Optimal Revenue-Sharing Double Auctions with Applications to Ad Exchanges

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1 Optiml Revenue-Shring Doule Auctions with Applictions to Ad Exchnges Rento Gomes Toulouse School of Economics Vh Mirrokni Google Reserch, ew York ABSTRACT E-commerce we-sites such s Ey s well s dvertising exchnges AdX such s DouleClick s, RightMedi, or AdEC work s intermediries who sell items e.g. pge-views on ehlf of seller e.g. pulisher to uyers on the opposite side of the mrket e.g., dvertisers. These pltforms often use fixed-percentge shring schemes, ccording to which i the pltform runs n uction mongst uyers, nd ii gives the seller constnt-frction e.g., 80% of the uction proceeds. In these settings, the pltform fces symmetric informtion regrding oth the vlutions of uyers for the item s in stndrd uction environment s well s out the seller s opportunity cost of selling the item. Moreover, pltforms often fce intense competition from similr mrket plces, nd such competition is likely to fvor uction rules tht secure high pyoffs to sellers. In such n environment, wht selling mechnism should pltforms employ? Our gol in this pper is to study optiml mechnism design in settings plgued y competition nd two-sided symmetric informtion, nd identify conditions under which the current prctice of employing constnt cuts is indeed optiml. In prticulr, we first show tht for lrge clss of competition gmes, pltforms ehve in equilirium s if they mximize convex comintion of seller s pyoffs nd pltform s revenue, with weight α on the seller s pyoffs which is proxy for the intensity of competition in the mrket. We generlize the nlysis of Myerson nd Stterthwite 1983, nd derive the optiml direct-reveltion mechnism for ech α. As expected, the optiml mechnism pplies reserve price which is decresing in α. ext, we present n indirect implementtion sed on shring schemes. We show tht constnt cuts re optiml if nd only if the opportunity cost of the seller hs power-form distriution, nd derive simple formul for computing the optiml constnt cut s function of the sellers distriution of opportunity costs, nd the mrket competition proxy α. Finlly, for completeness, we study the cse of seller s optiml uction with fixed Copyright is held y the Interntionl World Wide We Conference Committee IW3C. IW3C reserves the right to provide hyperlink to the uthor s site if the Mteril is used in electronic medi. WWW 14, April 7 11, 014, Seoul, Kore. ACM /14/04. profit for the pltform, nd derive the optiml direct nd indirect implementtions in this setting. Ctegories nd Suject Descriptors H.m [Informtion Systems]: Miscellneous; F.m [Theory of Computtion]: Miscellneous 1. ITRODUCTIO E-commerce we-sites such s Ey s well s d exchnges AdX such s Yhoo s RightMedi, Microsoft s AdEC, or Google s DouleClick Ad Exchnge sell items e.g. pgeviews on ehlf of seller e.g. pulisher y running n uction mongst certin numer of uyers e.g., dvertisers. In most cses, the e-commerce site or AdX pltform then follow fixed-percentge shring scheme, ccording to which the seller is given constnt cut from the uction revenue, nd the pltform keeps the remining proceeds from the uction. One typicl mechnism employed y d exchnges works s follows: i the pulisher declres reserve price p to e used in the uction, ii AdX runs secondprice uction with reserve price 1.5p, nd collects 0% of the finl revenue from the uction. The pulisher, in turn, gets the remining 80% of the revenue from the uction, mking sure tht it gets t lest its declred reserve price of p. In these settings, the pltform fces symmetric informtion regrding oth the vlutions of uyers for the item s in stndrd uction environment s well s out the seller s opportunity cost of selling the good. For exmple, in the cse of d exchnges, the opportunity cost for the pulisher seller is the profit tht the pulisher could otin y sending the pge-view to competing d exchnges or y using the pge-view to fulfill some gurnteed contrct previously signed with dvertisers. In such n environment, the optiml direct-reveltion mechnism following the oel-winning work y Myerson is such tht pyments to the seller depend on rther complex wy on the distriution of vlutions y uyers, nd the distriution of the opportunity cost of the seller. This contrsts with the current prctice of most Internet pltforms, which dopt simple fixed percentge or fixed fee models of revenue shre. Moreover, in prctice, pltforms often fce intense competition from similr mrket plces. This is specilly true in the cse of dvertising exchnges tht ply in highly competitive mrket plce with mny other d networks nd d exchnges. Such issues in these prcticl settings rise the following questions: wht selling mechnism should pltforms employ? In prticulr, is such mechnism sed on 19

