <, costatly equal to otherwise) ad cocave. Fially, we assume that i ( 0 ) i ( 00 ) apple

5 for all 0, That is, oe message ca hit at most oe receiver. This implies that i caot icrease at a rate greater tha oe ad that i (0) = 0. Oe way to thik about these assumptios is as reflectig a miimum level of itelligece or e ciecy of the commuicatio process. A extra message should have some chace of iformig. Cocavity captures itelligece i a subtler way. Ay strategy that ca help idetify uiformed receivers is exploited as soo as possible. These assumptios also rule out learig by doig. The above implicitly restricts attetio to commuicatio processes with homogeeous receivers i the sese that o receiver is ex-ate more likely to be iformed tha the others. Of course, that eed ot be true ex-iterim, sice each additioal message is a radom trial whose realizatio ca deped o the realizatio of the previous trials. For example, a uiformed receiver might be more likely to be reached by a give message as the umber of already iformed receivers icreases. I what follows we itroduce advertisers payo s ad characterize their choices as a fuctio of the techology employed ad the distributio of receivers accross outlets. Rather the deployig the more geeral model upfrot, we build our theory step by step. We first study the simplest degeerate case i which both outlets share the same set of receivers. We the move o to the more complex cases i which receivers partly overlap. The advertiser s (symmetric) dilemma. Idetical Outlets We cosider a situatio where o oe cosumer ca be idetified exclusively with a particular outlet. As metioed earlier, this might arise because each cosumer may cosume cotet o outlets utilizig ay ad etwork. Thus, we suppose that R = R with R i =ad i () = j () =: (). (figure ) 5

6 Chael Chael Fig. : Imperfect commuicatio with full overlap. A first observatio is that the fractio of the populatio iformed through (, ) messages is equal to oe mius the probability that a give receiver is ot iformed through either outlet. Uder idepedece, that is ( ( ))( ( )). Type v s choice is determied by: max v( ( ( ))( ( ))) p( + ). (), 0 Let (, )deoteasolutiototheaboveproblemad := + be the total umber of impressios purchased across all outlets. We say that a o-trivial ( > 0) solutio poit is characterized by maximum diversificatio if = ad maximum cocetratio if either or is equal to.cosiderthefamilyofoptimizatioproblemsparametrized by v ad let (v), (v) ad(v) := (v)+ (v) deotethemappigform v to the solutio. Propositio. For all values of v>0 there is a uique solutio 0. I additio (v) is mootoe icreasig i v. Propositio. a) If is log-cocave the there is maximum cocetratio at all solutio poits. b) If is log-liear the all elemets of {, 0: + = } are solutios. c) If is log-covex the there is maximum diversificatio. Propositio follows from the icreasig di ereces property of the objective fuctio. Higher types face a higher opportuity cost of ot iformig 6

7 receivers. This implies that higher types will trasmit more messages i total (if ay). To sketch the argumet used to prove Propositio it is useful to study how impressios o di eret outlets substitute for oe aother uder techology. Thatis,cosiderthefollowigproblem: max ( ( ( ))( ( ))) s.t. + =., 0 Acrucialobservatioisthatlog-cocavity,log-covexityadlog-liearity pi dow the sig of the rate of chage of the margial rate of substitutio of impressios o di eret outlets. I particular, if is log-cocave, the the margial rate of substitutio is icreasig (Figure, left). terms of the optimal allocatio problem preseted above log-cocavity calls for cocetratig all impressios o oe outlet. I A simple illustratio of a log-cocave commuicatio techology is perfect iteral trackig: () =if{, }. Uder log-liearity, impressios o di eret outlets are perfect substitutes whe it comes to impress the shared receivers. Figure, ceter, illustrates the level curves for () = e (the so-called Butters (977) o trackig techology). Fially, log-covexity implies strictly covex upper-cotour sets. (Figure, right). The solutio to this problem is characterized by a usual tagecy coditio betwee a level curve ad the budget set {, 0: + = }. To build ituitio, otice that there are two sources of missed ad wasted messages i this framework. First, additioal messages trasmitted through the same outlet ca reach a already iformed receiver ad hece get wasted. This is what we call withi outlet waste. Precisely, there is o withi outlet waste if () =if{, } for all 0. There is withi outlet waste otherwise. Secod, because of idepedece, two messages set o di eret outlets ca reach the very *same* receiver eve whe there is o withi outlet waste. This is what we call across outlet waste. Precisely,thereis I fact if some type v purchases a positive quatity, the all higher types will purchase a strictly higher quatity. Log(-x) is cocave. 7

