Bidding for Representative Allocations for Display Advertising

Transcription

1 Bidding for Repreentative Allocation for Diplay Advertiing Arpita Ghoh, Preton McAfee, Kihore Papineni, and Sergei Vailvitkii Yahoo! Reearch. {arpita, mcafee, kpapi, Abtract. Diplay advertiing ha traditionally been old via guaranteed contract a guaranteed contract i a deal between a publiher and an advertier to allocate a certain number of impreion over a certain period, for a pre-pecified price per impreion. However, a pot market for diplay ad, uch a the RightMedia Exchange, have grown in prominence, the election of advertiement to how on a given page i increaingly being choen baed on price, uing an auction. A the number of participant in the exchange grow, the price of an impreion become a ignal of it value. Thi correlation between price and value mean that a eller implementing the contract through bidding hould offer the contract buyer a range of price, and not jut the cheapet impreion neceary to fulfill it demand. Implementing a contract uing a range of price, i akin to creating a mutual fund of advertiing impreion, and require randomized bidding. We characterize what allocation can be implemented with randomized bidding, namely thoe where the deired hare obtained at each price i a non-increaing function of price. In addition, we provide a full characterization of when a et of campaign are compatible and how to implement them with randomized bidding trategie. 1 Introduction Diplay advertiing howing graphical ad on regular web page, a oppoed to textual ad on earch page i approximately a $24 billion buine. There are two way in which an advertier looking to reach a pecific audience (for example, 10 million male in California in July 2009) can buy uch ad placement. One i the traditional method, where the advertier enter into an agreement, called a guaranteed contract, directly with the publiher (owner of the webpage). Here, the publiher guarantee to deliver a prepecified number (10 million) of impreion matching the targeting requirement (male, from California) of the contract in the pecified time frame (July 2009). The econd i to participate in a pot market for diplay ad, uch a the RightMedia Exchange, where advertier can buy impreion one pageview at a time: every time a uer load a page with a pot for advertiing, an auction i held where advertier can bid for A full verion of thi paper appear in [6]

2 the opportunity to diplay a graphical ad to thi uer. Both the guaranteed and pot market for diplay advertiing now thrive ide-by-ide. There i demand for guaranteed contract from advertier who want to hedge againt future uncertainty of upply. For example, an advertier who mut reach a certain audience during a critical period of time (e.g around a forthcoming product launch, uch a a movie releae) may not want to rik the uncertainty of a pot market; a guaranteed contract inure the publiher a well againt fluctuation in demand. At the ame time, a pot market allow the advertier to bid for pecific opportunitie, permitting very fine grained targeting baed on uer tracking. Currently, RightMedia run over nine billion auction for diplay ad everyday. How hould a publiher decide which of her upply of impreion to allocate to her guaranteed contract, and which to ell on the pot market? One obviou olution i to fulfill the guaranteed demand firt, and then ell the remaining inventory on the pot market. However, pot market price are often quite different for two impreion that both atify the targeting requirement of a guaranteed contract, ince different impreion have different value. For example, the impreion from two uer with identical demographic can have different value, baed on different earch behavior reflecting purchae intent for one of the uer, but not the other. Since advertier on the pot market have acce to more tracking information about each uer 1, the reulting bid may be quite different for thee two uer. Allocating impreion to guaranteed contract firt and elling the remainder on the pot market can therefore be highly uboptimal in term of revenue, ince two impreion that would fetch the ame revenue from the guaranteed contract might fetch very different price from the pot market 2. On the other hand, imply buying the cheapet impreion on the pot 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, impreion in online advertiing have a common value component becaue advertier generally have different information about a given uer. Thi information (e.g. browing hitory on an advertier ite) i typically relevant to all of the bidder, even though only one bidder may poe thi information. In uch etting, price i a ignal of value in a model of valuation incorporating both common and private value, the price converge to the true value of the item in the limit a the number of bidder goe to infinity ([8, 11], ee alo [7] for dicuion). On average, therefore, the price on the pot market i a good indicator of the value of the impreion, and delivering 1 For example, a car dealerhip advertier may oberve that a particular uer ha been to hi webpage everal time in the previou week, and may be willing to bid more to how a car advertiement to induce a purchae. 2 Conider the following toy example: uppoe there are two opportunitie, the firt of which would fetch 10 cent in the pot market, wherea the econd would fetch only ɛ; both opportunitie are equally uitable for the guaranteed contract which want jut one impreion. Clearly, the firt opportunity hould be old on the pot market, and the econd hould be allocated to the guaranteed contract.

