Auctions with endogenous supply

Transcription

1 Auctons wth endogenous supply Serge Izmalkov, Dlyara Khakmova, Gleb Romanyuk Abstract We consder a mult-object prvate values settng wth quantty externaltes: a value to a bdder from an object may depend on the total number of objects sold. For example, a spectrum lcense s more valuable the less lcenses are beng allocated; the lkelhood a customer wll respond to an advertsement s hgher the fewer other advertsements are shown. We rase and solve the problem of fndng revenue maxmzng and effcently allocatng auctons n such a settng. We show that both optmal and effcent auctons have the property that the quantty of objects sold depends nontrvally on the whole profle of players valuatons. That s, the optmal quantty s determned endogenously, wthn the aucton. Nether optmal nor effcent auctons can be mplemented as smple (and currently used auctons) wth a reserve prce. Thus, we demonstrate that auctons currently used for allocatng advertsng postons and spectrum lcenses are suboptmal. 1 Introducton In most models of auctons, the quantty of goods for sale s explctly specfed and treated as gven, the players preferences as well as aucton mechansms consdered are descrbed relatve to specfcaton of goods. Importantly, the analyss of such models s done under the assumpton that the valuaton of a gven buyer for a gven good or a subset of goods s fxed and ndependent from the quantty of goods sold. Of course, the actual quantty of goods sold can be dfferent as, for nstance, sellers may choose not to sell at prces below certan levels. Valuatons of buyers do depend on the quantty of goods offered for sale n many crcumstances. In spectrum auctons, the value of a lcense to a gven buyer clearly depends on the total number of lcenses allocated, as future stream of profts depends on the structure of competton after the aucton. The more frms are gong to be on the market the lower would be the profts. A pece of art, a collectble con, a bottle of wne are more valuable the rarer they are. In sponsored search auctons, an advertsement poston may be more valuable for advertser when less other advertsements are shown. Whenever a seller can choose quantty to offer and ths quantty s not too large, a common practcal approach s to separate choces of the quantty and of the aucton format, selectng the aucton format optmally gven the quantty and choosng quantty based on Prelmnary and ncomplete. 1

2 the expected performance of the chosen format or by other external reasons. Ths s how governments have been choosng lcense quanttes for spectrum auctons and how search engnes have been choosng how many advertsements to show on a page. In ths paper we rase and solve the problem of fndng revenue maxmzng and effcently allocatng auctons n prvate values settngs wth quantty externaltes. We characterze both optmal and effcent auctons for a general mult-object settng based on exstng models of sponsored search auctons, see Edelman, Ostrovsky & Schwarz (2007) (from here on, EOS) and Varan (2007). In these auctons, the objects offered for sale are advertsng postons on the screen shown together wth generc search results n response to a user s search query. Advertsers are nterested n attractng the user to ther landng page. Postons have dfferent values to advertsers, as users are more lkely to clck on hgher ad postons and located above search results than on those located below search results or on the rght no matter what actual ads are shown. The value of a gven poston to a gven advertser s then a product of the advertser-specfc expected value generated followng a clck on the ad by a user and the poston-specfc CTR (clck through rate) a probablty that a user clcks on an ad at that poston. Gven the quantty and composton on the screen of postons offered, all advertsers have a common rankng of postons based on ther CTRs. We allow these CTRs to change dependng on the total number of postons offered and dfferently for dfferent postons. The ndustry standard for sponsored search allocatve mechansms s the generalzed second prce aucton (GSP), frst studed by Varan (2007) and EOS. In the smplest verson of ths aucton the advertsers submt bds. The ads on the lst are arranged accordng to ther bds, wth the hghest bd placed n the top slot. Advertsers pay only when a user clcks on ther ad and the prce per clck s the next hghest bd. EOS show that GSP has an equlbrum n whch advertsements are allocated effcently and payments are equal n expectaton to those of Vckrey-Clark-Groves (VCG) mechansm. Ostrovsky & Schwarz (2009) compute an optmal reserve prce for the GSP aucton and demonstrate that Yahoo! s revenues ndeed ncreased after addton of reserve prces to ts aucton. Naturally, the reserve prce decreases the length of the ad lst sometmes when not all slots are flled. Ths changes the confguraton of the page and mpacts the CTRs of the remanng ads. There s a strong emprcal evdence that CTR spllovers such as these may be sgnfcant. Jezorsk & Segal (2010) report that the top ad would receve 81% more clcks n a hypothetcal world n whch t faced no competton. Gven that most of revenue generated by search engnes comes from context advertsement, fndng the optmal and effcent sellng mechansms are mportant practcal problems. We show that both optmal and effcent auctons have the property that the quantty of objects sold depends non-trvally on the whole profle of players valuatons. That s, the optmal quantty s determned endogenously, as a part of aucton. Nether optmal nor effcent auctons can be mplemented as smple (and currently used auctons) wth a reserve prce. Thus, we demonstrate that auctons currently used for allocatng advertsng postons and spectrum lcenses are suboptmal. We then descrbe the optmal mechansm n detal for the case of only two postons avalable, n whch case an outcome when only one poston s shown can be characterzed as an exclusve dsplay outcome. Further, we provde numercal computatons for benefts of 2