8 Figure : Level curves of ( ( ))( ( ). From left to right is log-cocave, log-liear ad log-covex respectively. o across outlet waste wheever ( ( ))( ( )) = if{ +, }. There is across outlet waste otherwise. Notice that the assumptio that oe message hits at most oe receiver implies that there is always across outlet waste eve if () = for all apple. The curvature of the logarithm of captures the relatioship betwee withi outlet waste ad across outlet waste. This poit ca be illustrated formally through the equivalece betwee log liearity ad log additivity i this particular cotext. Lemma. ( ) is log-liear if ad oly if ( ) is log-additive, that is: log( ( + )) = log( ( )) + log( ( )) which is equivalet to: ( + )= ( ( ))( ( )) for all, 0 () Proof. By lemma 4 i appedix we have that ( ( ))( ( )) is costat over the set {, 0 : + = } for all 0. This coupled with the assumptio that (0) = 0 implies that for all we have 8

9 ( ())( (0)) = k =( ( ))( ( )) which is equivalet to (). Equatio () states the equivalece betwee the two sources of waste. Adaptig the same logic we ca characterize log-cocavity ad log-covexity. is log cocave (log-covex) if ad oly if ( + )isstrictlygreater (respectively strictly lower) tha ( ( ))( ( )) for all,. Usig this characterizatio we ca restate Propositio as follows: Propositio 3. a) If withi outlet waste is lower tha across outlet waste the there is maximum cocetratio at all solutio poits. b) If withi outlet waste is equal to across outlet waste the all feasible allocatios of are solutios. c) If withi outlet waste is greater tha across outlet waste the there is maximum diversificatio. This gives us a ew uderstadig of a driver i advertiser multi-homig behavior. For a give advertiser, v, admessageprice,p, theatureofthe commuicatio techology will drive whether that advertiser will wat to cocetrate their advertisig o all outlets, be idi eret or alteratively diversify maximally. If they choose to cocetrate their advertisig the, if impressios sold for di eret prices o each outlet, advertisers will choose to purchase impressios o the lowest price outlet. Thus, there will be direct competitio for advertisig busiess ad this price competitio will oly be mitigated by ay limitatios (as yet umodelled) o the capacity of a outlet to deliver ads. By cotrast, uder some commuicatio techologies where withi outlet waste is greater tha across outlet waste, advertisers will choose to diversify their impressios across outlets. I this situatio, it remais the case that a price di eretial ca cause advertisers to place more impressios o the low price outlet but the elasticity of the respose will be mitigated by the lower e ciecy of the margial impressio to that outlet. Whe might withi outlet waste be greater tha across outlet waste? This might arise if there is 9

10 s outlet outlet D l D s D l Figure 3: Imperfect commuicatio with partial overlap. o iteral trackig capability but a typical cosumer chooses to cosume oe outlet i the morig ad the other i the afteroo; that is, there is cotet diversificatio that aggregates to a attractive budle for cosumers compared with the cosumer choosig all the cotet of a particular outlet. This highlights the fact that part of the commuciatio techology could embed self-selectio by cosumers to tailored cotet.. Symmetric outlets with overlappig receivers We ow relax the assumptio R = R allowig outlets to have their ow captured set of receivers (see figure 3). Here receivers belog to two sets (or types): loyal of, ad switchers. We retai the assumptio that receivers are homogeeous i the sese that they oly di er with regard to the outlet they are coected to. With partial overlap, capturig this simple idea requires a extra igrediet. Switchers get messages from two di eret sources. I order to avoid switchers beig more likely to be iformed because of this, we assume that ay give switcher is twice less likely to receive a message set through a give outlet. 3 Specifically, suppose a total of messages are trasmitted through outlet. If loyals are twice as likely tha 3 For example, cosider the followig applicatio. All cosumers are edowed with two uits of time. Some cosumers choose to sped both uits o outlet. Some cosumers allocate both uits to outlet two. Some cosumers sped oe uit of time o each outlet. If messages are set at radom times the a loyal is twice more likely tha a switcher to be the recipiet of a give message. 0