3 cheapet impreion correpond to delivering the lowet quality impreion to the guaranteed contract 3. A publiher with acce to both ource of demand thu face a trade-off between revenue and fairne when deciding which impreion to allocate to the guaranteed contract; thi trade-off i further compounded by the fact that the publiher typically doe not have acce to all the information that determine the value of a particular impreion. Indeed, publiher are often the leat well informed participant about the value of running an ad in front of a uer. For example, when a uer viit a politic ite, Amazon (a an advertier) can ee that the uer recently earched Amazon for an ipod, and Target (a an advertier) can ee they earched target.com for coffee mug, but the publiher only know the uer viited the politic ite. Furthermore, the exact nature of thi trade-off i unknown to the publiher in advance, ince it depend on the pot market bid which are revealed only after the advertiing opportunity i placed on the pot market. The publiher a a bidder. To addre the problem of unknown pot market demand (i.e., the publiher would like to allocate the opportunity to a bidder on the pot market if the bid i high enough, ele to a guaranteed contract), the publiher act, in effect, a a bidder on behalf on the guaranteed contract. That i, the publiher now play two role: that of a eller, by placing hi opportunity on the pot market, and that of a bidding agent, bidding on behalf of hi guaranteed contract. If the publiher own bid turn out to be highet among all bid, the opportunity i won and i allocated to the guaranteed contract. Acting a a bidder allow the publiher to probe the pot market and decide whether it i more efficient to allocate the opportunity to an external bidder or to a guaranteed contract. How hould a publiher model the trade-off between fairne and revenue, and having decided on a trade-off, how hould he place bid on the pot market? An ideal olution i (a) eay to implement, (b) allow for a trade-off between the quality of impreion delivered to the guaranteed contract and hort-term revenue, and (c) i robut to the exact tradeoff choen. In thi work we how preciely when uch an ideal olution exit and how it can be implemented. 1.1 Our Contribution In thi paper, we provide an analytical framework to model the publiher problem of how to fulfill guaranteed advance contract in a etting where there i an alternative pot market, and advertiing opportunitie have a common value component. We give a olution where the publiher bid on behalf of it guaranteed contract in the pot market. The olution conit of two component: an allocation, pecifying the fraction of impreion at each price allocated to a contract, and a bidding trategy, which pecifie how to acquire thi allocation by bidding in an auction. 3 While allocating the cheapet inventory to the guaranteed contract i indeed revenue maximizing in the hort term, in the long term the publiher run the rik of loing the guaranteed advertier by erving them the leat valuable impreion.

4 The quality, or value, of an opportunity i meaured by it price 4. A perfectly repreentative allocation i one which conit of the ame proportion of impreion at every price i.e., a mix of high-quality and low quality impreion. The trade-off between revenue and fairne i modeled uing a budget, or average target pend contraint, for each advertier allocation: the publiher choice of target pend reflect her trade-off between hort-term revenue and quality of impreion for that advertier (thi mut, of coure, be large enough to enure that the promied number of impreion atifying the targeting contraint can be delivered.) Given a target pend 5, a maximally repreentative allocation i one which minimize the ditance to the perfectly repreentative allocation, ubject to the budget contraint. We firt how how to olve for a maximally repreentative allocation, and then how how to implement uch an allocation by purchaing opportunitie in an auction, uing randomized bidding trategie. Organization. We tart out with the ingle contract cae, where the publiher ha jut one exiting guaranteed contract, in Section 2; thi cae i enough to illutrate the idea of maximally repreentative allocation and implementation via randomized bidding trategie. We move on to the more realitic cae of multiple contract in Section 3; we firt prove a reult about which allocation can be implemented in an auction in a decentralized fahion, and derive the correponding decentralized bidding trategie, and comment on olution of the optimal allocation. Full detail, along with experimental validation of thee trategie appear in [6]. Related Work The mot relevant work i the literature on deigning expreive auction and clearing algorithm for online advertiing [9, 2, 10]. Thi literature doe not addre our problem for the following reaon. While it i true that guaranteed contract have coare targeting relative to what i poible on the pot market, mot advertier with guaranteed contract chooe not to ue all the expreivene offered to them. Furthermore, the expreivene offered doe not include attribute like relevant browing hitory on an advertier ite, which could increae the value of an impreion to an advertier, imply becaue the publiher doe not have thi information about the advertiing opportunity. Even with extremely expreive auction, one might till want to adopt a mutual fund trategy to avoid the inider trading problem. That i, if ome bidder poe good information about convertibility, other will till want to randomize their bidding trategy ince bidding a contant price mean alway loing on ome good impreion. Thu, our problem cannot be addreed by the ue of more expreive auction a in [10] the real problem i not lack of expreivity, but lack of information. 4 We emphaize that the aumption being made i not about price being a ignal of value, but rather that impreion do have a common value component given that impreion have a common value, price reflecting value follow from the theorem of Milgrom [8]. Thi aumption i commonly oberved in practice. 5 We point out that we do not addre the quetion of how to et target pend, or the related problem of how to price guaranteed contract to begin with. Given a target pend, we propoe a complete olution to the publiher problem.