3 runnng an optmal mechansm and present a smple endogenous supply mechansm (based on GSP) whch s better than GSP wth the approprately chosen reserve prce. In ths paper we focus on a specfc externalty to bdders valuatons stemmng from dependence of clck-trough-rates of ad lnks on the overall quantty or composton of ads shown. We take CTRs as gven, do not model users behavor, and assume that an advertser s expected value followng a clck on her ad s ndependent of a poston the ad s shown or on whch ads are shown at the other postons. Another knd of externalty s dentty-specfc externalty, whch has been studed extensvely n the lterature. Jehel, Moldovanu & Stacchett (1996) and Aseff & Chade (2008) study an optmal aucton wth dentty-dependent externaltes. In Aseff & Chade (2008), each bdder mposes a multplcatve externalty α j on bdder j. Unlke n our model, they have dentcal goods and dentty-specfc externaltes, whereas we have slot-specfc externaltes and ex ante dentcal bdders. Segal (1999) studes contractng wth externaltes, where buyers do not have sgnals and the seller can make ndependent offers to buyers. That s to say, the set of allocatons s a lattce, whch s not the case n our framework. Ülkü (2009) ntroduces the general model for optmal combnatoral mechansm desgn problem wth multple heterogenous objects. He allows for non-rsk-neutral bdders and general dependence of bdder s value functon on types of the other bdders. Therefore there are nformatonal and dentty-specfc externaltes n hs settng, but he does not address externaltes pertanng to the allocaton of objects, whch s the substance of our work. A vast lterature on the revenue-maxmzng mechansms exsts: Vckrey (1961), Myerson (1981), Mlgrom & Weber (1982). Edelman & Schwarz (2010) study the refnement of equlbra of the statc games of complete nformaton, whch s the standard way to model sponsored search aucton envronment, wth respect to ts dynamc counterpart. They analyze the underlyng dynamc game of ncomplete nformaton, and establsh an upper bound on the revenue of any equlbrum of any dynamc game n ths envronment. They then exclude equlbra of the correspondng statc game wth revenue that exceeds ths upper bound. They also fnd that the optmal mechansm s generalzed Englsh aucton (or GSP) wth a reserve prce that s ndependent of the number of bdders and clck-through rates. We use the same technques (gong back to Myerson (1981)) to derve the optmal aucton n the framework wth externaltes. We show that the optmal mechansm n our framework has a reserve prce that s stll ndependent of the number of bdders and clck-through rates. In addton, the optmal mechansm has rules for exclusve dsplay. Therefore, GSP wth reserve prce s not an optmal allocatve mechansm. There s an strand n the IO lterature on onlne advertsng. Jezorsk & Segal (2010) buld a dynamc model of utlty-maxmzng users and fnd large negatve externaltes between ads. Athey & Ellson (2011) have a structural model of users behavor where users look through the ads lst top to bottom. The resultng CTRs have a smple form of externalty ads above you nfluence your CTR whle the number of ads below you does not matter. Athey & Nekpelov (2010) argue that the GSP aucton turns out to be neffcent as an aucton for the sponsored search. The reason s that clck-through rates and qualtes scores vary across queres and a bdder s standng bd partcpates n many of dfferent queres because queres arrve faster than bdders can adjust ther bds. They test ther model on several popular search phrases and fnd however that the effcency dfferences are small. There s an addtonal lterature on exclusve dsplay. Ghosh & Sayed (2010) study 3

4 externaltes through competton for conversons. In ther model, the prvate value of an advertser and the bd are both two-dmensonal where the second component represents the value f shown exclusvely. Muthukrshnan (2009) proposes an aucton desgn n whch advertsers bd for the maxmum number of compettors they are wllng to be shown along wth. Jerath & Sayed (2011) study a settng where bdders submt two values one for beng shown exclusvely and the other for regular dsplay. They also consder the endogenous determnaton of the number of ads shown (though they do not use word endogenous). They study two possble extensons of GSP and fnd that they ncrease the revenue. An nterestng pont of note n ther framework (two dmensonal auctons), a larger number of bdders ncreases the revenue advantage to the search engne, whle n ours t s non-monotonc. Fnally, there s a broad CS lterature whch focuses on the effcent computablty of optmal outcomes. Our paper s organzed as follows. In Secton 2, we present a model of poston-specfc externaltes and fnd the optmal mechansm wthn ths framework. We study the optmal mechansm and develop ntuton for the specal case of two slots n Secton 4. In Secton 5, we fnd the equlbrum of the standard GSP aucton n our framework. Its expected revenue to the ad platform s compared to that of the optmal mechansm n Secton 6. Secton 7 concludes. 2 Model 2.1 Setup Consder a one shot game wth one seller (ad platform) and multple buyers (bdders, advertsers). The ad platform runs a sponsored search aucton to allocate postons n the sponsored lnk lst. The lst can have a maxmum of l postons. There are n l advertsers competng for the postons on the lst. The advertsers have prvately observed values per clck v R + whch are d dstrbuted wth absolutely contnuous cdf F. Reservaton value s normalzed to zero. Each advertser can get only one slot n the lst. The allocatve mechansms proceeds as follows. Advertser submts sngle bd z R, let z = (z 1,...z n ). The ad platform selects k l and allocates the top k slots to k bdders. We make the followng key assumpton, whch we mantan throughout the paper. We assume that the clck-through rates depend only on the poston of the slot (j) and the total number of ads shown on the page (k), and denote t by α j:k. Thus, every poston on the ad lst receves an dentcal stream of clcks regardless of whch ads are placed on the page. Users do not care about the denttes of advertsers and are not nterested n specal features of ndvdual ads. In a sense, users see only empty slots but do not see ther contents. Nevertheless the assumpton proves reasonable n practce. It s especally so n the keyword market where ads have smlar qualty (e.g., the text of the ads s smlar and does not nclude pctures or deep lnks) and there are no externaltes except postonng of the ad. There s an extensve lterature that studes dentty-specfc externaltes; we shut down these effects to focus on postonal externaltes. Addtonally, search engnes use qualty scores to rank the ads, multplcatve qualty scores can be easly added nto ths framework wthout a sgnfcant changes n the workng. 4