2 fixed-percentge shring scheme? If so, wht is the optiml fixed-percentge shring scheme? Finlly, how re shring schemes ffected y the degree of mrket competition? Our gol in this pper is to study optiml mechnism design in settings plgued y competition nd two-sided symmetric informtion, nd identify conditions under which the current prctice of employing constnt cuts is indeed optiml. 1.1 Our Contriutions. Equilirium Strtegy for Competing Ad Exchnges. Competition mong d exchnges is likely to fvor uction rules tht secure high pyoffs to sellers, so tht sellers continuously trde through the d exchnge. Intuitively, competition hs the effect of mking d exchnges internlize t lest prtilly the sellers pyoffs in their ojective functions. In this pper, we first show tht for lrge clss of competition gmes etween intermediries, the equilirium est responses of ech intermediry mximizes convex comintion of seller s pyoffs nd intermediry s revenue. The weight ssigned in the seller s profit, denoted y α, is proxy for the degree of competition in the mrket - nd depends on the primitives of the competition gme etween pltforms. We refer to this mximiztion prolem s the α-optiml prolem. Chrcterizing α-optiml Mechnisms. As first step towrd understnding the optiml doule uction in the presence of competing exchnges, we derive the optiml directreveltion mechnism ssocited to ech α-optiml prolem. As expected, the α-optiml mechnism lloctes the good to the uyer with the highest vlution if nd only if the highest vlution is greter thn some reserve price. This reserve price is decresing in α: s the weight on the seller profits increse, the pltform withholds the good less often, nd mkes greter trnsfers to the seller. In prticulr, when α 1, the α-optiml mechnism coincides with the mechnism tht the seller would run if he owned the pltform s in Myerson To understnd this result, notice tht the pltform s ojective function puts wekly higher weight on the seller s pyoff thn on its own profits. Becuse of the seller s incentive constrints, y giving up one dollr in profits, the pltform cn dd more thn on dollr to the seller s pyoff. As consequence, the pltform mximizes its ojective y following the seller s optiml mechnism tht is, the Myerson uction. On the other hnd, when α < 1, the pltform puts strictly more weight on its profits thn on the seller s pyoffs. As consequence, it is willing to introduce distortions on the seller side of the mrket therefore reducing the seller s pyoffs in order to generte higher profits. The lloctive distortions generted y the seller s informtionl rents re proportionl to 1 α 1 α, which is decresing in α. As expected, when α = 0, the α-optiml mechnism coincides with the pltform profitmximizing mechnism derived in [8]. As second step towrd understnding α-optiml mechnisms, we study how to indirectly implement the α-optiml direct-reveltion mechnism y mens of shring schemes. Such shring schemes work in the following mnner: i the seller sumits reserve price r to the pltform, ii the pltform runs second-price uction mong sellers with reserve price r, iii the pltform gives frction θ α r of the uction proceeds to the seller, nd keeps the rest. Such shring schemes in which the frction θ α r is function of the reported reserve price implement the α-optiml directreveltion mechnism for ny distriutions of the seller opportunity costs nd the uyers vlutions tht stisfy mild regulrity conditions. Constnt Cuts. As discussed erlier, in prctice, pltforms use constnt shring schemes, in which the frction or cut ssigned to the seller does not depend on his reported reserve price. ext, we turn to nswering the following question: Under wht conditions, cn constnt shring scheme implement the α-optiml direct-reveltion mechnism? Interestingly, we show tht constnt cuts re optiml if nd only if the distriution of seller opportunity cost, denoted y G, hs the power form v0 k Gv 0 =, where k > 0 nd is the superior limit of the support of the seller s opportunity cost distriution. The result ove cn e understood in the light of monopolistic price theory. Intuitively, power-form distriution functions hve constnt price elsticity of supply which mesures how mny more percentge points of inventory sellers re willing to offer for one percentge increse in expected revenue. mely, power-form distriution with prmeter k hs price elsticity of supply equl to k for ll opportunity costs in the support. As it turns out, constnt shring schemes re optiml provided tht the price elsticity of supply is constnt. In prticulr, we derive n esy-to-implement formul tht reltes the price elsticity of supply k nd the degree of competition α to the constnt seller s shre: θ α,k k = k + 1 α. 1 α Dt Anlysis. As the price elsticity k increses, the seller s revenue shre goes up. Intuitively, s the distriution of the seller s opportunity costs ecome concentrted on high vlues, the