5 Another area of reearch focue on electing the optimal et of guaranteed contract. In thi line of work, Feige et al. [5] tudy the computational problem of chooing the et of guaranteed contract to maximize revenue. A imilar problem i tudied by in [3, 1]. We do not addre the problem of how to elect the et of guaranteed contract, but rather take them a given and addre the problem of how to fulfill thee contract in the preence of competing demand from a pot market. 2 Single contract We firt conider the implet cae: there i a ingle advertier who ha a guaranteed contract with the publiher for delivering d impreion. There are a total of d advertiing opportunitie which atify the targeting requirement of the contract. The publiher can alo ell thee opportunitie via auction in a pot 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 landcape. That i, for every unit of upply, the highet bid from all external bidder,which we refer to a the price, i drawn i.i.d from the ditribution 6 f. We aume that the upply and the bid landcape f are known to the publiher 7. Recall that the publiher want to decide how to allocate it inventory between the guaranteed contract and the external bidder in the pot market. Due to penaltie a well a poible long term cot aociated with underdelivering on guaranteed contract, we aume that the publiher want to deliver all d impreion promied to the guaranteed contract. An allocation a(p) i defined a follow: a(p)/ i the proportion of opportunitie at price p purchaed on behalf of the guaranteed contract (the price i the highet (external) bid for an opportunity.) That i, of the f(p)dp impreion available at price p, an allocation a(p) buy a fraction a(p)/ of thee f(p)dp impreion, i.e., a(p)f(p)dp impreion. For example, a contant bid of p mean that for p p, a(p) = 1 with the advertier alway winning the auction, and for p > p, a(p) = 0 ince the advertier would never win. Generally, we will decribe our olution in term of the allocation a(p)/, which mut integrate out to the total demand d: a olution where a(p)/ i larger for higher price correpond to a olution where the guaranteed contract i allocated more high-quality impreion. A another example, a(p)/ = d/ i a perfectly repreentative allocation, integrating out to a total of d impreion, and allocating the ame fraction of impreion at every price point. Not every allocation can be purchaed by bidding in an auction, becaue of the inherent aymmetry in bidding a bid b allow every price below b and rule out every price above; however, there i no way to rule out price below a certain value. That i, we can chooe to exclude high price, but not low price. Before decribing our olution, we tate what kind of allocation a(p)/ can be purchaed by bidding in an auction. 6 Specifically, we do not conider adverarial bid equence; we alo do not model the effect of the publiher own bid on other bid. 7 Publiher uually have acce to data neceary to form etimate of thee quantitie.

6 Propoition 1. A right-continuou allocation a(p)/ can be implemented (in expectation) by bidding in an auction if and only if a(p 1 ) a(p 2 ) for p 1 p 2. Proof. Given a right-continuou non-increaing allocation a(p) (that lie between 0 and 1), define H(p) := 1 a(p). Let p := inf {p : a(p) < }. Then, H i monotone non-decreaing and i right-continuou. Further, H(p ) = 0 and H( ) = 1. Thu, H i a cumulative ditribution function. We place bid drawn from H (the probability of a trictly poitive bid being a(0)/). Then the expected number of impreion won at price p i then exactly a(p)/. Converely, given that bid for the contract are drawn at random from a ditribution H, the fraction of upply at price p that i won by the contract i imply 1 H(p), the probability of it bid exceeding p. Since H i non-decreaing, the allocation (a a fraction of available upply at price p) mut be non-increaing in p. Note that the ditribution H ued to implement the allocation i a different object from the bid landcape f againt which the requiite allocation mut be acquired in fact, it i completely independent of f, and i pecified only by the allocation a(p)/. That i, given an allocation, the bidding trategy that implement the allocation in an auction i independent of the bid landcape f from which the competing bid i drawn. 2.1 Maximally repreentative allocation Ideally the advertier with the guaranteed contract would like the ame proportion of impreion at every price p, i.e., a(p)/ = d/ for all p. (We ignore the poibility that the advertier would like a higher fraction of higher-priced impreion, ince thee cannot be implemented according to Propoition 1 above.) However, the publiher face a trade-off between delivering high-quality impreion to the guaranteed contract and allocating them to bidder who value them highly on the pot market. We model thi by introducing an average unit target pend t, which i the average price of impreion allocated to the contract. A maller (bigger) t deliver more (le) cheap impreion. A we mentioned before, t i part of the input problem, and may depend, for intance, on the price paid by the advertier for the contract. Given a target pend, the maximally repreentative allocation i an allocation a(p)/ that i cloet (according to ome ditance meaure) to the ideal allocation d/, while repecting the target pend contraint. That i, it i the olution to the following optimization problem: ( ) inf a( ) p u a(p), d f(p)dp.t. p a(p)f(p)dp = d (1) pa(p)f(p)dp td p 0 a(p) 1. The objective, u, i a meaure of the deviation of the propoed fraction, a(p)/, from the perfectly repreentative fraction, d/. In what follow, we will