11 switchers to receive a message the the expected fractio of that goes to them is l := D l /(D l + D s ). (Oe way to thik about this is as if each loyal receiver of outlet i were edowed with two mailboxes, each switcher receiver with oe mailbox ad messages o outlet i were radomly allocated to each mailbox i a uiform way; i the sese that o *mailbox* is ex-ate more or less likely to get a give message). Sice by defiitio the argumet of the fuctio is the average (per receiver) umber of messages set, the the correspodig fractio of D l iformed whe sedig a total of messages through chael i is equal to l = D l. (3) D l + D s The above simplifies to ( )wheoutletsaresymmetric: D l = D l =: D l. Iteratig the same reasoig we fid that switchers receive a total umber of ( l )= D s /(D l + D s ) from outlet oe. Uder symmetry, D l +D s =sothetotalumberis D s.thatis messages per switcher. Symmetrically, if messages are set through outlet the per switcher reaches these latter. Sice messages set through di eret outlets are statistically idepedet, the expected umber of switchers reached whe sedig (, )messagesis ( ( ))( ( )). The seder s dilemma (ormalizig by v the price p) is: max, 0 Dl ( ( )+ ( ))+D s ( ( ( ))( ( ))) ( + )p/v (4) Note that if D s =(sothatd l =0)wearebacktotheprevioussectio s case. It is useful to decompose this problem i two sub-problems. First, give a total umber of impressios to allocate across outlets, which of all possible allocatios maximizes reach? Secod, give the solutio to the above problem (deoted with star decoratios), how may impressios should a advertiser of type v purchase if the price is the same for both

12 = Figure 4: Optimal allocatio for D s =0 outlets? Cosider the first stage problem. max, 0 Dl ( ( )+ ( ))+D s ( ( ( ))( ( ))) s.t. + =. (5) Propositio 4. If is log-cocave there exists a threshold 0 < D s < such that the solutio to the optimal allocatio problem etails maximum cocetratio if D s > D s ad full diversificatio if D s < D s.if is logliear, for all D s < the solutio to the optimal allocatio problem etails full diversificatio. Cosider firstly the case log-cocave. If D s were equal to zero the decreasig margial returs o loyals would imply a eve allocatio of impressios across outlets: = = /. Ideed the level curves of ( )+ ( )arealwayscovextowardstheorigi(strictlycovexif 00 < 0) ad the solutio ca be easily visualized as the tagecy poit at = (figure 4). O the cotrary if D s is equal to oe the we kow that the optimal allocatio calls for maximum cocetratio from the previous sectio. The threshold D s is such that withi outlet waste equals accross outlet waste so that the level curves of the objective fuctio (5) have costat slope. I

13 this case, the optimal allocatio problem has ifiite solutios. Fially, give the optimal allocatio ( (), ()) the optimal total umber of messages purchased is foud by solvig: max 0 Dl ( ())+D l ( ())+D s ( ( ( ()))( ( ()))) p/v (6) How does the choice of the seder chage with D s? Propositio 5. Suppose : R ++! [0, ] is cotiuous, a.e. di eretiable, icreasig, cocave, bouded above by oe ad satisfies the followig boudary coditios: (0) = 0, lim! () =. If ( ) is log-liear the advertisers choices do ot deped o cosumer switchig. Thus, whe ( ) is log-cocave, advertisers multi-homig choices iteract with the share of cosumers who are switchers. If there are o switchers, advertisers will choose maximum diversificatio because there is techically o across outlet waste as this depeds o there beig switchers. As the umber of switchers rises, ay across outlet waste will rise ad weake advertiser icetives to multi-home. Iterestigly, this does ot occur whe ( ) islog-liearaswithiadacrossoutletwastearethesameregardless of the umber of switchers. If ( ) islog-cocavewecojecturethat there exists a threshold ṽ such that advertisers demad for impressios decreases with D s i (p, ṽ) adicreaseswithd s i (ṽ, ). We have o yet demostrated this cojecture for the geeral case here but it does emerge i the specific model i Athey, Calvao ad Gas (0). 3 The advertiser s asymmetric dilemma Suppose R R (Figure 5). That is, oe set of receivers fully cotais the other. I lie with the above otatio let D s = R,D l = R \ R ad 3