5 Note that our framework assumes that the advertser s value-per-clck s ndependent both of the other bdders, and of the realzed length of the ad lst. Ths may mply n partcular that users do not sort out between postons by any unobserved varable and don t clck more when there are more ads. All advertsers are dentcal along dmensons other than per-clck value, and n partcular have dentcal clck-through rates n the same postons. Ths mples n partcular the same converson rate for advertsers n the same poston. It s reasonable to consder only α j:k 0 for all j and k. Furthermore, wthout loss of generalty order CTR for every k n descendng order; that s, let α j:k α j+1:k. Indeed, even f t turns out that the CTR of the upper slot s less than that of the lower slot, we can renumerate slots n such a manner that CTRs decrease n j. Introduce a notatonal devce n whch α j:k := 0 for k + 1 j l. α 1:1 α 1:2 α 1:3 α 1:4 0 α 2:2 α 2:3 α 2:4 0 0 α 3:3 α 3: α 4:4 Table 1: Clck-through-rates matrx Clck-through-rates α j:k are exogenous and are common knowledge. In the baselne model wthout externaltes (lke n EOS), α 1:1 = α 1:2 = α 1:3 = α 1:4, α 2:2 = α 2:3 = α 2:4, etc. Advertser s payoff from beng n poston j out of k s 2.2 Optmal Mechansm α j:k v (payment to the seller). In ths secton we characterze the mechansm whch maxmzes the seller s revenue. From here on, we wll refer to ths as the optmal mechansm. The analyss of the optmal mechansm s based on the framework from Myerson (1981). We look for a drect revelaton mechansm that nduces truth tellng. We are phrasng our problem n order to use the logc of Myerson (1981). Consder a drect mechansm (Q, M) where Q s the allocaton rule and M s the payment rule. An allocaton rule Q = {Q j:k (z)} s a collecton of functons where Q j:k (z) [0, 1] s the jont probablty that bdder gets poston j on the lst and the lst length s k. We wll refer to each such functon as an allocaton ndcator. In generc cases (no tes among bds), Q j:k (z) {0, 1}. The allocaton ndcators are consstent, f for all and z, the probablty that the ad of bdder s shown somewhere, j,k Qj:k (z) 1, and f for all k and z, the total number of postons shown never exceeds avalable postons,,j Qj:k (z) k. The key step n our analyss s to defne the realzed CTR for bdder when vector of bds z s submtted, Q (z) = j,k α j:k Q j:k (z). (1) Agan, n generc cases, Q (z) {0, α 1:1, α 1:2, α 2:2,...}. Wth that defned, the sponsored search multple object aucton wth externaltes lke n (1), can be represented as a sngle 5

6 unt aucton n whch the probablty of bdder gettng the object s Q (z). Therefore, the search for the revenue-maxmzng aucton (or welfare-maxmzng aucton) s smplfed. The dfference s that Q (z) s not probablty but clck-through-rate; furthermore, specal care should be taken towards verfyng the ncentve compatblty of the mechansm we wll address these ssues n ths secton. Defne expected CTR for bdder as ˆ q (z ) = Q (z)df (z ). (2) The payment rule s gven by the collecton of payment functons M = {M j:k (z)}. Payment functon M j:k (z) s the realzed prce-per-clck for bdder n poston j when k slots are allocated by the ad platform. In the manner of allocaton functons, defne expected payment as ˆ m (z ) = α j:k M j:k (z)df (z ). j,k Ths defnes the nterm expected payment of bdder f he submts z. Assume for clarty of exposton that zero value advertsers stay out; that s, m (0) = 0 for all. (3) Lemma 1. (Q, M) s ncentve compatble f and only f () expected CTR q (z ) s nondecreasng, and () payments are chosen such that the expected payment s m (v ) = q (v )v v 0 q (t )dt. The proof s standard, see Krshna (2009). Defne vrtual valuaton functon as ψ(v ) = v 1 F (v ). f(v ) Usng realzed CTRs {Q (z)} we can wrte down the revenue of the ncentve compatble mechansm smlar to Myerson (1981). The result s recorded n the followng. Proposton 1. The ncentve compatble drect revelaton mechansm gven by Q generates revenue ˆ R(Q) = ψ(z )Q (z)f(z)dz. The optmal mechansm s then gven by the allocaton rule whch solves s.t. max k,q R(Q) Q s ncentve compatble; and the payment rule gven by Lemma 1. 6

7 Proof. The proof essentally parallels one n Myerson (1981). The revenue of the mechansm s the sum of ex-ante expected payments from buyers, R(Q) = Em (v ). Denote the space of values by V = V 1 V 2... V n, v V = [0, ). To compute the ex-ante expected payment of one buyer, use (??), ˆ ( ˆ v ) Em (v ) = q (v )v q (t )dt f (v )dv V 0 ˆ ˆ ˆ ˆ v = Q (v, v )f (v )dv v f (v )dv q (t )dt f (v )dv, V 0 0 V where the expresson (2) s substtuted n for q (v ). Now change the order of ntegraton n the second ntegral and agan use (2), ˆ ˆ ˆ Em (v ) = Q (v)v f(v)dv q (t ) f(v )dv dt V 0 t ˆ ˆ = v Q (v)f(v)dv q (t ) 1 F (t ) f(t )dt V 0 f(t ) ˆ ( = v 1 F (v ) ) Q (v)f(v)dv. f(v ) V It turns out that after properly defnng varables Q (z) many results from Myerson (1981) also go through. In partcular, the search for the optmal mechansm can be made sgnfcantly easer after we mpose regularty condtons standard for the mechansm desgn lterature. Accordngly, we are gong to assume that vrtual valuaton ψ(z) s nondecreasng. The next result uses the specfcs of the externaltes framework and does not follow drectly from the prevous lterature. Proposton 2. The optmal mechansm (Q, M) solves the unconstraned maxmzaton problem ˆ argmax k,q ψ(z )Q (z)f(z)dz (4) where M s gven by Lemma 1. Proof. Proposton 1 gves the formula for ex-ante expected revenue. We need to show that the IC constrant can be omtted. From Lemma 1, t follows that t suffces to show that expected CTR q (z ) s nondecreasng. We wll show that realzed CTR Q (z, z ) s nondecreasng n z and the result wll follow. Frst note that we can t apply standard comparatve statcs technques here. The ad platform optmzes allocaton for each vector of bds z, R(z) = ψ(z )Q {Q } A max. 7