7 conider the L 2 meaure ( a(p) u, d ) = ( a(p) d ) 2, 2 in [6] we alo conider the Kullback-Leibler (KL) divergence. 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 probability ditribution only KL divergence i tatitical [4]. The firt contraint in (1) i imply that we mut meet the target demand d, buying a(p)/ of the f(p)dp opportunitie of price p. The econd contraint i the target pend contraint: the total pend (the pend on an impreion of price p i p) mut not exceed td, where t i a target pend parameter (averaged per unit). A we will hortly ee, the value of t trongly affect the form of the olution. Finally, the lat contraint imply ay that the proportion of opportunitie bought at price p, a(p)/, mut never go negative or exceed 1. Optimality condition: Introduce Lagrange multiplier λ 1 and λ 2 for the firt and econd contraint, and µ 1 (p), µ 2 (p) for the two inequalitie in the lat contraint. The Lagrangian i ( a(p) L = u, d ) ( ) ( f(p)dp + λ 1 d a(p)f(p)dp + λ 2 + µ 1 (p)( a(p))f(p)dp + µ 2 (p)(a(p) )f(p)dp. ) pa(p)f(p)dp td By the Euler-Lagrange condition for optimality, the optimal olution mut atify ( a(p) u, d ) = λ 1 λ 2 p + µ 1 (p) µ 2 (p), where the multiplier µ atify µ 1 (p), µ 2 (p) 0, and each of thee can be nonzero only if the correponding contraint i tight. Thee optimality condition, together with Propoition 1, give u the following: Propoition 2. The maximally repreentative allocation for a ingle contract can be implemented by bidding in an auction for any convex ditance meaure u. The proof follow from the fact that u i increaing for convex u. L 2 utility In thi ubection, we derive the optimal allocation when u, the ditance meaure, i the L 2 ditance, and how how to implement the optimal allocation uing a randomized bidding trategy. In thi cae the bidding trategy turn out to be very imple: to a coin to decide whether or not to bid, and, if bidding, draw the bid value from a uniform ditribution. The coin toing

8 probability and the endpoint of the uniform ditribution depend on the demand and target pend value. Firt we give the following reult about the continuity of the optimal allocation; thi will be ueful in deriving the value that parameterize the optimal allocation. See [6] for the proof. Propoition 3. The optimal allocation a(p) i continuou in p. Note that we do not aume a priori that a( ) i continuou; the optimal allocation turn out to be continuou. The optimality condition, when u i the L 2 ditance, are: a(p) d = λ 1 λ 2 p + µ 1 (p) µ 2 (p), where the nonnegative multiplier µ 1 (p), µ 2 (p) can be non-zero only if the correponding contraint are tight. The olution to the optimization problem (1) then take the following form: For 0 p p min, a(p)/ = 1; for p min p p max, a(p)/ i proportional to C p, i.e., a(p)/ = z(c p); and for p p max, a(p)/ = 0. To find the olution, we mut find p min, p max, z, and C. Since a(p)/ i continuou at p max, we mut have C = p max. By continuity at p min, if p min > 0 then 1 z(c p min ) = 1, o that z = p max p min. Thu, the optimal allocation a(p) i alway parametrized by two quantitie, and ha one of the following two form: 1. a(p)/ = z(p max p) for p p max (and 0 for p p max ). When the olution i parametrized by z, p max, thee value mut atify pmax 0 pmax 0 z(p max p)f(p)dp = d (2) zp(p max p)f(p)dp = td (3) Dividing (2) by (3) eliminate z to give an equation which i monotone in the variable p max, which can be olved, for intance, uing binary earch. 2. a(p)/ = 1 for p p min, and a(p)/ = pmax p p max p min for p p max (and 0 thenceforth). When the olution i parametrized by p min, p max, thee value mut atify pmin 0 F (p min ) + pf(p)dp + pmax (p max p) f(p)dp = d (4) p min p max p min pmax p min p (p max p) p max p min f(p)dp = td. (5) Note that the optimal allocation can be repreented more compactly a a(p) = min{1, z(p max p)}. (6)