pltform hs to increse the seller s revenue shre to mke sure tht the seller is willing to prticipte in the trding mechnism with high enough proility. In turn, for fixed k, the seller revenue shre increses s the degree of competition α increses. Figure 1 represents the α-optiml constnt shring rule s function of α for the following price elsticities of supply: k = 0.4, 0.5, 0.6, 1, 1.5. Furthermore, we explore wy of djusting the design of the AdX mechnism to incorporte the ove chrcteriztion, nd pply it on rel dt sets. In prticulr, we use dt on the distriution of pulishers opportunity costs to estimte i the polynomil distriution tht etter pproximtes its rndom generting process, nd ii the ojective function pursued y intermediries s function of the revenue shre given to pulishers. Our nlysis suggests tht revenue shre of 80% which is the current prctice of Google s AdEx is consistent with ssigning weight of 47% to pulishers revenues on the pltform s ojective. See detils in Section 4.1. Constnt Fee. Finlly, we nlyze n lterntive model of competition etween uction pltforms. mely, we ssume tht competition drives down expected profits to equl the qusi-fixed of cost of intermediting ech pulisher s trns- 0

3 Figure 1: The α-optiml revenue shring rules s function of α when the price elsticity of supply equls k = 0.4, 0.5, 0.6, 1 nd 1.5 from lower to higher curves respectively. ction. We give chrcteriztion of optiml doule uctions in this setting, nd present n indirect implementtion in terms of shring schemes. 1. Relted Work Doule Auctions. Optiml doule uctions hve een widely studied in the economics literture nd more recently CS literture. Following the oel-prize winning work of [7], [8] generlized the design of optiml uctions for two-sided settings, nd chrcterized the pltform-optiml doule uction. This rticle generlizes the nlysis of [8] y i studying settings in which the pltform s ojective function is convex comintion of the seller s profit nd the pltform s profit, ii nlyzing indirect implementtion y shring schemes, nd iii providing necessry nd sufficient condition under which constnt shring schemes indirectly implement the optiml mechnism. More recently, [1] studied multidimensionl vrints of the Byesin optiml doule uctions nd present polynomiltime pproximtion lgorithms for the prolem. For the non-byesin prior-free setting, [] studied revenue-mximizing doule uctions when the uctioneer hs no prior knowledge out ids. Unlike this pper, ll the ove work focus on the pltform-optiml uctions, nd do not consider other ojective functions or fixed-percentge or fixed-cost uctions. Competing Mechnisms. There is lrge literture in economics studying competing mechnism design. The seminl work of [6] llows principls to post ritrry trding mechnisms, ut ssumes tht principls devitions do not ccount for their effects on the pyoffs of gents who visit other principls this is known s the lrge mrket ssumption. In this setting, McAfee shows tht there exists n equilirium in which principls post second-price uctions. Dispensing with the lrge mrket ssumption, ut ssuming tht principls re only llowed to post second-price uctions, [10] derive symptotic reserve prices when the numer of principls grows lrge. Allowing for oth ritrry mechnisms nd lso dispensing with the lrge mrkets ssumption, [9] provides sufficient condition for the equilirium outcome to e qusiefficient in the sense tht inefficiencies re due to exclusion, ut not to mislloctions. Under this condition, sellers employ second-price uctions in equilirium. In contrst to the ove cited ppers, in our model, pulishers choose which pltform to ptronize efore knowing the relized opportunity cost of ech impression. This ssumption cptures the notion tht the pulishers choice of pltform is often long-term choice, while the opportunity cost of ech impression is short-term contingency. Other E-commerce Applictions. Other thn d serving systems, the results of this pper pply to vrious online nd offline retilers nd e-commerce wesites like Amzon nd Ey. For exmple, Ey pplies similr revenueshring uctions to the ones studied in this pper when it serves s roker etween set of uyers nd seller. Roughly speking, Ey tkes 9% cut on ech sle, referred to s finl vlue fee, nd lso fixed fee for listing n item, referred to s n insertion fee 1. They lso pply convex cost function for the fixed fee s the numer of listings, nd mximum of $50 for the finl vlue fee. A very recent pper y [4] studies EBy s doule uction prolem, ut their setting is different from this pper s they consider multiple sellers nd one uyer, nd explore pproximtely optiml pricing schemes for this setting.. MODEL Before studying competition, we will introduce the model s nottion y considering the prolem of single pltform populted y uyers indexed y i = 1,..., nd one seller whose index is 0. The seller