8 Here set A s the set of all feasble consstent allocatons of ads to postons. A s not a lattce therefore monotone comparatve statcs does not apply. We need to check the monotoncty drectly. We wll consder the generc case. In the optmal allocaton, for fxed, only one of { Q j:k } j,k equals one and the others equal zero. Therefore the optmal number of slots k (z) l s non-random gven z as well as the allocaton of bdders n these k (z) slots. Consder two bds of, z < z. By the assumpton of regularty of vrtual valuatons, ψ(z ) ψ(z ). Suppose the vector of bds s ordered n the descendng order. Let the rank of bd z n ths orderng be r and the rank of bd z be r. The optmal number of slots changes from k = k (z, z ) to k = k (z, z ). Let (ψ 1,... ψ n) be the ordered vrtual valuatons when z was submtted and (ψ 1,... ψ n) when z was submtted. Note that ψ r = ψ(z ) and ψ r = ψ(z ). The optmalty condton requres that the ad platform chooses k when z s submtted and chooses k when z s submtted, ψ j α j:k ψ jα j:k ψ j α j:k ψ j α j:k Expand the sums, ψ jα j:k + ψ(z )α r :k ψ jα j:k + ψ(z )α r :k j r j r ψ j α j:k + ψ(z )α r :k ψ j α j:k + ψ(z )α r :k j r j r Note that here clck-through-rates are Q (z, z ) = α r :k, Q (z, z ) = α r :k and therefore we need to show that the clck-through rate does not go down, that s, α r :k α r :k. Wthout loss of generalty, consder the margnal case where r = r or r = r + 1. That s, when bdder rases hs bd, he ether outbds one compettor or retans the poston. The general case s done by analogy but wll be notatonally cumbersome. Then vectors (ψ 1,... ψ n) and (ψ 1,... ψ n) concde except for postons r and r, ψ j = ψ j for j r and j r ; moreover, ψ r = ψ r. So n the case of r = r + 1, the ordered vrtual valuaton vectors look lke ψ 1 ψ 2 ψ 3... ψ r ψ r = ψ(z )... ψ n ψ 1 ψ 2 ψ 3... ψ r = ψ(z ) ψ r... ψ n Add the nequaltes above to obtan ψ r α r :k + ψ(z )α r :k + ψ r α r :k + ψ(z )α r :k ψ r α r :k + ψ(z )α r :k + ψ r α r :k + ψ(z )α r :k (ψ r ψ(z ))(α r :k α r :k ) (ψ r ψ(z ))(α r :k α r :k ) (ψ r ψ(z ))(α r :k α r :k ) + (ψ r ψ(z ))(α r :k α r :k ) (ψ r ψ(z ))(α r :k α r :k ) + (ψ r ψ(z ))(α r :k α r :k ) 8

9 Use the fact that the just outbdded bdder does not change hs bd, ψ r = ψ r to obtan (ψ(z ) ψ(z ))(α r :k α r :k ) (ψ r ψ(z ))(α r :k α r :k ) + (ψ(z ) ψ r )(α r :k α r :k ) All expressons n brackets n the rght-hand sde are nonnegatve, ψ(z ) ψ(z ) 0 by regularty; therefore, α r :k α r :k 0. Incentve compatble mechansms requre ncreasng expected CTR. Ths alone does not allow to pn down the relaton between a bdder s bd and the number of slots. If we make the natural assumpton that CTR of a poston goes down when new slots are added 1, we can guarantee the monotoncty of the number of ads n a bdder s bd. When bdder ncreases the bd, he has to get more clcks. Ths happens ether n the case when k does not change or goes down and bdder retans poston or moves up, or n the case when he moves up but k goes up. The latter case wll not realze because t wll volate the revenue maxmzaton condton. The formal result s gven below. For the next result we are gong to assume that the clck-through-rate decreases n the number of slots, α j:k α j:k+1 for j k. Corollary 1. Condtonal on advertser s ad beng shown, () s poston s nondecreasng n z ; () the optmal number of slots k (z, z ) s nonncreasng n z. Proof. Part () follows from the fact that n the optmal allocaton, no matter what the number of slots s, greater bds get hgher postons. Now we prove part (). In the course of the proof of Proposton 2 t was shown that realzed CTR ncreases wth bd, Q (z, z ) Q (z, z ) when z < z. Agan we wll consder a margnal ncrease n bd. There are two cases. In the frst case, s rank does not change after the bd change. Suppose ths rank s j. Then the result follows drectly from the premse of the corollary. In the second case, after the bd change, outbds the bdder above hm. Fx the bd above z and suppose a contnuous ncrease n bd z. For z < z n some neghborhood of z, the revenue-maxmzng number of slots s k: t j ψ tα t: k + ψ(z )α j: k t j ψ tα t:k + ψ(z )α j:k for all k. Vrtual valuaton s contnuous from the left (otherwse, pck a varant of the pdf whch s). Therefore n the lmt z z, the nequalty holds, and the number of slots s stll k. Snce bds are the same, we can swap bdder wth the bdder above hm nothng changes, ths s just a relabelng. Further margnal ncrease n bd does not change the rank of bdder and we are back to case 1. The corollary descrbes the ncentves for bdders n the optmal mechansm. Each bdder has a one-dmensonal bd and by ths bd he can sgnal that he wants to be shown alone. Indeed, by bddng more, a bdder get rd of the compettors ads and attans more exclusvty. Ths mechansm s also optmal for the ad platform. If z s hgh relatve to the next bd, t s proftable to put alone and charge hm a hgh prce for beng shown exclusvely; f z s low relatve to the next bd, he can t be charged a lot; therefore, t s proftable to place both ads and charge a medum prce. 1 Emprcally, ths s not always the case, larger block of ads can attract more attenton and ncrease ndvdual CTR of the slots. Moreover, there are effects related to the sze of the computer screen and the vsblty of the algorthmc search results. Nevertheless, n a vast majorty of cases, the assumpton s true. 9