14 Outlet S Outlet R R Fig. 5: Asymmetric outlets with D l =0 D l = 0. Assume oce more p = p = p. The the advertiser s dilemma is (dividig everythig by v): max, D l D l +D s D l + D s D l D s D l + D s D s D s (7) The first term are the expected reveues o loyals of outlet. D l /(D l + D s ) is the expected fractio of the total umber of messages that lads o the loyals of. This quatity is the ormalized by the amout of loyals D l. That gives the average umber of messages that lad o loyals; i.e., o receivers i R \ R. It follows that the expected reach amog these is D l ( /(D l + D s )). The secod term represets expected reveues o shared receivers. It ca be easily derived resortig to the same logic. As will be clear later o, for our purpose it is coveiet to operate a chage of variables ad rewrite the above objective i terms of the average umber of impressios. So suppose the cotrol variables are ( + )p/v. ñ := D l + D s / ad ñ := D s /. The (7) ca be rewritte as follows: max D l (ñ )+D s ñ,ñ ñ ñ ñ (D l + D s /) + ñ (D s /) p/v. (8) Propositio 6. If is log-liear the ñ (p/v) =ñ (p/v) ad decreases with p/v. 4

15 Note that despite the asymmetric model there is a symmetric characterizatio of the equilibrium advertisig policy provided the focus is shifted to the average umber of messages set to each subgroup. Next we look at the case where accross outlet waste is ot equal to withi outlet waste. To gai isight o the problem we assume the followig fuctioal form: () =e /( + e ), (9) ad specific parameters p =,v =0adoeoutletassumedtobetwice as large as the other. That is D l =/ add s =/. The correpodig optimal choices are ñ.9whileñ 3.4. So the smaller outlet commads ahigherdemadiproportiototheumberofitscustomers. Thehigher margial returs o shared customers iduce a higher average demad for messages to be delivered this subgroup. More geerally, i the appedix, we show that if () =e /( + e ) ad the outlets are asymmetric (D l 6= D)theñ l 6=ñ wheever p = p. We expect this property to hold for ay log-cocave fuctio but we have ot proved it as yet. The result here mirrors the maget cotet discussio i Athey, Calvao ad Gas (0). Similar to this case, there oe outlet oly attracted switchers while the other outlet could have some exclusive cosumers. The cotext we had i mid were sites like Facebook that had a large reach but did ot ecessarily capture the exclusive attetio of cosumers. We demostrated that as the reach of the high reach outlet icreased, it attracted more demad from sigle-homig advertisers who chose to advertise exclusively o the high reach outlet so as to elimiate across outlet waste. They could do this because the share of missed cosumers would be relatively low. We saw this as idicative of the impact that imperfect trackig techologies ca have o the type of cotet supplied by outlets; that is, whether the cotet emphasized reach versus attetio per se. 5

16 4 Receivers welfare ad e ciet commuicatio Thusfar, we have focussed o advertiser icetives to sed messages ad, i particular, their allocatio across outlets. The primary cost to advertisers of wasted messages was the additioal paymets for those messages. However, there is also a additioal cost that is ot ecessarily icurred by the advertisers; the attetio of receivers. I this sectio we order the di eret advertisig policies accordig to how wasteful these policies are from the poit of view of the receivers. We sidestep the issue of comparig welfare chages accross seders ad receivers by characterizig those policies that miimize the burde o receivers while keepig the seders welfare fixed at some predetermied level. Total waste (or duplicated impressios) equals the di erece betwee the total umber of messages set ad the total umber of receivers hit by at least oe message. If this latter quatity is deoted r(, )thetotal waste is equal to: + r(, ) (0) With this measure at had we ca ask if there is ay wedge betwee what the seder chooses ad what the recievers would like them to choose coditioal o leavig them at least o the same utility level. That is, coditioal o keepig the expected reach of the campaig equal or higher. We say that a advertisig policy is k-e ciet give prices (p,p )if(, ) miimizes (0) subject to iformig at least a expected fractio of the populatio k (0, ). Formally, give the followig choice set: {(, ) R + : r(, ) k} () a advertisig policy (, )belogigtotheabovesetisk-e ciet if 6