cn otin price v 0 for his good y trding it outside of the pltform for exmple, y using the impression to fulfill gurnteed contrcts. The vlue of v 0 is privte informtion of the seller. From the perspective of the pltform, v 0 is drw from the distriution G with support [, ] nd density g. Ech uyer i hs vlution v i for the good offered y the seller. The vlution of ech uyer is his privte informtion. From the perspective of the pltform, the vlution of ech uyer is n independent drw from the distriution F with support [, ] nd density f. As stndrd in mechnism design, we ssume tht F possesses wekly incresing hzrd rte, nd tht G possesses wekly decresing reverse hzrd rte. Denote y v v 0, v 1,..., v the profile of types including the uyer nd ll sellers. Following the Reveltion Principle, we will confine ttention to truthful direct-reveltion mechnisms. A direct-reveltion mechnism consists of i n lloction rule q iv i=0,1,..., tht mps v into proilities tht ech gent is ssigned the good q 0v stnds for the proility tht the seller keeps the good, nd ii pyment rule p iv i=0,1,..., tht mps v into pyments for ech uyer i nd the seller. We denote mechnism y M = q iv i=0,1,...,, p iv i=0,1,...,. 1 For detils see nd the chrging tle t fees.html#if_uction. 1

4 To recp, direct-reveltion mechnisms disply the follow extensive form: 1. The seller is sked to report his opportunity cost v 0 nd uyers re sked to report their vlutions v i,. The pltform delivers the good ccording to the lloction rule q iv i=0,1,...,, 3. The pltform chrges p iv to ech uyer nd trnsfers p 0v to the seller. An lloction is fesile if i=0 qiv = 1 for ll v. Denote y Q iv i E v i [q iv] the interim proility tht gent i is ssigned the good nd y P iv i E v i [p iv] the interim pyment of ech gent i. A mechnism is individully rtionlir nd incentive comptileic for uyers if nd only if for ll i 1,..., U iv = Q iv v P iv mx Q iˆv v P iˆv, 0. for ll v, ˆv [, ]. A mechnism is individully rtionl nd incentive comptile for the seller if nd only if U 0v = Q 0v v P 0v mx Q 0ˆv v P 0ˆv, v. for ll v, ˆv [, ]. It is stndrd to show tht mechnism is incentive comptile if nd only if for ll i 0, 1,...,, Q i is wekly incresing nd U iv = U i + v Q iˆvdˆv = U i v Q iˆvdˆv. 1 The pltform s expected profits from mechnism M re then ΠM P ivdf v + P 0vdGv. The seller s ex-nte expected pyoff from mechnism M is ΓM U 0vdGv. As we will show in Section.1, the optiml est response strtegy for the pltform is to solve the following α-optiml prolem: The α-optiml prolem denoted P α is to choose q iv, p iv i=0,1,..., to P α : mx α k ΓM + 1 α k ΠM, where α [0, 1], suject to IR, IC, the fesiility constrint, nd the pltform s rek-even constrint Π 0, 3 which sttes tht the pltform mkes non-negtive profits. As will e discussed in Section.1, the prmeter α cptures the intensity of competition in the mrket: in more competitive mrkets, the pltform should ssign higher weight in its ojective function to the sellers pyoffs. We refer to the mechnism tht solves prolem P α s the α-optiml mechnism, nd denote it y q α i v, p α i v i=0,1,...,..1 Pltforms Competition nd Best Responses In this section, we study some generl properties of competition gmes etween etween K pltforms, indexed y k 1,..., K. It is often the cse tht pulishers often hve ex-nte preferences over which pltform to join. These ex-nte preferences cn e conveniently summrized y vector h 1,..., h K, where ech component h k descries the pure utility gin or loss tht pulishers ssocite with joining pltform k. Consider the following extensive-form competition gme etween K pltforms. 1. Ech pltform k simultneously post mechnism M k consisting of n lloction rule q k i v i=0,1,..., nd pyment rule p k i v to ech uyer nd trnsfers p k 0v to the seller.. Before oserving her opportunity costs v 0, ut knowing her horizontl preferences h 1,..., h K, the seller chooses the pltform k tht mximizes her ex-nte utility ΓM k + h k. All uyers join ll pltforms. 3. The uyers nd the seller ply the mechnism proposed t stge 1 y the pltform selected y the seller. otice tht the clss of gmes descried ove ccommodtes ny form of horizontl differentition l Hotelling [3]. Consider the set I of ll pirs Π, Γ of expected pyoffs to the pltform nd seller induced y ll implementle mechnisms. The Preto frontier of I, denoted y E, consists of ll pirs Π, Γ such tht there exists no implementle mechnism tht leds to expected pyoffs Π, Γ such tht Π > Π nd Γ Γ, or Π Π nd Γ > Γ. If pir Π, Γ elongs to the Preto frontier E, we sy tht Π, Γ is Preto efficient. It is importnt to note tht the set of implementle pyoffs I is convex set. To see why, tke two implementle mechnisms M nd M tht led to expected pyoffs Π, Γ nd Π, Γ, respectively. ow consider the rndom mechnism M β M + 1 β M. It follows directly from the incentive nd rtionlity constrints tht the mechnism M is implementle. Moreover, the expected pyoffs of mechnism M re β Π + 1 β Π, β Γ + 1 β Γ. The previous oservtion is key to prove the following result. Proposition 1. Let M 1,..., M K e sh equilirium of the competition gme descried ove, nd let Π k, Γ k e the expected pyoff of mechnism M k. Then there exists vector α 1,..., α K [0, 1] K such tht ech mechnism M k solves P αk : mx M α k ΓM + 1 α k ΠM. Moreover, if the equilirium is symmetric, α k = α for ll k. The proof of this proposition is n immedite ppliction of the Support Hyperplne Theorem [5] together with the fct tht, in ny equilirium, the pir of expected pyoffs is Preto efficient. The result ove formlizes the intuition