10 Proposton 3. The optmal mechansm has a cutoff reserve prce ndependent of clckthrough-rates and the number of bdders and s gven by r = ψ 1 (0). Proof. Ths almost mmedately follows from (4). Recall that the revenue of the seller s R = max ψ j α j:k (5) k where ψ 1 ψ 2... ψ n are order statstcs of {ψ(v )} n =1. Suppose towards contradcton that bdder wth ψ(v ) < 0 was placed on the ad lst. Under the decreasng CTR s assumpton, t s proftable for the ad platform to cut off bdder (and everyone else wth negatve vrtual valuatons). Frst, s negatve nput s elmnated, second, postve vrtual values get hgher weghts n the sum (5). Note that the decreasng CTR s assumpton s crucal here. It s easy to construct a case such that α 1:1 ψ 1 < α 1:2 ψ 1 + α 2:2 ψ 2, ψ 2 < 0. If one ad alone attracts too few clcks, then t s better to put another one to ncrease revenue from the frst slot by sacrfcng some revenue on the second slot. j 3 Effcent mechansm In ths Secton we present an effcent mechansm allocatng objects n the presence of quantty externaltes. The constructon s rather straghtforward, as t s a Vckrey-Clark-Groves mechansm constructed for the consdered envronment. Importantly, the actual quantty of objects sold s non-trvally determned wthn the mechansm based on submtted reports. 4 Case of Two Slots In ths secton we consder a specal case l = 2 wth two slots. Ths wll help us to develop ntuton regardng the optmal mechansm for the case wth arbtrary number of slots. 4.1 Setup Consder an aucton n whch the seller has two postons for sale. We adopt specal notaton for ths secton. Clck-through-rates of the postons wll be denoted as: 1 + β := α 1:1, 1 = α 1:2, α := α 2:2. Therefore, the second poston receves α tmes of the frst poston, and the top poston when shown exclusvely attracts β more clcks. In case of two slots there are three possble allocatons: one poston allocated (exclusve dsplay, ED): there s subject to Q = 1 + β, Q j = 0 j ; two postons allocated (multple dsplay, MD): there are 1, 2 subject to Q 1 = 1, Q 2 = α, Q j = 0 j 1, j 2 ; no postons allocated (no dsplay, ND): Q = 0 for all. 10

11 Fgure 1: [Insert Fgure 1 here.] Suppose vector of basc valuatons z was submtted. Denote the order statstcs of the vrtual valuatons by ψ (n) = max ψ (z ), ψ (n 1),... Also denote by Y 1 the hghest competng bd. The reservaton prce s denoted by r. Use Proposton (2) and formula (4) to fnd that the optmal allocaton requres (see Fgure 1 for graphcal llustraton): Exclusve dsplay f { (1 + β)ψ (n) ψ (n) + αψ (n 1) ψ (n) > 0 Multple dsplay f { (1 + β)ψ (n) < ψ (n) + αψ (n 1) ψ (n) + αψ (n 1) > 0 No dsplay, otherwse. Note that f β α, exclusve dsplay domnates multple dsplay n terms of revenue. Thus, n ths case, the optmal mechansm wll be a SPA whch sells the top poston only. For the rest of the dscusson assume α > β. The allocaton rule for the revenue-maxmzng mechansm s then: 1 + β, ψ = ψ (n), ψ > 0, βψ αψ (n 1) 1, ψ = ψ (n), ψ > 0, βψ < αψ (n 1) Q (z, z ) = α, ψ = ψ (n 1), ψ (n) > 0, αψ (n) < βψ 0, otherwse Frst, we brefly dscuss the case β < 0. When β < 0 the allocaton rule s pecular: a bdder can be allocated the second poston even f hs vrtual value s below zero, that s hs bd s less than the reserve prce r = ψ 1 (0). Imagne that the second-hghest bd s just below r and therefore hs vrtual valuaton s slghtly below 0. Thus, sellng hm the object adds about zero to the seller s revenue, but ths s nevertheless better than sellng a sngle poston to the hghest bd wth low CTR. 4.2 Optmal Mechansm Descrpton Here we wll focus on the leadng case β 0. Defne the boundary bds as follows. Denote by y ED (z ) the mnmal amount bdder has to bd to attan the exclusve dsplay gven z, the bds of the other players; y MD1 (z ) s n the same manner the mnmal amount bdder has to bd to get the frst poston n the multple dsplay (to be placed frst together wth someone n the second slot); y MD2 (z ) s the mnmal amount bdder has to bd to get the second poston. ( ) Defne functon h(y) = ψ 1 α ψ(y). Then h 1 (y) = ψ ( 1 β ψ(y)). Note that the reserve β α prce s a soluton of equaton h(y) = y. 11

12 Fgure 2: [Insert Fgure 2 here.] Proposton 4. Assume the regularty condton holds. Assume also α > β > 0. Then the optmal mechansm has two allocaton rules. The frst s a fxed reserve prce r = ψ 1 (0). The second s the rule for exclusve dsplay, Defne boundary bds as ψ(z (n) ) > α β ψ(z (n 1)) (6) y ED (z ) = max {h(y 1 ), r} y MD1 (z ) = max {Y 1, r} y MD2 (z ) = max { Y 2, h 1 (Y 1 ), r } y ED y MD1 y MD2. The allocaton rule s then (see Fgure 2 for graphcal llustraton) 1 + β, z > y ED 1, y MD1 Q (z, z ) = α, y MD2 0, z y MD2 < z y ED < z y MD1 (7) The payment rule s (1 + β)y ED (y ED y MD1 ) α(y MD1 y MD2 ), z > y ED y MD1 α(y MD1 y MD2 ), y MD1 M (z, z ) = αy MD2, y MD2 0, z y MD2 < z y ED < z y MD1 (8) The prce-per-clck s M (z, z )/Q (z, z ). To make formulas more tractable consder separately two cases. The frst case s Y 1 > r. Then y MD1 (z ) = Y 1 y ED (z ) = h(y 1 ) y MD2 (z ) = max { Y 2, h 1 (Y 1 ) } Note that reserve prce r does not enter the prces the bdders end up payng. Ths s because the second poston pays at least h 1 (Y 1 ) whch s greater than r because Y 1 > r. The second case s Y 1 r. Then y ED = y MD1 = y MD2 = r and the wnner of the exclusve dsplay pays r. 12

