Bidding for Representative Allocations for Display Advertising

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1 Bidding for Rereentative Allocation for Dilay Advertiing Arita Ghoh Preton McAfee Kihore Paineni Sergei Vailvitkii Abtract Dilay advertiing ha traditionally been old via guaranteed contract a guaranteed contract i a deal between a ubliher and an advertier to allocate a certain number of imreion over a certain eriod, for a re-ecified rice er imreion. However, a ot market for dilay ad, uch a the RightMedia Exchange, have grown in rominence, the election of advertiement to how on a given age i increaingly being choen baed on rice, uing an auction. A the number of articiant in the exchange grow, the rice of an imreion become a ignal of it value. Thi correlation between rice and value mean that a eller imlementing the contract through bidding hould offer the contract buyer a range of rice, and not jut the cheaet imreion neceary to fulfill it demand. Imlementing a contract uing a range of rice, i akin to creating a mutual fund of advertiing imreion, and require randomized bidding. We characterize what allocation can be imlemented with randomized bidding, namely thoe where the deired hare obtained at each rice i a non-increaing function of rice. In addition, we rovide a full characterization of when a et of camaign are comatible and how to imlement them with randomized bidding trategie. 1 Introduction Dilay advertiing howing grahical ad on regular web age, a ooed to textual ad on earch age i aroximately a $24 billion buine. There are two way in which an advertier looking to reach a ecific audience (for examle, 1 million male in California in July 29) can buy uch ad lacement. One i the traditional method, where the advertier enter into an agreement, called a guaranteed contract, directly with the ubliher (owner of the webage). Here, the ubliher guarantee to deliver a reecified number (1 million) of imreion matching the targeting requirement (male, from California) of the contract in the ecified time frame (July 29). The econd i to articiate in a ot market for dilay ad, uch a the RightMedia Exchange, where advertier can buy imreion one ageview at a time: every time a uer load a age with a ot for advertiing, an auction i held where advertier can bid for the oortunity to dilay a grahical ad to thi uer. Both the guaranteed and ot market for dilay advertiing now thrive ide-by-ide. There i demand for guaranteed contract from advertier who want to hedge againt future uncertainty of uly. For examle, an advertier who mut reach a certain audience during a critical eriod of time (e.g around a forthcoming roduct launch, uch a a movie releae) may not want to rik the uncertainty of a ot market; a guaranteed contract inure the ubliher a well againt fluctuation in demand. At the ame time, a ot market allow the advertier to bid for ecific oortunitie, ermitting very fine grained targeting baed on uer tracking. Currently, RightMedia run over nine billion auction for dilay ad everyday. How hould a ubliher decide which of her uly of imreion to allocate to her guaranteed contract, and which to ell on the ot market? One obviou olution i to fulfill the guaranteed demand firt, and then ell the remaining inventory on the ot market. However, ot market rice are often quite different for two imreion that both atify the targeting requirement of a guaranteed contract, ince different imreion have different value. For examle, the imreion from two uer with identical demograhic can have different value, baed on different earch behavior reflecting urchae intent for one of the uer, but not the other. Since advertier on the ot market have acce to more tracking information about each Yahoo! Reearch {arita,mcafee,kai,ergei}@yahoo-inc.com 1

2 uer 1, the reulting bid may be quite different for thee two uer. Allocating imreion to guaranteed contract firt and elling the remainder on the ot market can therefore be highly ubotimal in term of revenue, ince two imreion that would fetch the ame revenue from the guaranteed contract might fetch very different rice from the ot market 2. On the other hand, imly buying the cheaet imreion on the ot market to atify guaranteed demand i not a good olution in term of fairne to the guaranteed contract, and lead to increaing hort term revenue at the cot of long term atifaction. A dicued above, imreion in online advertiing have a common value comonent becaue advertier generally have different information about a given uer. Thi information (e.g. browing hitory on an advertier ite) i tyically relevant to all of the bidder, even though only one bidder may oe thi information. In uch etting, rice i a ignal of value in a model of valuation incororating both common and rivate value, the rice converge to the true value of the item in the limit a the number of bidder goe to infinity ([7, 1], ee alo [6] for dicuion). On average, therefore, the rice on the ot market i a good indicator of the value of the imreion, and delivering cheaet imreion correond to delivering the lowet quality imreion to the guaranteed contract 3. A ubliher with acce to both ource of demand thu face a trade-off between revenue and fairne when deciding which imreion to allocate to the guaranteed contract; thi trade-off i further comounded by the fact that the ubliher tyically doe not have acce to all the information that determine the value of a articular imreion. Indeed, ubliher are often the leat well informed articiant about the value of running an ad in front of a uer. For examle, when a uer viit a olitic ite, Amazon (a an advertier) can ee that the uer recently earched Amazon for an iod, and Target (a an advertier) can ee they earched target.com for coffee mug, but the ubliher only know the uer viited the olitic ite. Furthermore, the exact nature of thi trade-off i unknown to the ubliher in advance, ince it deend on the ot market bid which are revealed only after the advertiing oortunity i laced on the ot market. The ubliher a a bidder. To addre the roblem of unknown ot market demand (i.e., the ubliher would like to allocate the oortunity to a bidder on the ot market if the bid i high enough, ele to a guaranteed contract), the ubliher act, in effect, a a bidder on behalf on the guaranteed contract. That i, the ubliher now lay two role: that of a eller, by lacing hi oortunity on the ot market, and that of a bidding agent, bidding on behalf of hi guaranteed contract. If the ubliher own bid turn out to be highet among all bid, the oortunity i won and i allocated to the guaranteed contract. Acting a a bidder allow the ubliher to robe the ot market and decide whether it i more efficient to allocate the oortunity to an external bidder or to a guaranteed contract. How hould a ubliher model the trade-off between fairne and revenue, and having decided on a tradeoff, how hould he lace bid on the ot market? An ideal olution i (a) eay to imlement, (b) allow for a trade-off between the quality of imreion delivered to the guaranteed contract and hort-term revenue, and (c) i robut to the exact tradeoff choen. In thi work we how reciely when uch an ideal olution exit and how it can be imlemented. 1.1 Our Contribution In thi aer, we rovide an analytical framework to model the ubliher roblem of how to fulfill guaranteed advance contract in a etting where there i an alternative ot market, and advertiing oortunitie have a common value comonent. We give a olution where the ubliher bid on behalf of it guaranteed contract in the ot market. The olution conit of two comonent: an allocation, ecifying the fraction of imreion at each rice allocated to a contract, and a bidding trategy, which ecifie how to acquire thi allocation by bidding in an auction. 1 For examle, a car dealerhi advertier may oberve that a articular uer ha been to hi webage everal time in the reviou week, and may be willing to bid more to how a car advertiement to induce a urchae. 2 Conider the following toy examle: uoe there are two oortunitie, the firt of which would fetch 1 cent in the ot market, wherea the econd would fetch only ɛ; both oortunitie are equally uitable for the guaranteed contract which want jut one imreion. Clearly, the firt oortunity hould be old on the ot market, and the econd hould be allocated to the guaranteed contract. 3 While allocating the cheaet inventory to the guaranteed contract i indeed revenue maximizing in the hort term, in the long term the ubliher run the rik of loing the guaranteed advertier by erving them the leat valuable imreion. 2