5 tht competition mong pltforms hs the effect of mking ech pltform internlize t lest prtilly the sellers pyoffs in its ojective function. Oviously, the vector of weights α 1,..., α K is endogenous to the equilirium, nd crucilly depends on the horizontl preferences of the seller over pltforms. Ech α k cptures the equilirium intensity of competition fced y pltform k: in more competitive mrkets, ech pltform should ssign higher weight in its ojective function to the sellers pyoffs. Proposition 1 llows us to strct from the detils of the competition gme plyed y firms, nd derive properties of equilirium mechnisms of ny competition gme. Specificlly, in light of this proposition, we cn chrcterize the equilirium ply of ech firm y studying its respective P αk - prolem. As stted erlier, for some ritrry α, we refer to the mechnism tht solves prolem P α s the α-optiml mechnism, nd denote it y q α i v, p α i v i=0,1,...,. Oviously, when α = 0, the pltform s prolem is tht of mximizing profits s in Myerson nd Stterthwite In turn, when α = 1, the pltform mximizes the seller s pyoff suject to ttining wekly positive profits which, s will e cler soon, is equivlent to the prolem considered in Myerson For α 0, 1, the pltform ehves s if it were prtilly owned y sellers. 3. THE α-optimal MECHAISM In this section, we derive the direct-reveltion mechnism tht solves ech prolem P α. Proposition. The α-optiml Direct-Reveltion Mechnism Let us choose indexes such tht v i = mx j 1,..., v j. The α-optiml direct-reveltion mechnism is descried elow. 1. Let α 1. Then qα i v i = 1 if v i 1 F vi fv i v 0 1 α 1 α Gv0 gv 0 0. Otherwise, no sle occurs: q α 0 v = 1.. Let α > 1. Then qα i v i = 1 if v i 1 F vi fv i v 0 0. Otherwise, no sle occurs: q α 0 v = 1. In turn, the seller s pyoff cn e written s [,] +1 1 U 0 U 0 U 0v 0dGv 0 = v 0 Q 0ˆvgv 0dˆvdv 0 = Gv 0 Q 0v 0dv 0 = U 0 Gv 0 q iv fv i dv i dv 0 = The pltform ojective is then U 0 + Eṽ 0+ q iv Gv0 df v idgv 0. 5 gv 0 [,] +1 q iv [ 1 α v i 1 F vi ] v 0 1 α Gv0 fv i gv 0 df v idgv 0 1 α U i + α Eṽ 0 + α 1 U 0. 6 If α 1, the pltform s ojective is decresing in Ui nd U 0. This implies tht, t the α-optimum, the individul rtionlity constrints hve to ind for every uyer of type nd every seller of type : U i = 0 nd U 0 =. Mximizing the integrl ove point-wise leds to the following ng-ng solution: qi α v i = 1 for i 1,..., if v i 1 F vi fv i v 0 1 α 1 α Gv0 mx v j 1 F vj v 0 1 α j 1,..., fv j 1 α Gv0 gv, 0 0 gv = 0, 7 nd q0 α v = 1 if there is no j 1,..., tht stisfies the equlity ove. In turn, if α > 1, the pltform s ojective is decresing in U i nd incresing in U 0. This implies tht, t the α-optimum, U i = 0, nd U 0 is set to stisfy the rekeven constrint with equlity: Proof. Applying the envelope formul 1, the pltform s profits cn e rewritten s [,] +1 q iv v i 1 F vi v 0 Gv0 fv i gv 0 df v idgv 0 U i U 0. 4 v i 1 F vi fv i [,] +1 U 0 = q iv v 0 Gv0 gv 0 df v idgv 0. 3

6 Plugging 8 into the ojective 6 leds to [ α v i 1 F vi ] v 0 fv i [,] +1 q iv df v idgv 0 +α Eṽ 0. 8 Mximizing the integrl ove point-wise leds to the following ng-ng solution: qi α v i = 1 for i 1,..., if v i 1 F vi fv i v 0 = mx v j 1 F vj v 0, 0, j 1,..., fv j nd q α 0 v = 1 if there is no j 1,..., tht stisfies the equlity ove. Becuse F hs n incresing hzrd rte nd G hs decresing reverse hzrd rte, it follows tht q α i v i, v i is wekly incresing in v i for every α [0, 1] i.e., the α- optiml mechnism is implementle. Q.E.D. When α 1, the α-optiml mechnism coincides with the mechnism tht the seller would run if he owned the pltform derived in Myerson To understnd this result, notice tht the pltform s ojective function puts wekly higher weight on the seller s pyoff thn in its own profits. Becuse of the seller s incentive constrints, y giving up one dollr in profits, the pltform cn dd more thn on dollr to the seller s pyoff. As consequence, the pltform mximizes its ojective y following the seller s optiml mechnism tht is, the Myerson uction. When α < 1, the pltform puts strictly more weight on its profits thn on the seller s pyoffs. As consequence, it is willing to introduce distortions on the seller side of the mrket