13 1 2 0 Fgure 3: Optmal allocaton rule, β > 0. The mechansm runs as follows. There s a reserve prce r = ψ 1 (0). If no bd exceeds t, no poston s allocated. If the hghest bd s above the reserve prce but Y 1 < r, the wnner attans exclusve dsplay and pays r per clck. If two hghest bds exceed the reserve prce, there are two cases. If bds dffer a lot, namely ψ(x (n) ) > αψ(x β (n 1)), or X (n) > h(x (n 1) ), the the hghest bdder s placed alone. If the bds are closer than that, both postons are allocated. See Fgure (3). As usually, the payment s talored such that to make a bdder ndfferent between wnnng and losng when he tes. The standard one unt optmal mechansm s defned by a reserve prce. In the multple object sponsored search aucton wth no externaltes, VCG wth reserve prce or GSP wth reserve prce are optmal auctons (Edelman & Schwarz (2010)). When there are externaltes, there has to be also rule for exclusve dsplay (6). Therefore, GSP wth reserve prce s not an optmal mechansm; moreover, as we wll show n Secton 5, n GSP n equlbrum, bdders bd as f there are no externaltes. A natural queston to ask s the sze of the gan of ntroducng an addtonal rule whch would take nto account the externaltes. We make the frst step and assess the revenue gan va smulatons n Secton 6. The optmal mechansm s mpractcal because even n the two-slot case the rule for exclusve dsplay (6) s hard to mplement, as t depends on the dstrbuton of values n greater extent the reservaton prce does. The soluton for applcatons would be to approxmate rules for exclusve dsplay. The obvous start s lnear approxmaton. Lnear approxmaton suggests comng up wth two parameters, the slope and the orgn. It s not that bad compared to the standard case wth a sngle parameter, the reserve prce. Examples below provdes some deas on how t can be done. For example the lnear approxmaton of rule for exclusve dsplay turns out to be exact for unform dstrbuton of values and very close for lognormal dstrbuton. 13

14 1 2 0 Fgure 4: Optmal allocaton rule, unform values per clck 4.3 Example: Unform Values Consder the case of unformly dstrbuted values per clck, F (x) = x. Then ψ(x) = 2x 1, ψ 1 (x) = x+1, 2 h(y) = 1 ( 1 α ) + α 2 β β y Note that h s lnear. Consder only the non-trval case Y 1 > r. Accordng to (8) the payments are: (1 + β)h(y 1 ) (h(y 1 ) Y 1 ) α(y 1 y MD2 ), z > h(y 1 ) Y 1 α(y 1 y MD2 ), Y 1 < z h(y 1 ) M (z, z ) = αy, y MD2 < z Y 1 0, z y MD2 max { (1 + β)y 1, αy 2 + Y 1 + 1, z 2 > h(y 1 ) max { Y = } 2 1), αy 2 + (1 α)y 1, Y1 < z h(y 1 ) max { αy 2, βy 1 +, y MD2 2 < z Y 1 0, z y MD2 The rule for exclusve dsplay becomes 2X (n) 1 > α(2x β (n 1) 1). See Fgure (4). Ths can be rewrtten as X (n) r X (n 1) r > α (9) β Therefore, the rule for exclusve dsplay reads: the hghest bd has to be α tmes greater β than the second hghest bd, both reserve prce deducted. Rato α does not depend on β the dstrbuton of values but only on the CTRs. Ad platforms know reserve prces for keyword markets; they also estmate CTRs of the postons. Therefore rule (9) s readly mplementable. For arbtrary dstrbuton equaton (6) doesn t gve lnear rule but can be approxmated n a way. To make the rule even more smpler the auctoneer may ntroduce the threshold 14

15 Fgure 5: [Insert Fgure 5 here.] for the dfference between the hghest bds, X (n) X (n 1) > K, or the threshold for ther proporton, X (n) /X (n 1) > K. Note that the latter, the proporton approxmaton, s a good approxmaton of (9) n the case of low reserve prce r. 4.4 Log-normal dstrbuton In ths secton we consder log-normal dstrbuton for basc values whch s the most popular choce n applcatons. For example, Ostrovsky & Schwarz (2009) fnd that log-normal dstrbuton of values s the good approxmaton and use t n ther feld experment. But mnd the unpleasant feature of the log-normal dstrbuton that vrtual valuaton s not monotone for σ The lnear approxmaton n fashon of (9) would work perfect. Indeed, the relevant part of vrtual valuaton functon above the reserve prce s essentally lnear (see Fgure??). Therefore the rule for exclusve dsplay (6) s essentally lnear, too (see Fgure 5). Notce also that for low σ 2 the exclusve dsplay border s almost vertcal. Ths mples that the border does not le far from the reserve prce threshold z (n 1) = r. Therefore n the case of low varance we expect the gans from the optmal mechansm wth respect to VCG wth reserve prce to be mnmal. 4.5 Example: β = 0 In ths case h(y) = h 1 (y) r, y MD2 rule s Payments are M(z, z ) = = max {r, Y 2 } and y MD1 1, z > y MD1 Q(z, z ) = α, y MD2 < z y MD1 0, z y MD2 αy MD1 αy MD2 + (1 α)y MD2, z > y MD1, y MD2 0, z y MD2 = max {r, Y 1 }. Allocaton < z y MD1 The rules of the mechansm are straghtforward: the hghest or the second hghest bdder gets the good only f he outbds the reserve prce. The bdder who receved the second good pays αr or α tmes the next hghest bd (that s, hs prce per clck s maxmum of r and the next hghest bd). The hghest bdder pays the reserve prce r f he s the only wnner, and the adjusted second prce Y 1 α(y 1 y MD2 ) f there s a second wnner. Ths case of β = 0 s no externaltes case and therefore payments are those of GSP wth reserve prce. We wll address general case of equlbrum of GSP n the presence of externaltes n Secton 5. 15