3 The quality, or value, of an oortunity i meaured by it rice 4. A erfectly rereentative allocation i one which conit of the ame roortion of imreion at every rice i.e., a mix of high-quality and low quality imreion. The trade-off between revenue and fairne i modeled uing a budget, or average target end contraint, for each advertier allocation: the ubliher choice of target end reflect her trade-off between hort-term revenue and quality of imreion for that advertier (thi mut, of coure, be large enough to enure that the romied number of imreion atifying the targeting contraint can be delivered.) Given a target end 5, a maximally rereentative allocation i one which minimize the ditance to the erfectly rereentative allocation, ubject to the budget contraint. We firt how how to olve for a maximally rereentative allocation, and then how how to imlement uch an allocation by urchaing oortunitie in an auction, uing randomized bidding trategie. Organization. We tart out with the ingle contract cae, where the ubliher ha jut one exiting guaranteed contract, in Section 2; thi cae i enough to illutrate the idea of maximally rereentative allocation and imlementation via randomized bidding trategie. We move on to the more realitic cae of multile contract in Section 3; we firt rove a reult about which allocation can be imlemented in an auction in a decentralized fahion, and derive the correonding decentralized bidding trategie. Next we olve for the otimal allocation when there are multile contract. Finally, in Section 4, we validate thee trategie by imulating on data derived from real world exchange Related Work The mot relevant work i the literature on deigning exreive auction and clearing algorithm for online advertiing [8, 2, 9]. Thi literature doe not addre our roblem for the following reaon. While it i true that guaranteed contract have coare targeting relative to what i oible on the ot market, mot advertier with guaranteed contract chooe not to ue all the exreivene offered to them. Furthermore, the exreivene offered doe not include attribute like relevant browing hitory on an advertier ite, which could increae the value of an imreion to an advertier, imly becaue the ubliher doe not have thi information about the advertiing oortunity. Even with extremely exreive auction, one might till want to adot a mutual fund trategy to avoid the inider trading roblem. That i, if ome bidder oe good information about convertibility, other will till want to randomize their bidding trategy ince bidding a contant rice mean alway loing on ome good imreion. Thu, our roblem cannot be addreed by the ue of more exreive auction a in [9] the real roblem i not lack of exreivity, but lack of information. Another area of reearch focue on electing the otimal et of guaranteed contract. In thi line of work, Feige et al. [5] tudy the comutational roblem of chooing the et of guaranteed contract to maximize revenue. A imilar roblem i tudied by in [3, 1]. We do not addre the roblem of how to elect the et of guaranteed contract, but rather take them a given and addre the roblem of how to fulfill thee contract in the reence of cometing demand from a ot market. 2 Single contract We firt conider the imlet cae: there i a ingle advertier who ha a guaranteed contract with the ubliher for delivering d imreion. There are a total of d advertiing oortunitie which atify the targeting requirement of the contract. The ubliher can alo ell thee oortunitie via auction in a ot market to external bidder. The highet bid from the external bidder come from a ditribution F, with denity f, which we refer to a the bid landcae. That i, for every unit of uly, the highet bid 4 We emhaize that the aumtion being made i not about rice being a ignal of value, but rather that imreion do have a common value comonent given that imreion have a common value, rice reflecting value follow from the theorem of Milgrom [7]. Thi aumtion i eaily jutifiable ince it i commonly oberved in ractice. 5 We oint out that we do not addre the quetion of how to et target end, or the related roblem of how to rice guaranteed contract to begin with: given a target end (reumably choen baed on the rice of guaranteed contract and other conideration), we rooe a comlete olution to the ubliher roblem. 3