17 (, ) also miimizes (0) for all (, ) belogig to the set; i.e., a self-iterested ad message-adverse receiver who therefore miimizes (0) subject to () would choose the same elemet of the set. Propositio 7. Suppose that the advertiser s optimal choice at prices (p,p ) is (, ) ad the associated expected fractio of the populatio reached is k. The (, ) is k-e ciet if ad oly if p = p. E ciecy requires the margial reach to be equalized across () Ituitively, if this coditio were violated the it would be possible to reduce the total umber of messages set + while keepig the total reach of the campaig fixed by subsitutig messages across outelts. optimal policy give uit prices p ad p solves: The seder i = p i v i =,. (3) That is, it equates the margial reach to the uit price ormalized by the value of iformig a receiver. So the seder s optimal choice satisfies (3) if ad oly if p = p.soitfollowsthataygapiuitpriceswillresultia wasteful allocatio of messages from the perspective of receivers. The ext sectio supplies oe (obvious) reaso for why the prices, which we itepret as market clearig prices, eed ot be equal: log-cocave trackig coupled with asymmetries across outlets. 5 Allocatig messages through a simple market mechaism We itroduce here a simple market mechaism that allocates messages (ad therefore the receivers attetio) to a populatio of idetical seders of mass 7

18 . I lie with the above, we model receivers as passive ad do ot study the more complex problem i which the total amout of attetio supplied to each outlet depeds o the amout of messages (i.e. advertisig) provided. Formally, each loyal receiver geerates a ivetory of messages whereas each switcher geerates a ivetory of / messagesoeachoutlet. Sothe total supply of messages o outlet i is equal to (Di l + D s /). If messages are uit prices the the market clearig prices, deoted p ad p solve: (p,p )= (D l + D s /) (p,p )= (D l + D s /) (4) Or, alteratively, i average terms: ñ (p,p )= ñ (p,p )=. (5) Let := (p,p )ad := (p,p )deoteacompetitiveallocatio, that is the seders demad evaluated at the market clearig prices. A corollary of Propositio 6 is that if the techology is log-liear the the correspodig market clearig prices must be equal. By Propositio 7 the the competitive allocatio must be k-e ciet. However the same eed ot hold if is log-cocave. For istace, give the log-cocave fuctio (9) we have that ñ ever equals ñ whe p = p. Because market clearig requires ñ =ñ,thecombiatio of log-cocave trackig techlogies ad asymmetries accross outlets ca potetially iduce a wedge i the equilibrium prices ad, therefore, a ie ciet allocatio of the outlets ivetory. Note that the market clearig prices reflect oly (relative) scarcity cosideratios. Because the receivers attetio is a upriced resource, there is o market mechaism at work to restore e ciecy. This is despite there beig a simple coicidece of iterest betwee seders ad receivers: the 8

19 lower the waste the higher the amout of iformatio that ca be pushed to receivers. 6 Coclusios To be doe. 7 Refereces Ambrus, A., E. Calvao ad M. Reisiger (0): Either or Both Compeitio: a two-sided theory of advertisig with overlappig viewerships, Mimeo, Duke Uiversity. Aderso, S. ad S. Coate (005): Market provisio of broadcastig: a Welfare Aalysis, The Review of Ecoomic Studies, 7, Athey, S.C., E. Calvao ad J.S. Gas (0), The Impact of the Iteret o Advertisig Markets for News Media, mimeo, Rotma School of Maagemet, Uiversity of Toroto. Butters, G. (977): Equilibrium Distributios of Sales ad Advertisig Prices, The Review of Ecoomic Studies, 44,