therefore reducing the seller s pyoffs in order to generte higher profits. The lloctive distortions generted y the seller s informtionl rents re cptured y the term 1 α 1 α Gv0 gv 0, which is decresing in α. As expected, when α = 0, the α-optiml mechnism coincides with the pltform profitmximizing mechnism derived in Myerson in Stterthwite IDIRECT IMPLEMETATIO: SHAR- IG RULES AD COSTAT CUTS The direct-reveltion mechnism derived ove offers little insight on the ctul prctice of d-exchnges, s the pyment functions tht implement the optiml lloctions re not very intuitive. In order to shed light on the ctul mechnisms used y dvertising pltforms, we will now study one nturl nd widely dopted indirect implementtion: Revenue-shring mechnisms. They work s follows: 1. The seller is sked to report reserve price.. The pltform runs second-price uction with reserve price. 3. The pltform gives to the seller frction θ of the proceeds of the uction we cll θ the shring rule. Before proceeding, let us define the function 1 α if α 1 hα = 1 α 0 if α > 1. ote tht the function hα is continuous t α = 1. The next proposition shows how to define the shring rule θ α tht implements the α-optiml mechnism identified in Proposition. To this end, let us define the function r α such tht r α v 0 1 F rα v 0 fr α v 0 nd the inverse v α 0 r ccording to r 1 F r fr = v 0 + hα Gv0 gv 0, = v α 0 r + hα Gvα 0 r gv α 0 r. 1 F r Define the minimum reserve price r ccording to r = fr if this eqution does not dmit solution in [, ], set r =. Denote y Rr, the expected revenue of second-price uction with idders nd reserve price r: Rr, r v 1 F v fv fv F 1 vdv. Proposition 3. α-optiml Revenue-Shring Mechnism Consider the revenue-shring mechnism descried y the shring rule v θ α 0 α F F rˆvdˆv v0 = α for [r, ] R, 9 nd θ α = 0 for < r. This mechnism indirectly implements the α-optiml direct-reveltion mechnism. Proof. By the Envelope formul 1, incentive comptiility implies tht the expected pyment to seller with vlue v 0 under the α-optiml mechnism is given y v 0 F r α v 0 v 0 F r α ˆvdˆv. 10 Rther thn sking sellers to report v 0, the revenue-shring mechnism considered here requires tht sellers report some reserve price r. Let us posit tht the seller equilirium reserve price strtegy is given y r α v 0. We will derive the expected pyments induced y incentive comptiility under this strtegy, nd then rgue tht sumitting reserve prices ccording to r α v 0 is indeed profit-mximizing for the seller. Becuse F hs n incresing hzrd rte, G hs decresing reverse hzrd rte, nd h 0, we know tht the function r α v 0 is strictly incresing. Therefore we cn rewrite the expected pyments of seller with vlue v 0 s implied y IC in terms of his sumitted reserve price = r α v 0: v α 0 F v α 0 F r α ˆvdˆv. We will define the shring rule θ α such tht the expected revenue tht the seller otins from the second-price uction run t stge equls the expected pyments implied y incentive comptiility. This is equivlent to θ α R, = v α 0 F v α 0 F r α ˆvdˆv. 4

7 After rerrnging, we get the formul 9. We will now rgue tht sumitting reserve prices ccording to r α v 0 is profit-mximizing for the seller. To see why, notice tht the seller prolem cn e written s mx v 0 F + θ α R,. By construction, the selection r α v 0 stisfies the envelope formul 1. Becuse the seller s ojective stisfies strictly incresing differences, we cn use the Constrint Simplifiction Theorem Milgrom 004, pge 105 to conclude tht r α v 0 mximizes the seller s pyoff mong ll reserve prices in the rnge [r, ] if nd only if r α is wekly incresing. As rgued efore, ecuse F hs n incresing hzrd rte, G hs decresing reverse hzrd rte, nd h 0, we know tht the function r α v 0 is indeed strictly incresing. Becuse it is clerly suoptiml to sumit reserve price lower thn r since θ α = 0 for < r or greter thn, we conclude tht sumitting reserve prices ccording to r α v 0 is profit-mximizing for the seller. Q.E.D. An importnt clss of revenue-shring mechnisms is descried y constnt shring rules, where θ α is constnt cross ll possile reserve prices. Such constnt-shring schemes re firly simple to descrie nd re widely dopted in prctice y Ad Exchnges nd internet uction sites. The next proposition provides necessry nd sufficient condition for the α-optiml mechnism to employ constnt shring rule. Proposition 4. α-optiml Constnt Shring Rule The α-optiml direct-reveltion mechnism cn e implemented y revenue-shring mechnism tht employs constnt shring rule if nd only if v0 k Gv 0 =, where k > 0. In this cse, θ α r = k k + hα nd θ α = 0 for < r. for ll r [r, ], Before going through the proof, we present the following exmple confirming the ove result for the cse of uniform distriution nd α = 0, i.e,. the pltform-optiml mechnism. Interestingly, under uniform distriution, the pltform-optiml mechnism cn e indirectly