4 from all external bidder,which we refer to a the rice, i drawn i.i.d from the ditribution 6 f. (An examle of uch a denity een in a real auction for advertiing oortunitie i hown in ection 4.) We aume that the uly and the bid landcae f are known to the ubliher 7. Recall that the ubliher want to decide how to allocate it inventory between the guaranteed contract and the external bidder in the ot market. Due to enaltie a well a oible long term cot aociated with underdelivering on guaranteed contract, we aume that the ubliher want to deliver all d imreion romied to the guaranteed contract. An allocation a() i defined a follow: a()/ i the roortion of oortunitie at rice urchaed on behalf of the guaranteed contract (the rice i the highet (external) bid for an oortunity.) That i, of the f()d imreion available at rice, an allocation a() buy a fraction a()/ of thee f()d imreion, i.e., a()f()d imreion. For examle, a contant bid of mean that for, a() = 1 with the advertier alway winning the auction, and for >, a() = ince the advertier would never win. Generally, we will decribe our olution in term of the allocation a()/, which mut integrate out to the total demand d: a olution where a()/ i larger for higher rice correond to a olution where the guaranteed contract i allocated more high-quality imreion. A another examle, a()/ = d/ i a erfectly rereentative allocation, integrating out to a total of d imreion, and allocating the ame fraction of imreion at every rice oint. Not every allocation can be urchaed by bidding in an auction, becaue of the inherent aymmetry in bidding a bid b allow every rice below b and rule out every rice above; however, there i no way to rule out rice below a certain value. That i, we can chooe to exclude high rice, but not low rice. Before decribing our olution, we tate what kind of allocation a()/ can be urchaed by bidding in an auction. Prooition 1. A right-continuou allocation a()/ can be imlemented (in exectation) by bidding in an auction if and only if a( 1 ) a( 2 ) for 1 2. Proof. Given a right-continuou non-increaing allocation a() (that lie between and 1), define H() := 1 a(). Let := inf { : a() < }. Then, H i monotone non-decreaing and i right-continuou. Further, H( ) = and H( ) = 1. Thu, H i a cumulative ditribution function. We lace bid drawn from H (the robability of a trictly oitive bid being a()/). Then the exected number of imreion won at rice i then exactly a()/. Converely, given that bid for the contract are drawn at random from a ditribution H, the fraction of uly at rice that i won by the contract i imly 1 H(), the robability of it bid exceeding. Since H i non-decreaing, the allocation (a a fraction of available uly at rice ) mut be non-increaing in. Note that the ditribution H ued to imlement the allocation i a different object from the bid landcae f againt which the requiite allocation mut be acquired in fact, it i comletely indeendent of f, and i ecified only by the allocation a()/. That i, given an allocation, the bidding trategy that imlement the allocation in an auction i indeendent of the bid landcae f from which the cometing bid i drawn. 2.1 Maximally rereentative allocation Ideally the advertier with the guaranteed contract would like the ame roortion of imreion at every rice, i.e., a()/ = d/ for all. (We ignore the oibility that the advertier would like a higher fraction of higher-riced imreion, ince thee cannot be imlemented according to Prooition 1 above.) However, the ubliher face a trade-off between delivering high-quality imreion to the guaranteed contract and allocating them to bidder who value them highly on the ot market. We model thi by introducing an average unit target end t, which i the average rice of imreion allocated to the contract. A maller (bigger) t deliver more (le) chea imreion. A we mentioned before, t i art of the inut roblem, and may deend, for intance, on the rice aid by the advertier for the contract. 6 Secifically, we do not conider adverarial bid equence; we alo do not model the effect of the ubliher own bid on other bid. 7 Publiher tyically have acce to data neceary to form etimate of thee quantitie; thi i alo dicued briefly in the concluion 4

5 Given a target end, the maximally rereentative allocation i an allocation a()/ that i cloet (according to ome ditance meaure) to the ideal allocation d/, while reecting the target end contraint. That i, it i the olution to the following otimization roblem: ( ) inf a( ) u a(), d f()d.t. a()f()d = d (1) a()f()d td a() 1. The objective, u, i a meaure of the deviation of the rooed fraction, a()/, from the erfectly rereentative fraction, d/. In what follow, we will conider the L 2 meaure a well a the Kullback-Leibler (KL) divergence ( a() u, d ) ( a() u, d ) = ( a() d ) 2 2 = a() log a() d. Why the choice of KL and L 2 for cloene? Only Bregman divergence lead to a election that i conitent, continuou, local, and tranitive [4]. Further, in R n only leat quare i cale- and tranlation- invariant, and for robability ditribution only KL divergence i tatitical [4]. Indeed, KL i more aroriate in our etting. However, a leat quare i more familar, we dicu KL in Aendix C. The firt contraint in (1) i imly that we mut meet the target demand d, buying a()/ of the f()d oortunitie of rice. The econd contraint i the target end contraint: the total end (the end on an imreion of rice i ) mut not exceed td, where t i a target end arameter (averaged er unit). A we will hortly ee, the value of t trongly affect the form of the olution. Finally, the lat contraint imly ay that the roortion of oortunitie bought at rice, a()/, mut never go negative or exceed 1. Otimality condition: Introduce Lagrange multilier λ 1 and λ 2 for the firt and econd contraint, and µ 1 (), µ 2 () for the two inequalitie in the lat contraint. The Lagrangian i ( a() L = u, d ) ( ) ( ) f()d + λ 1 d a()f()d + λ 2 a()f()d td + µ 1 ()( a())f()d + µ 2 ()(a() )f()d. By the Euler-Lagrange condition for otimality, the otimal olution mut atify ( a() u, d ) = λ 1 λ 2 + µ 1 () µ 2 (), where the multilier µ atify µ 1 (), µ 2 (), and each of thee can be non-zero only if the correonding contraint i tight. Thee otimality condition, together with Prooition 1, give u the following: Prooition 2. The maximally rereentative allocation for a ingle contract can be imlemented by bidding in an auction for any convex ditance meaure u. The roof follow from the fact that u i increaing for convex u. 5