20 Appedix A Proofs A. Proof of Propositios ad. We shall split the problem ito two subproblems as follows. Problem : max v ( ( ( )) ( ( ))) p ( + ) s.t. + =., 0 Let () ad () = () bethesolutioofproblem. Problem: max 0 v ( ( ( ())) ( ( ()))) p. a) Cosider the case whe is log-cocave. By the property of logcocave fuctios for ay ad : ( ( )) = = + 0 ( ()) ( (0)) =( ()) so: Similar result holds for : ( ( )) ( ()) (6) ( ( )) ( ()) (7) Multiply both parts of (6) by ( ( )) (it is strictly positive) ad use iequality (7): 0

21 ( ( )) ( ( )) ( ()) ( ( )) ( ()) ( ()) =( ( + )) So, ( ( )) ( ( )) ( ( + )) Multiply by bothpartsadadd: ( ( )) ( ( )) apple ( + ) Hece, for ay give full cocetratio gives higher profit tha ay level of diversificatio. So, i the solutio either or is equal to 0. Now cosider the secod problem that i the case of full cocetratio ca be writte as: FOC of this problem is: max v () p 0 It has uique solutio. Derive w.r.t. v: v 0 () =p 0 ()+v @v = 0 () v 00 () > 0 Last iequality holds by properties of fuctio. b) Cosider the case of log-liear. We apply similar argumet as i

22 case a) but for log-liear fuctio. Now: So: ( ( )) = = + 0 = =( ()) ( (0)) =( ()) ad ( ( )) = ( ( + )) So, for ay ad we have: ( ( )) = ( ( + )) ( ( )) ( ( )) = ( ( + )) ( ( + )) = = ( ( + )) = ( + ) Hece, for ay fixed ad [0,]thefollowigequalityholds: (8) () = ( ( )) ( ( )) The secod part of the proof is idetical to case a). c) Cosider the case of log-covex. By the same argumet as before: ( ( )) ( ( )) () So, for ay ad [0,], ay degree of diversificatio is more optimal tha full cocetratio. Fid the optimal degree of diversificatio for a fixed :

23 FOC is: max ( (k)) ( (( k) )) k(0,) ( (( k) )) 0 (k) =( (k)) 0 (( k) ) or 0 (k) 0 ( (k)) = (( k) ) ( (( k) )) (9) By properties of log-covex fuctios, 0 (x) (x) is mootoe icreasig i x, so the LHS of (9) is mootoe icreasig with k whe it s RHS is mootoe decreasig. Hece it has a uique solutio for k, because trivial solutio k = always holds, it is uique solutio. So maximum diversificatio is always optimal i this case. We ca rewrite the fial problem as follows: FOC is: Derive w.r.t. v: max v 0 v 0 = p p 0 + v =0 by = 0 v 00 0 > 0 3

24 Proof of Propositio 4 Start with log-cocave case. For ay give i case of full cocetratio advertiser gets: C = D l + D s whe i case of diversificatio he gets: D =D l ()+D s () The di erece C D is: C where () D = D s = D s () () D s D l () = () > 0forlog-cocavecase,sothedi erece is liearly icreasig with D s from () < 0to () > 0. So, for every there is a poit D s () where he is idi eret betwee two cases: D s () = () 3 () Ad for all D s < D s () diversificatioisoptimal,wheforalld s > D () cocetratio is optimal. I log-liear case: C D = D s () D l () = D l () < 0 4

25 so, for every, so diversificatio always brigs higher profit. Proof of Propositio 5 By Propositio 4, full diversificatio is always optimal i log-liear case, so advertiser maximizes D.Butilog-liearcasefrom(8): D =D l ()+D s =D l ()+D s () = () that does ot deped o D s. Proof of Propositio 6 By properties of log-liear fuctio the followig equality holds: ñ so we ca rewrite the problem as follows: max D l (ñ )+D s ñ,ñ FOCs of this problem are: D l 0 (ñ )+ Ds ñ = ñ ñ + ñ D l + Ds D s 0 ñ + ñ = 0 ñ + ñ = Ds p v D l + Ds ñ + ñ +ñ D s p v p v Obviously, ñ = ñ s.t. 0 (ñ )= p is a solutio of this problem. Ad v because is cocave, solutio is decreasig with p. v 5