16 5 GSP The standard verson of GSP desgn does not have exclusve dsplay as an outcome, because all postons are allocated unless there are nsuffcent bds that exceed the reserve prce 2. We are gong to demonstrate that bdders ncentves for obtanng the exclusve dsplay are not exploted by GSP. Moreover, the strateges are the same as n the no externaltes framework wth the maxmal number of postons. Consder a GSP aucton wth cutoff reserve prce r, whch s chosen optmally r = ψ 1 (0) for smulatons, l slots and n 2 bdders. As n EOS and Varan (2007) we use statc framework wth complete nformaton and s lookng for locally envy-free equlbrum. Consder the Englsh aucton representaton of locally envy-free equlbrum. Imagne that n generalzed Englsh aucton prce starts off from r. The equlbrum number of postons s ˆk = mn{l, #{ : b > r}}, the CTR of poston j s α j:ˆk n equlbrum. Bd functons are β(v > l bdders left) = v β(v j l bdders left) = v α j:ˆk (v max {r, b j+1 }) (10) α j 1:ˆk where b j+1 s the bd of the prevous player (last bd some player qut the Englsh aucton). In ths settng the truncaton of the lst occurs only by chance; bdders are not ncentvzed to bd more and they do not. In the complete nformaton equlbrum the number of slots ˆk s known and therefore we are back to the no externaltes case. Indeed, changes n CTRs wth k ˆk do not alternate bdders behavor (10). 6 Revenue Comparson GSP wth reserve prce s optmal n the no externaltes framework (Edelman & Schwarz (2010)). Howerver, as was shown n Secton 5, t s not optmal n the presence of externaltes. A natural queston to ask s how much the ad platform loses by usng GSP rather than the revenue maxmzng mechansm. In ths secton we consder unform values per clck v U[0, 1] and l = 2 slots and α =.77. We compare revenues from the optmal mechansm (4), whch takes nto account externaltes, wth GSP wth the optmal reserve prce r = 1 2. In the case of l = 2, formulas (10) yeld β(v > 2 bdders left) = v β(v 2 bdders left) = v α(v max {r, b 3 }) β(v 1 bdder left) = undef 2 Note however that Google somewhat has the exclusve dsplay. Top ads n each aucton that proved to be of the hght qualty (by some non-transparent rules that nvolve analyzng text of the ad, the bd, the advertser s webste qualty and the hstory of the bdder) are promoted from the rght-hand sde slots to the top-of-page poston. Varan (2007) fnds that ths mght be an explanaton of sharp ncrease n the expendture functon n the hgh bds range. 16

17 Fgure 6: [Insert Fgure 6 here.] Then the revenue s gven by ˆ R GSP (n) = (1 + β)r dv + v (n 1) <r v (n) >r ˆ v (n 1) >r (b (n 1) + α max {r, b 3 })dv We compare ths wth the revenue of the optmal aucton (4). The results are shown n Fgure 6 for a range of β. The fgure shows the revenue generated by the optmal mechansm along wth the shares that come from the sale of the exclusve dsplay and from the sale of multple dsplay. The fgure also shows the probabltes of these outcomes: the ED allocated, MD allocated or no dsplay s allocated. For n = 2, large share of the revenue comes from the sale of the exclusve dsplay whch s sold half of the tmes or more often. Conversely, for larger n and medum β, the exclusve dsplay s sold pretty rarely and therefore generates the small share of revenue. As β grows, however, t gets sold more often and for β hgh enough (about.7) the ED consttutes the man source of revenue. When the number of partcpants s hgh the dfference between two hghest bds s unlkely to be large, n whch case the sale of ED does not occur. Furthermore, f β s low, the optmal mechansm requres larger spread between the frst hghest bd and the second hghest bd thus lowerng the probablty of ED allocaton. In the no externaltes case, the ntroducton of a reserve prce ncreases the revenue but the revenue gan decreases n the number of bdders. That s, n compettve markets, the reserve prces do not make much of a dfference. Conversely, the ntroducton of rule for exclusve dsplay does not have ths property when the reserve prce s already set. See Fgure (7). Observaton 1. The revenue gan wth respect to GSP wth reserve prce s not monotone n the number of bdders. It ncreases for medum n and converges to zero as n. When the number of bdders goes up from small to medum number, GSP wth reserve prce starts allocatng all l slots because a lot of bdders have postve vrtual valuatons. The optmal aucton on the other hand stll truncates. When the number of bdders ncreases further, top bds are dense (no bg gaps between bds), therefore the optmal aucton does not truncate the lst but allocates all l slots. 17

18 n=2 n=3 n=4 n=7 n=10 Revenue Increase, unt unfrom dstrbuton Max revenue / GSP revenue β Fgure 7: Revenue gan of the optmal aucton wth respect to GSP wth reserve prce for a range number of bdder n, d U[0,1] values per clck, 2 slots, α =.77. The exclusve dsplay CTR bonus β s along the x-axs. Note that the revenue gan s not monotone n n. 18