6 2.1.1 L 2 utility In thi ubection, we derive the otimal allocation when u, the ditance meaure, i the L 2 ditance, and how how to imlement the otimal allocation uing a randomized bidding trategy. In thi cae the bidding trategy turn out to be very imle: to a coin to decide whether or not to bid, and, if bidding, draw the bid value from a uniform ditribution. The coin toing robability and the endoint of the uniform ditribution deend on the demand and target end value. Firt we give the following reult about the continuity of the otimal allocation; thi will be ueful in deriving the value that arameterize the otimal allocation. See Aendix B for the roof. Prooition 3. The otimal allocation a() i continuou in. Note that we do not aume a riori that a( ) i continuou; the otimal allocation turn out to be continuou. The otimality condition, when u i the L 2 ditance, are: a() d = λ 1 λ 2 + µ 1 () µ 2 (), where the nonnegative multilier µ 1 (), µ 2 () can be non-zero only if the correonding contraint are tight. The olution to the otimization roblem (1) then take the following form: For min, a()/ = 1; for min max, a()/ i roortional to C, i.e., a()/ = z(c ); and for max, a()/ =. To find the olution, we mut find min, max, z, and C. Since a()/ i continuou at max, we mut have 1 C = max. By continuity at min, if min > then z(c min ) = 1, o that z = max min. Thu, the otimal allocation a() i alway arametrized by two quantitie, and ha one of the following two form: 1. a()/ = z( max ) for max (and for max ). When the olution i arametrized by z, max, thee value mut atify max max z( max )f()d = d (2) z( max )f()d = td (3) Dividing (2) by (3) eliminate z to give an equation which i monotone in the variable max, which can be olved, for intance, uing binary earch. 2. a()/ = 1 for min, and a()/ = max max min for max (and thenceforth). When the olution i arametrized by min, max, thee value mut atify min F ( min ) + f()d + max ( max ) f()d = d (4) min max min max We how how to olve thi ytem in Aendix A. Note that the otimal allocation can be rereented more comactly a a() min ( max ) max min f()d = td. (5) = min{1, z( max )}. (6) Effect of varying target end: Varying the value of the target end, t, while keeing the demand d fixed, lead to a tradeoff between rereentativene and revenue from elling oortunitie on the ot market, in the following way. The minimum oible target end, while meeting the target demand (in exectation) 6

7 i achieved by a olution where min = max and a()/ = 1 for le equal thi value, and for greater. The value of min i choen o that min f()d = d min = F 1 ( d ). Thi olution imly bid a flat value min, and correond to giving the cheaet oible inventory to the advertier, ubject to meeting the demand contraint. Thi give the minimum oible total end for thi value of demand, of min wherea the exected revenue obtained by elling all oortunitie on the exchange i E[]. td = f()d = F ( min )E[ min ] A the value of t= increae de[ above min ] t, min decreae and max increae, until we reach min =, at which oint we move into the regime of the other otimal form, with z = 1. A t i increaed further, z decreae from 1, and (Note that the maximum oible total end that i maximally rereentative while not overdelivering i R = f()d = de[] = d.) max increae, until at the other extreme when the end contraint i eentially removed, the olution i a() A the value of t increae above t, min decreae and max increae, until we reach min =, at which oint we move into the regime of the other otimal = d form, for all ; i.e., a erfectly rereentative with z = 1. A t i increaed further, z decreae allocation acro rice. from 1, and max increae, until at the other extreme when the end contraint i eentially removed, the olution i a() Thu = d the forvalue all ; of i.e., t arovide erfectlya rereentative dial by which allocation to move from acrothe rice. cheaet Thu the value of t rovide a dial by allocation which to to move the from erfectly the cheaet rereentative allocation allocation. to thefigure erfectly 1 illutrate rereentative the allocation. Figure 1 illutrate effect theof effect varying of varying target end targeton end the on otimal the otimal allocation. allocation. a() 1 t < t 1 < t 2 < t 3 < t = t d t = c t 3 t 2 t 1 u Figure 1: Effect of target end on L 2 -otimal allocation Figure 1: Effect of target end on L 2 -otimal allocation 3.2 Finding the arameter value 2.2 Randomized We nowbidding dicu olving trategie for the comlete olution, i.e., finding the value of z, max or min, max. The quantity a()/ When i the an olution otimal i allocation, arametrized i.e., by a z, recommendation max, thee value to mut the ubliher atify a to how much inventory to allocate to a guaranteed contract at every rice. However, recall that the ubliher need to acquire thi inventory max on behalf of the guaranteed contract by bidding in the ot market. The z( max )f()d = d zf ( max ) ( max E[ max )]) = d following theorem how how to do thi when u i the L 2 ditance. max Theorem 1. The otimal z( allocation max )f()d for the L 2 ditance td meaure can be imlemented (in exectation) in an auction by the following random trategy: to a coin to decide whether or not to bid, and if bidding, draw the bid from a uniform ditribution. 7 7