implemented y fifty-fifty revenue shring rule. Exmple 1. Assume α = 1, nd F = G U[0, ], in which cse rv 0 = + v0 nd r 1 =. Then the seller s revenue under some shring rule λ which priori is not necessrily constnt is given y λ v 1 F v fv F 1 vdv = fv = λ λ v 1 v 1 dv v v 1 dv = λ [ ] v v = λ [ In turn, incentive comptiility nd the envelope formul deliver nother expression for the seller s revenue: = = = + +1 r 1 F r 1 + ṽ0 dṽ 0 [ ṽ0 F rˆvdˆv dˆv ]. ] = [ ] = [ Compring 11 nd 1 leds to λ = 1 for ll [, ] This shows tht the only revenue shring uction tht implements the pltform-optiml mechnism shres the revenue from the uction evenly etween the pltform nd the seller The proof elow estlishes tht constnt shring rules implement the α-optiml mechnisms if nd only if the distriution of the seller s outside options hs power-form distriution. Proof of Proposition 4. rewritten s θ α r Rr, = v α 0 r F r ]. The expression 9 cn e v α 0 r F r α ˆvdˆv. Differentiting with respect to r leds to the following liner differentil eqution. θ α r Rr, + θ α r R r, = r v0 α r F 1 r fr. Therefore, θ α r = 0 for ll r if nd only if θ α r R r r, = vα 0 r F 1 r fr. ote tht t the α-optiml mechnism = r 1 F r fr = v0 α r + hα Gvα 0 r gv0 αr R r, r fr F 1 r fr F 1 r. 5

8 Therefore, θ α r = 0 for ll r if nd only if v0 α r + hα Gvα 0 r gv0 αr θ α r = v0 α r, which cn e rewritten s 1 θ α r Gv0 α r = hα θ α r v0 αr gvα 0 r. The function v α 0 r is strictly incresing in r. Therefore, the expression ove is constnt for every r if nd only if xgx Gx is constnt, wht leds to Gv 0 = v 0 k. Finlly, note tht if Gv 0 = v 0 k, then 1 θ α r θ α r = hα k, wht leds to θ α r = k for ll r [r, ]. Q.E.D. k+hα The result ove cn e understood in the light of monopsonistic price theory. Intuitively, polynomil distriution functions hve constnt price elsticity of supply which mesures how mny more percentge points of inventory sellers re willing to offer for one percentge increse in expected revenue. mely, distriution of the form Gv 0 = v0 k hs price elsticity of supply equl to k for ll opportunity costs v 0. As it turns out, constnt shring schemes re optiml provided tht the price elsticity of supply is constnt. As the price elsticity k increses, the seller s revenue shre goes up. Intuitively, s the distriution of the seller s opportunity costs ecome concentrted on high vlues, the pltform hs to increse the seller s revenue shre to mke sure tht the seller is willing to prticipte in the trding mechnism with high enough proility. In turn, for fixed k, the seller revenue shre increses s the weight α on the seller s pyoffs increses. The next section pplies the results ove to dt in order to estimte the ojective function tht rtionlizes constntshring schemes commonly used in prctice. 4.1 Dt Anlysis: Estimting Pltform s Ojective Function In this section we ring the theory developed ove to dt. We pply the results ove to nswer the following questions: i Given n ojective function in mind for the pltform, wht is the est constnt cut shring rule to use? nd ii Wht ojective function rtionlizes the revenueshring rule used y n Ad Exchnge AdX pltform, which uses constnt shring rule tht ssigns for exmple 80% of the uction revenue to sellers, nd 0% to the pltform? To nswer the ove questions, min methodologicl chllenge is the estimtion of the distriution of opportunity costs of pulishers s dt on the vlue of gurnteed contrcts signed y pulishers is not ville. In order to overcome this difficulty, we will mke the ssumption tht the distriution of opportunity costs of pulishers coincides with the distriution of revenue in the Ad Exchnge. This ssumption cptures the ide tht pulishers should e indifferent etween selling impressions in the Ad Exchnge nd selling impressions through gurnteed contrcts s otherwise, pulishers re expected to sell ll of their inventory through one of these chnnels. Under this ssumption, we cn use the following procedure to estimte the pltform-optiml mechnism: get the empiricl distriution of revenues otined y given pulisher on prticulr d slot, remove the oservtion where AdX pltform ws not le to sell the pulisher s d slot, estimte the power distriution tht est fits the empiricl revenue distriution, with coefficient k use this estimte to pin down the weight tht the AdX pltform gives to sellers pyoffs, denoted α, y solving the eqution 0.8 = k k + hα. In order to estimte the polynomil distriution tht est fits the dt, we use the mximum likelihood method. This method consists in choosing the distriution prmeter k tht mximizes the theoreticl proility tht the relized smple ws generted y power distriution with prmeter k. We pplied the technique descried ove to five d slots, nd otined the following results. #os #sles id support k α 710, ,509 [0,0.35] % 4,037 37,15 [0,7.99] % 9,813 1,919 [0,1] % ,58 [0,7.07] % 6,059 3,616 [0,54.38] % Our