19 7 Concluson We presented a model of poston-specfc clck-through-rate externaltes n sponsored search auctons and found the revenue-maxmzng mechansm of sellng the ads n the presence of externaltes. The optmal mechansm ncludes () fxed reserve prce ndependent of the number of bdders, ads and CTRs; () rules for exclusve dsplay that truncate the ad lst further after the reservaton prce threshold has been met. We compared the revenue advantage of the optmal mechansm aganst the GSP wth a reserve prce, whch s the optmal aucton when there are no externaltes. In our smulatons of two-slot auctons, we found that the proposed optmal mechansm ncreases the revenue by a sgnfcant amount for emprcally reported externalty levels. We observed that the revenue gan n non-monotone n the number of bdders. The next step of our research s to obtan real world data from an ad platform to estmate the dstrbutons of values per clck and clck-through-rates, n order to assess the revenue gan of ntroducton of the optmal aucton on the real keyword markets. For the optmal mechansm to be mplemented n practce, t should deally have smple rules and robust desgn. The desgner should also to take nto account that real-world advertsers adopt slowly to rules changes. In partcular, ths seems to be an obstacle for the ntroducton of VCG nstead of GSP n VCG the bdders have to submt hgher bds and can be reluctant to do so. Therefore, the aucton that takes nto account CTR externaltes should be an adjusted verson of GSP. In a locally envy-free equlbrum of GSP, bdder s prce per clck depends on the bds of the bdders below hm (ths allows to generate the equlbrum va the generalzed Englsh aucton). In the externalty-recognzng verson of GSP prce per clck would depend on all bds. References Aseff, J. & Chade, H. (2008), An optmal aucton wth dentty-dependent externaltes, The RAND Journal of Economcs 39(3), pp Athey, S. & Ellson, G. (2011), Poston auctons wth consumer search, The Quarterly Journal of Economcs 126(3), pp Athey, S. & Nekpelov, D. (2010), A structural model of sponsored search advertsng auctons, n Sxth Ad Auctons Workshop. Edelman, B., Ostrovsky, M. & Schwarz, M. (2007), Internet advertsng and the generalzed second-prce aucton: Sellng bllons of dollars worth of keywords, Amercan Economc Revew 97(1), Edelman, B. & Schwarz, M. (2010), Optmal aucton desgn and equlbrum selecton n sponsored search auctons, Amercan Economc Revew 100(2), Ghosh, A. & Sayed, A. (2010), Expressve auctons for externaltes n onlne advertsng, n Proceedngs of the 19th nternatonal conference on World wde web, WWW 10, ACM, New York, NY, USA, pp

20 Jehel, P., Moldovanu, B. & Stacchett, E. (1996), How (not) to sell nuclear weapons, The Amercan Economc Revew 86(4), pp Jerath, K. & Sayed, A. (2011), Exclusve dsplay n sponsored search advertsng. Carnege Mellon Unversty, mmeo. Jezorsk, P. & Segal, I. (2010), What makes them clck: Emprcal analyss of consumer demand for search advertsng. the Johns Hopkns Unversty, Department of Economcs, Workng Paper No Krshna, V. (2009), Aucton theory, Academc press. Mlgrom, P. R. & Weber, R. J. (1982), A theory of auctons and compettve bddng, Econometrca 50(5), pp Muthukrshnan, S. (2009), Bddng on confguratons n nternet ad auctons, n Proceedngs of the 15th Annual Internatonal Conference on Computng and Combnatorcs, COCOON 09, Sprnger-Verlag, Berln, Hedelberg, pp Myerson, R. B. (1981), Optmal aucton desgn, Mathematcs of Operatons Research 6(1), pp Ostrovsky, M. & Schwarz, M. (2009), Reserve prces n nternet advertsng auctons: A feld experment. Segal, I. (1999), Contractng wth externaltes, The Quarterly Journal of Economcs 114(2), pp Ülkü, L. (2009), Optmal combnatoral mechansm desgn, Economc Theory pp Varan, H. R. (2007), Poston auctons, Internatonal Journal of Industral Organzaton 25(6), Vckrey, W. (1961), Counterspeculaton, auctons, and compettve sealed tenders, The Journal of Fnance 16(1), pp

21 (a) β 0 (b) 1 <β<0 Fgure 1: Optmal allocaton rule. 1 ndcates that n ths range of varables one poston has to be allocated, 2 for two postons and 0 for no postons allocated.

22 (a) ψ(y 1 ) 0 (b) ψ(y 1 ) < 0 Fgure 2: Optmal allocaton rule, β>0. The frst row ndcates how to allocate (exclusve dsplay, multple dsplay or no dsplay); the second row gves the values of Q (z,z ), the realzed CTR for bdder ; the captons below the axs defne the thresholds between the allocaton cases, Y 1 stands for the hghest competng bd.

23 Vrtual valuaton s2.01 s2.1 s2.5 s2 1 X n (a) Vrtual valuaton X n 1 (b) Rule for exclusve dsplay, compare wth Fgure 3 Fgure 5: Log-normal dstrbuton for a range of σ 2,μ =0

24 Revenue, unt unfrom dstrbuton, n=2 sales of ED sales of MD Opt mech (ED+MD) GSP wth r Probablty of allocatons, unt unfrom dstrbuton, n=2 ED MD ND ED n GSP Revenue 0.4 Probablty β β (a) n =2 (b) n = Revenue, unt unfrom dstrbuton, n=4 sales of ED sales of MD Opt mech (ED+MD) GSP wth r Probablty of allocatons, unt unfrom dstrbuton, n=4 ED MD ND ED n GSP Revenue Probablty β (c) n = β (d) n =4 Fgure 6: Expected revenue and probabltes of allocatons, d U[0,1] values per clck, 2 slots, α =.77. The exclusve dsplay CTR bonus β s along the x-axs. The left column shows graphs of revenue for the number of bdders n =2, 4, 10. Sales of ED s the part of the optmal mechansm revenue generated by sales of ED; Sales of MD s that of MD; Opt mech s the sum of these two, that s the total revenue of the optmal mechansm; GSP wth r s revenue generated by GSP aucton wth r = ψ 1 (0). The rght column depcts graphs of correspondng probabltes. ED s the probablty of ED allocaton n the optmal mechansm; ED n GSP s that n GSP; MD s the probablty of MD allocaton n the optmal mechansm.

25 1.5 Revenue, unt unfrom dstrbuton, n=10 sales of ED sales of MD Opt mech (ED+MD) GSP wth r Probablty of allocatons, unt unfrom dstrbuton, n=10 ED MD ND ED n GSP Revenue Probablty β (e) n = β (f) n =10 Fgure 6: ctd.