8 Proof. From (6) that the otimal allocation can be rereented a a() = min{1, z( max )}. By Prooition 1, an allocation a() = min{1, z( max )} can be imlemented by bidding in an auction uing the following randomized bidding trategy: with robability min{z max, 1}, lace a bid drawn uniformly at random from the range [max{ max 1 z, }, max]. The otimal allocation for KL-divergence decay exonentially with rice, and the bidding trategy involve drawing bid from an exonential ditribution; ee Aendix C for detail. 3 Multile contract We now tudy the more realitic cae where the ubliher need to fulfill multile guaranteed contract with different advertier. Secifically, uoe there are m advertier, with demand d j. A before, there are a total of 8 d j advertiing oortunitie available to the ubliher. An allocation a j ()/ i the roortion of oortunitie urchaed on behalf of contract j at rice. Of coure, the um of thee allocation cannot exceed 1 for any, which correond to acquiring all the uly at that rice. A in the ingle contract cae, we are firt intereted in what allocation a j () are imlementable by bidding in an auction. However, in addition to being imlementable, we would like allocation that atify an additional ractical requirement, exlained below. Notice that the ubliher, acting a a bidding agent, now need to acquire oortunitie to imlement the allocation for each of the guaranteed contract. When an oortunity come along, therefore, the ubliher need to decide which of the contract (if any) will receive that oortunity. There are two way to do thi: the ubliher ubmit one bid on behalf of all the contract; if thi bid win, the ubliher then elect one amongt the contract to receive the oortunity. Alternatively, the ubliher can ubmit one bid for each contract; the winning bid then automatically decide which contract receive the oortunity. We refer to the former a a centralized trategy and the latter a a decentralized trategy. There are ituation where the ubliher will need to chooe the winning advertier rior to eeing the rice, that i, the highet bid from the ot market. For examle, to reduce latency in lacing an advertiement, the auction mechanim may require that the bid be accomanied by the advertiement (or it unique identifier). A decentralized trategy automatically fulfill thi requirement, ince there i one bid for each contract and the highet bid win, o that the choice of winning contract doe not deend uon knowing the rice. In a centralized trategy, thi requirement mean that the relative fraction won at rice, a i ()/a j (), are indeendent of the rice when thi haen, the choice of advertier can be made (by chooing at random with robability roortional to a j ) without knowing the rice. A before, we will be intereted in imlementing otimal (i.e., maximally rereentative) allocation. For uch an allocation, a we will how in Section 3.2, if the relative fraction are indeendent of the rice, they can alo be decentralized. We will, therefore, concentrate on characterizing allocation which can be imlemented via a decentralized trategy. 3.1 Decentralization In thi ection, we examine what allocation can be imlemented via a decentralized trategy. Note that it i not ufficient to imly ue a ditribution H j = 1 aj() a j() a in Prooition 1, ince thee contract comete amongt each other a well. Secifically, uing the ditribution 1 aj() a j() will lead to too few oortunitie being urchaed for contract j, ince thi ditribution i deigned to comete againt f alone, rather than 8 In general, not all of thee oortunitie might be uitable for every contract; we do not conider thi here for clarity of reentation. However the ame idea and method can be alied in that mot general cae; the reult are alo qualitatively imilar. 8

9 againt f a well a the other contract. We need to how how to chooe ditribution in uch a way that lead to a fraction a j ()/ of oortunitie being urchaed for contract j, for every j = 1,..., m. Firt, we argue that a decentralized trategy with given ditribution H j will lead to allocation that are non-increaing, a in the ingle contract cae. A decentralized imlementation ue ditribution H j to bid for imreion, i.e., it draw a bid randomly from the ditribution H j to lace in the auction on behalf of j-th contract. Then, contract j win an imreion at rice with robability a j () = H k (x) h j (x)dx, k j ince to win, the bid for contract j mut be larger than and larger than the bid laced by each of the remaining m 1 contract. Since all the quantitie in the integrand are nonnegative, a j i non-increaing in. Now aume that a j are differentiable almot everywhere (a.e.) and non-increaing. Let A() := j be the total fraction of oortunitie at rice that the ubliher need to acquire. Clearly, a j mut be uch that A(),. Let := inf{ : A() < }. Now define { R e H j () := a j (x)/( A(x))dx > (7) ele Then, H j () and i continuou. Since a j () i non-increaing, H j() i monotone non-decreaing. Further, H( ) = 1 and H j ( ) =. Thu, H j i a ditribution function. Now we verify that bidding according to H j will reult in the deired allocation: Note that which imlie o that or and hence log( A(x)) = a j () h j () = d d H j() = H j () a j () A() A () A() = j a j () A (x) A(x) dx = log( A()) = j A() = j j log(h j ()) H k () = A(). Then, the fraction of imreion at that are won by contract j i H k (x) h j (x)dx = k j k ( ) h j (x) H k (x) H j (x) dx = k h j () H j (), h j (x) H j (x) dx = j ( A(x)) h j(x) H j (x) dx = log(h j (x)) a j(x)dx = a j() Thu, we contructed ditribution function H j () which imlement the given non-increaing (and a.e. differentiable) allocation a j (). If any a j i increaing at any oint, the et of camaign cannot be decentralized. We ummarize thi in the following theorem, whoe ecial cae for the ingle contract cae i Prooition 1: 9