estimte of α is roughly constnt in the five dt sets considered. Interestingly, our estimte suggests tht n AdX pltform tht uses 0% s constnt cut puts roughly 46% of weight on the seller s pyoffs when designing the uction rules of AdX. Also given desirle ojective function, nd k for ech pulisher, the d exchnge cn decide out the declred fixed percentge if it needs to negotite them. 5. SELLER-OPTIMAL MECHAISM WITH A MIIMUM PROFIT CODITIO In the previous sections, we studied the mechnism design prolem of n Ad Exchnge tht wishes to mximize convex comintions of its own profits nd the seller s pyoffs. This formultion cptures in reduced form competition etween Ad Exchnges: s more Ad Exchnges re present in the mrket, the higher is the weight ssigned to the sellers pyoff in the Ad Exchnges ojective function. In this section, we follow n lterntive pproch to study competition etween Ad Exchnges. mely, we ssume tht Ad Exchnges mximize the sellers pyoff suject to ttining miniml profit level which should e used to cover operting costs, for exmple. Such reduced-form formultion is the outcome of Bertrnd competition gme etween Ad Exchnges under the ssumption tht sellers cn multi-home nd hve non-inding cpcity constrints. Formlly, the Ad Exchnge prolem is to choose mechnism q iv, p iv i=0,1,..., to mximize the seller s profit suject to ttining minimum profit level of π. mx U 0vdGv, 13 6

9 suject to IR, IC, the fesiility constrint, nd P ivdf v + P 0vdGv π. The next proposition chrcterizes the solution to this prolem. Proposition 5. The π-optiml Direct-Reveltion Mechnism Let us choose indexes such tht v i = mx j 1,..., v j. The π-optiml direct-reveltion mechnism sets q π i v i = 1 if nd only if v i 1 F vi fv i v 0 λπ Gv0 gv 0 0, nd sets q0 π v = 1 otherwise. The function λ : [0, π ] [0, 1] is strictly incresing in π, nd stisfies λ0 = 0 nd λπ = 1, where π = v 1 F v v 0 Gv0 fv fv gv 0 rv 0 F 1 vgv 0dvdv 0, with rv 0 1 F rv 0 frv 0 = v 0 + Gv 0 gv 0. For ech π, λπ solves sv 0,λπ π = v 1 F v v 0 λπ Gv0 fv gv 0 fv F 1 vgv 0dvdv 0, 14 with sv 0, λ 1 F sv 0,λ fsv 0,λ = v 0 + λ Gv 0 gv 0. Proof. The pltform s profits cn e rewritten s q iv v i 1 F vi v 0 Gv0 fv i gv 0 gv 0 df v i U i + [ U 0]. In turn, the pltform s ojective which is the seller s pyoff rewrites U 0 + Eṽ 0 q iv Gv0 gv 0 gv 0 df v i. Expressing the pltform s constrined mximiztion prolem in Lgrngin form leds to q iv µ v i µ 1 F vi µ v µ Gv0 fv i gv 0 gv 0 df v i µ U i + µ [ U 0] + U 0 + Eṽ 0, where µ > 0. It is immedite tht t the optimum U i = 0 nd U 0 =. Mximizing the integrl ove point-wise leds to the following ng-ng solution: qi π v i = 1 if µ v i µ 1 F vi 1 + µ v 0 Gv0 0, fv i gv 0 nd q0 π v = 1 otherwise. Defining λ µ leds to the 1+µ sttement in the proposition. Finlly, since the profit constrint is lwys inding, the vlue of λ is given y 14. Q.E.D. In order to otin insight on the ctul prctice of dexchnges, the next proposition derives the Revenue-shring mechnisms tht implement the direct-reveltion mechnism descried ove. Define the function v 0r, λ ccording to r 1 F r fr = v 0r, λ + λ Gv0r, λ gv 0r, λ. Then the following result is true. The proof is identicl to tht of Proposition 4.1, nd is therefore omitted. Proposition 6. π-optiml Indirect Implementtion The following trding procedure indirectly implements the π-optiml mechnism: 1. The seller is sked to report reserve price.. The pltform runs second-price uction with reserve price. 3. The pltform gives to the seller frction θ π of the proceeds of the uction, where the shring rule θ π stisfies θ π = v0, λπ F v 0,λπ F rˆvdˆv R, for [r, ] Interestingly, constnt shring schemes re not le to implement the π-optiml mechnism. This suggests tht pltforms might need to dopt non-constnt shring schemes if competition in the d exchnge mrket increses, therefore pproching Bertrnd competition, nd opertionl costs per trnsction re ounded y some π greter thn zero. 6. FUTURE RESEARCH This pper studies the optimlity of constnt shring schemes in settings plgued y two-sided symmetric informtion nd competition. We provide necessry nd sufficient conditions under which the optiml mechnism cn e implemented y constnt cuts for the seller, nd nlyze how such cuts re ffected y the degree of competition in the mrket. One fruitful direction of future reserch pertins the reltive performnce of constnt shring mechnisms when the necessry nd sufficient conditions identified in this pper fil. In such settings, wht frction of the revenue ssocited with the optiml mechnism does constnt shring schemes chieve? The widespred use of constnt shring rules renders this n importnt question with first-order prcticl relevnce. Acknowledgements. We thnk our collegues Ggn Goel, Eyl Mnor, nd Mrtin Pl, for very helpful discussions nd comments on the pper. 7

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