26 Proof of Propositio 7 I geeral case, optimal allocatio is a solutio for: max D l (ñ )+D l (ñ )+ ñ,ñ + D s ñ ñ p ñ D l + Ds + p ñ D l + Ds v where ñ i = ad k-e i D l i + Ds ciet allocatio is a solutio for: mi ñ D l + Ds +ñ D l + Ds + ñ,ñ appled l (ñ )+D l (ñ )+D s ñ ñ s.t. D l (ñ )+D l (ñ )+D s ñ ñ = k Compare FOCs for optimal allocatio: ad for e D l 0 (ñ )+ Ds D l 0 (ñ )+ Ds ciet oe: ñ ñ 0 ñ = 0 ñ = D l + Ds D l + Ds p v p v ( + (k)) appled l 0 (ñ )+ Ds ñ 0 ñ = D l + Ds 6

27 ( + (k)) appled l 0 (ñ )+ Ds ñ 0 ñ = D l + Ds For these two systems of equatios to coicide we eed: p v = + (k) p v = + (k) or, p = p. So, for every k there are prices p = p = v + (k), such that solutio of the optimal allocatio problem coicides with solutio of k -e ciecy problem. For every p = p = p we setk : k = D l (ñ )+D l (ñ )+D s ñ ñ whereñ ad ñ are solutios of the optimal allocatio problem, ñ ad ñ will be also solutios for e ciecy problem with k. To prove it, suppose per cotra that solutio of e ciecy problem (ñ k, ñ k )isotequaltoñ, ñ,sowecaguarateek with smaller ñ D l + Ds +ñ D l + Ds : D l + Ds D l + Ds D l + Ds D l + Ds ñ +ñ > ñ k +ñ k But the, for the value fuctio of optimal allocatio : 7

28 (ñ, ñ )=k ñ D l + Ds +ñ D l + Ds <k ñ k D l + Ds +ñ k D l + Ds p v < p v = ñk, ñ k that cotradicts the fact that ñ ad ñ are solutios of this problem. So, (ñ k, ñ k )isequalto(ñ, ñ ). B Istrumetal Results (used i proofs) B. Log-liear ad log-additive trackig fuctios Defiitio. A positive real valued fuctio f : X! R + is log-cocave if log f is cocave, log-covex if log f is covex ad log-liear if log f is liear. Straightforward calculus delivers the followig equivalece (proof omitted): Lemma. Cosider a positive, real valued cotiuous ad twice di eretiable fuctio f : X! R + The followig are equivalet:. f is log-cocave,. f 0 /F is mootoe decreasig. Aalogously, log-covexity is equivalet to f 0 /F icreasig ad log-liearity to f 0 /F costat. A importat result which will be key i the first propositio is the followig. Suppose : R ++! [0, ] is cotiuous, a.e. di eretiable, icreasig, cocave, bouded above by oe ad satisfies the followig boudary coditios: (0) = 0, lim! () =. Lemma 3. ( ()) is log-liear if ad oly if ( ( ))( ( )) is costat over the set {, 0: + = } for all 0. 8

29 Proof. Take ay (ˆ, ˆ ) {, > 0: + = >0} ad let k>0be the value of ( ( ))( ( )) computed at that poit. Cosider the followig (o empty) level set A := {, 0:( ( )( ( )) = k}. Note that A is symmetric i the sese that if ( = a, = b) A the ( = b, = a) A. Strict mootoicity implies that for each there is at most oe such that (, ) A which, by cotiuity ad surjectivity of, alwaysexists. Let ( ) > 0betheimplicitfuctio,whichisa.e. di eretiable with slope equal to 0 ( ) 0 ( ( )) / ( ( )) ( ( )). (0) By lemma, log-liearity of is equivalet to (0) beig costat. Symmetry of A ad cotiuity of ( )implythatthepoit = also belogs to A. (0)computedi = equals -, which the implies that ( )= ( (ˆ ))( (ˆ )) = k.. That is (ˆ, ˆ ) {(, ) > 0: + = } implies Asecodimportatad,asweshallsee,ituitivepropertyofourtrackig fuctio is the followig: Defiitio. A positive real valued fuctio f : X! R + is log-additive if log f is additive. That is: log f(x + y) =logf(x)+logf(y) for all x, y X; f is log-superadditive if the former relatio holds with > ad f is logsubadditive if the former relatio holds with <. 9