10 Theorem 2. A et of allocation a j () can be imlemented in an auction via a decentralized trategy if and only if each a j () i non-increaing in, and j a j()/ 1. Having determined which allocation can be imlemented by bidding in an auction in a decentralized fahion, we turn to the quetion of finding uitable allocation to imlement. A in the ingle contract cae, we would like to imlement allocation that are maximally rereentative, given the end contraint. 3.2 Otimal allocation for multile contract A in the ingle contract cae, every contract would ideally like an equal roortion of oortunitie at every rice. However, every contract ha a er unit target end which limit the fraction of oortunitie that can be urchaed at higher rice. In addition to the target end, the allocation i alo contrained by the fact that the total fraction of oortunitie bought at every rice mut not exceed one. The maximally rereentative allocation i the allocation cloet to the ideal allocation that atifie the target end contraint, and uch that the collective allocation doe not exceed the uly at any rice. That i, it i the olution to the otimization roblem below with the L 2 ditance meaure in the objective. We ue j to index the m contract. min.t. m 2 j=1 ( aj() a j()f()d = d j a j()f()d t j d a j (), j m j=1 a j() dj )2 f()d Oberve that the allocation for individual contract are couled only by the lat contraint. Otimality condition: Introduce Lagrange multilier λ j 1 and λj 2 for the firt and econd contraint, and µ j 1 (), µ 2() for the lat two inequalitie. The otimality condition are a j () j j d j = λj 1 λj 2 + µj 1 () µ 2(), where λ j 2, µj 1 and µ 2 mut be nonnegative and can be non-zero only when the correonding contraint i tight. Note that µ 2 i a contract-indeendent multilier, correonding to the couling contraint. Suoe a j ( ) j and j a j( ) < for ome >. Then, µ j 1 ( ) = µ 2 ( ) =. It follow that for >, a j () < a j ( ). Let = inf { : j a j() < }. Then,, each a j decay linearly with loe λ j 2 until it become. If =, the olution decoule, a µ 2 (). In thi cae, we can olve for the a j indeendently of one another. However, if >, we have < : Together with j a j() =, thi imlie Denoting λ 2 := 1 m a j () d j = λj 1 λj 2 µ 2(). µ 2 () = 1 m 1 d j 1 λ j 1 m m + 1 λ j 2 m. j λj 2, we ee that a j () j = c j (λ j 2 λ 2 ), <. Therefore, at leat one a j will have a oitive loe below unle λ j 2 = λk 2, j, k. That i, decentralization i not alway guaranteed. In cae the target end are uch that > and λ j 2 = λk 2, j, k, the otimal allocation a j tay flat until and then decay with identical loe until each become, a hown in Figure 2. Thu, the otimal allocation i decentralizable in two cae: j j (8) 1

11 a j() z 3 z 2 z 1 min max 1 max 2 max 3 Figure 2: Couled decentralizable allocation. 1.9 Emirical CDF Log Normal Model CDF Figure 3: Emirical cumulative denity function of the bid on the exchange and a log-normal fit to the ditribution. 1. = : The target end are uch that the olution decoule. In thi cae the allocation for each contract i indeendent of the other; we olve for the arameter of each allocation a in Section > : The target end are uch that, for all j, k, i indeendent of. In thi cae we need to a j() a k () olve for the common loe and min, and the contract ecific value j max, which together determine the allocation. Thi can be done uing, for intance, Newton method. When the target end are uch that the allocation i not decentralizable, the vector of target end can be increaed to reach a decentralizable allocation 9. One way i to cale u the target end uniformly until they are large enough to admit a earable olution; thi ha the advantage of reerving the relative ratio of target end. The minimum multilier which render the allocation decentralizable can be found numerically, uing for intance binary earch. 4 Exerimental Validation Our algorithm for obtaining rereentative allocation are randomized, and all of the reult are derived in exectation. In thi ection we imulate the erformance of the algorithm, and verify that the randomization doe not lead to under-delivery for a realitic choice of bid landcae f. To imulate the bid landcae, data wa collected from live auction conducted by the RightMedia exchange. RightMedia run the larget ot market for dilay advertiing, with billion of auction daily. Winning bid were collected from aroximately 4, auction over the coure of a day for a ecific ubliher. The cdf of the emirical bid ditribution i lotted in Figure 3. (The cale on the x-axi i omitted for rivacy concern.) The emirical ditribution i well aroximated by a log-normal ditribution, a een in Figure 3. For the exerimental evaluation, we, therefore, draw bid from a log-normal ditribution. The mean of the 9 We do not invetigate the aroach of finding the bet ubotimal allocation that can be decentralized, i.e., an aroximately otimal decentralizable allocation, in thi aer. 11

12 Allocation Mean Send Variance Variance Allocation.55.5 Mean Send Variance Variance Allocation Mean Send Variance Variance (a) Allocation achieved for different exchange ditribution (b) Target end achieved for different exchange ditribution with ideal target.25,.5 and.75 of the maximum with ideal allocation of.25,.5 and.75 fraction of the total. feaible end. Figure 4: Exeriment evaluation ditribution i et to, and the variance arameter i changed to invetigate the enitivity of our algorithm to the variance of the bid landcae. To tudy the effectivene of the algorithm in winning the right number of imreion on the exchange, we fix the target fraction, d at.25,.5 and.75, and comute the max neceary to achieve the allocation, yet minimize the total end. For each etting of the variance of the exchange ditribution, we run 15 trial, each with 1, auction total. The reult are lotted in Figure 4(a). We erform a imilar exeriment to invetigate the deendency of the target end on the variance of the bid ditribution. In thi cae, we fix the allocation fraction to.8 and target end to.25µ,.5µ and.75µ, where µ = f()d i the maximum achievable target end. For each etting of the variance arameter we run 15 trial each with 1, auction. The reult are lotted in Figure 4(b). In both imulation, the ecific min and max for each variance etting vary greatly to achieve the deired allocation and target end. However, the change in the reulting a and d themelve are minimal the algorithm rarely underdeliver or under/overend by more than 1%; it i robut to variation in the variance of the underlying bid ditribution. 5 Concluion Moving guaranteed contract into an exchange environment reent a variety of challenge for a ubliher. Randomized bidding i a ueful comromie between minimizing the cot and maximizing the quality of guaranteed contract. It i akin to the mutual fund trategy common in the caital aet ricing model. We rovide a readily comutable olution for ynchronizing an arbitrary number of guaranteed camaign in an exchange environment. Moreover, the olution we detail aear table with real data. There are many intereting direction for further reearch. We aumed throughout that the uly i known to the ubliher. A more realitic model aume either an unknown or a tochatic uly (a trawman olution i to ue algorithm in thi aer uing a lower bound on the uly in lace of ). Another intereting avenue i analyzing the trategic behavior by other bidder on the ot market in reone to uch randomized bidding trategie. Reference [1] Mohe Babaioff, Jaon Hartline, and Robert Kleinberg. Selling banner ad: Online algorithm with buyback. In 4th Workho on Ad Auction,

13 [2] C. Boutilier, D. Parke, T. Sandholm, and W. Walh. Exreive banner ad auction and model-baed online otimization for clearing. In National Conference on Artificial Intelligence (AAAI), 28. [3] Florin Contantin, Jon Feldman, S Muthukrihnan, and Martin Pal. Online ad lotting with cancellation. In 4th Workho on Ad Auction, 28. [4] I. Cizar. Why leat quare and maximum entroy? an axiomatic aroach to interference for linear invere roblem. Annal of Statitic, 19(4): , [5] Uriel Feige, Nicole Immorlica, Vahab S. Mirrokni, and Hamid Nazerzadeh. A combinatorial allocation mechanim with enaltie for banner advertiing. In Proceeding of ACM WWW, age , 28. [6] R. Preton McAfee and John McMillan. Auction and bidding. Journal of Economic Literature, 25(2): , [7] Paul R. Milgrom. A convergence theorem for cometitive bidding with differential information. Econometrica, 47(3): , May [8] David Parke and Tuoma Sandholm. Otimize-and-diatch architecture for exreive ad auction. In 1t Workho on Ad Auction, 25. [9] Tuoma Sandholm. Exreive commerce and it alication to ourcing: How we conducted $35 billion of generalized combinatorial auction. 28(3):45 58, 27. [1] Robert Wilon. A bidding model of erfect cometition. Rev. Econ. Stud., 44(3): , Aendix A Solving for min and max Calling min and max x and y reectively, we want to olve the ytem of equation: f(y, x) = d [ ] 1, t where f(y, x) = [ F (x) + y y x x f()d x f()d + y x y y x f()d ]. We will how that the derivative matrix i invertible o we can ue Newton method to converge to the olution. The derivative of f i: [ y x y ] y f x (y x) f()d = 2 x (y x) f()d 2 y x x (y x) f()d y 2 x y (y x) f()d 2 [ y 1 x = ( x)f()d y ] (y )f()d x (y x) 2 y ( x)f()d y (y )f()d x x [ ] F (y) F (x) E x y E = (y x) 2, E 2 xe ye E 2 where we have defined E = E[ x y]. 13

14 It i eay to check that E x, y E and E 2 xe are oitive ince x y. For the final term, oberve that ye 2 = E(y 2 ) = E(y ). We note that E 2 = (E) 2 + σ 2 where σ 2 i the variance of conditioned on x y, and therefore can comute the determinant to be (y x)σ 2. Therefore, rovided that y > x and F i non-degenerate, the matrix i invertible; which in turn imlie that we can ue Newton method to find the olution. B Proof of Continuity L 2 Given the (Lebegue) integral in the objective, a i not aumed continuou a riori. We how that the otimal olution, however, i. Let t be the average target end er unit. inf a( ) 2 ( a() d )2 f()d.t. a()f()d = d (9) a()f()d td a() 1. We will ignore the nonnegativity contraint for imlicity. The Lagrangian i L = 1 (a() d) 2 f()d + µ()[a() ]f()d 2 ( ) + z a()f()d d ( ) + λ a()f()d td with µ( ) and λ. Note that µ() = if a() <. By Euler-Lagrange, Then, a d + µ() + z + λ = a()f()d = d z = ( µ() λ)f()d and a() = d z µ() λ Now uoe there i a uch that a( ) <. Then, µ( ) = and Then, for >, we have Thu, a() i monotone non-increaing. Let Note that [, F 1 (d/)]. Then, a( ) = d z λ < a() = d z λ µ() d z λ (becaue µ() ) < d z λ = a( ) := inf{ : a() < } a() = < = d z λ m = m 14

15 Here, m i uch that d z λ m =. We now exre z in term of λ,, m rather than in λ, µ( ): z(λ,, m ) = F ( ) + d(f ( m ) F ( ) 1) m (F ( m ) F ( )) Similarly for the Lagrangian: L(λ, ) = ( d)2 F ( ) + λf()d 2 1 m [ 2 z 2 2 λ λd]f()d 2 It can be verified that z = f( )( a( )) (F ( m ) F ( )) L = f( ) 2 ( a( ) 2 λf()d We ee that either the otimum with reect to i achieved on the boundary ( = or = F 1 (d/)) or that a( ) = at the otimum. We are not intereted in the trivial cae = F 1 (d/). We thu have two cae: Cae 1. Cae 2. a() = d z λ < a() = < = (1 + λ( ) m = m We could combine the two cae by allowing to be negative. C KL divergence We want to minimize the KL divergence between a()f()/d and f(): a()f() log a()f()/d d = f() a() a() log d f() d d d which i equivalent to minimizing f()a() log a()d Given the (Lebegue) integral in the objective, a i not aumed continuou a riori. We how that the otimal olution, however, i. Let t be the average target end er unit. Thu we have inf a( ).t. a() log a() f()d a()f()d = d a()f()d td a() 1. (1) Here, t i the average target end er unit. For feaibility, t := F 1 (d/) f()d. If t := f()d, the otimal olution i a() d. 15

16 The Lagrangian i L = a() log a() f()d + µ()[a() ]f()d ( ) ( ) + γ a()f()d d + λ a()f()d td with µ( ) and λ. Note that µ() = if a() <. By Euler-Lagrange, 1 + log a + µ() + γ + λ = which give Then, a = e γ 1 e µ() λ a()f()d = d d = e γ 1 e µ() λ f()d which lead to a() = d Z(λ, µ( )) e µ() λ Now uoe there i a uch that a( ) <. Then, µ( ) = and Then, for >, we have Thu, a() i monotone non-increaing. Let Note that [, F 1 (d/)]. Then, a( ) = d z e λ < a() = d z e λ µ() d z e λ (becaue µ() ) < d z e λ = a( ) := inf{ : a() < } a() = < = d z e λ We now exre z in term of λ, rather than in λ, µ( ): Similarly for the Lagrangian: z(λ, ) = d e λ f()d d F ( ) from which follow d L(λ, ) = F ( ) log + log z(λ, ) + λ f()d λtd L = f( )[log a( ) a() + 1] a()f()d 16

17 When the olution i arametrized by min, max, thee value mut atify min F ( min ) + f()d + max ( max ) f()d = d min max min max min ( max ) max min f()d td. We ee that either the otimum with reect to i achieved on the boundary ( = or = F 1 (d/)) or that a( ) = 4 at the KLotimum. utility We are not intereted in the trivial cae = F 1 (d/). We have two cae for the form of the olution in the KL cae. We have two cae for the form of the olution in the KL cae. Cae 1. Cae 1. a() = d z(λ) e λ < a() = d z(λ) e λ < Cae 2. Cae 2. a() = < a() = = e λ( ) < = e λ( ) Figure 5 how the effect of varying target end on the otimal allocation. Figure 2 how the effect of varying target end on the otimal allocation. a() 1 t < t 1 < t 2 < t 3 < d t = t d t = c t 3 t 2 t 1 u Figure 2: Effect of target end on KL-otimal allocation Figure 5: Effect of target end on KL-otimal allocation Parametric uly 4.1 ditribution Parametric Auly an illutration, ditribution we conider the cae when the uly ditribution f i exonential: f() = γe γ. Note that = 1 γ. A the budget decreae, a tranition from Cae 1 to Cae 2 occur at a certain A an budget. illutration, Until then, we conider the cae when the uly ditribution f i exonential: f() = γe γ a() = d z. Note that e λ where d = 1 z, with equality at the tranitional budget. Demand contraint γ. A the budget decreae, a tranition from Cae 1 to Cae 2 occur γ at d a certain budget. Until then, a() = d z e λ where d z, with equality at the z e (λ+γ) d = d tranitional budget. Demand contraint give γz d = γ e (λ+γ) d = d λ + γ At the otimum, end equal budget: td = γ 8 d(λ + γ) e (λ+γ) d = γ d λ + γ which lead to λ = 1 t γ and z = γt. Again, note that until the tranition haen, d γt t d γ = d. The otimal KL-divergence for d t i given by KLot = γt 1 log γt, that i, which i when t = and i d 1 log d at the tranitional budget. 17