9 Effect of varying target pend: Varying the value of the target pend, t, while keeping the demand d fixed, lead to a tradeoff between repreentativene and revenue from elling opportunitie on the pot market, in the following way. The minimum poible target pend, while meeting the target demand (in expectation) i achieved by a olution where p min = p max and a(p)/ = 1 for p le equal thi value, and 0 for greater. The value of p min i choen o that pmin 0 f(p)dp = d p min = F 1 ( d ). Thi olution imply bid a flat value p min, and correpond to giving the cheapet poible inventory to the advertier, ubject to meeting the demand contraint. Thi give the minimum poible total pend for thi value of demand, of td = pmin 0 pf(p)dp = F (p min )E[p p p min ] = de[p p p min ] (Note that the maximum poible total pend that i maximally repreentative while not overdelivering i R = pf(p)dp = de[p] = d p.) A the value of t increae above t, p min decreae and p max increae, until we reach p min = 0, at which point we move into the regime of the other optimal form, with z = 1. A t i increaed further, z decreae from 1, and p max increae, until at the other extreme when the pend contraint i eentially removed, the olution i a(p) = d for all p; i.e., a perfectly repreentative allocation acro price. Thu the value of t provide a dial by which to move from the cheapet allocation to the perfectly repreentative allocation. 2.2 Randomized bidding trategie The quantity a(p)/ i an optimal allocation, i.e., a recommendation to the publiher a to how much inventory to allocate to a guaranteed contract at every price p. However, recall that the publiher need to acquire thi inventory on behalf of the guaranteed contract by bidding in the pot market. The following theorem how how to do thi when u i the L2 ditance. Theorem 1. The optimal allocation for the L 2 ditance meaure can be implemented (in expectation) 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. Proof. From (6) that the optimal allocation can be repreented a a(p) = min{1, z(p max p)}. By Propoition 1, an allocation a(p) = min{1, z(p max p)} can be implemented by bidding in an auction uing the following randomized bidding trategy: with probability min{zp max, 1}, place a bid drawn uniformly at random from the range [max{p max 1 z, 0}, p max].

10 3 Multiple contract We now tudy the more realitic cae where the publiher need to fulfill multiple guaranteed contract with different advertier. Specifically, uppoe there are m advertier, with demand d j. A before, there are a total of d j advertiing opportunitie available to the publiher. 8 An allocation a j (p)/ i the proportion of opportunitie purchaed on behalf of contract j at price p. Of coure, the um of thee allocation cannot exceed 1 for any p, which correpond to acquiring all the upply at that price. A in the ingle contract cae, we are firt intereted in what allocation a j (p) are implementable by bidding in an auction. However, in addition to being implementable, we would like allocation that atify an additional practical requirement, explained below. Notice that the publiher, acting a a bidding agent, now need to acquire opportunitie to implement the allocation for each of the guaranteed contract. When an opportunity come along, therefore, the publiher need to decide which of the contract (if any) will receive that opportunity. There are two way to do thi: the publiher ubmit one bid on behalf of all the contract; if thi bid win, the publiher then elect one amongt the contract to receive the opportunity. Alternatively, the publiher can ubmit one bid for each contract; the winning bid then automatically decide which contract receive the opportunity. We refer to the former a a centralized trategy and the latter a a decentralized trategy. There are ituation where the publiher will need to chooe the winning advertier prior to eeing the price, that i, the highet bid from the pot market. For example, to reduce latency in placing an advertiement, the auction mechanim may require that the bid be accompanied by the advertiement (or it unique identifier). A decentralized trategy automatically fulfill thi requirement, ince the choice of winning contract doe not depend upon knowing the price. In a centralized trategy, thi requirement mean that the relative fraction won at price p, a i (p)/a j (p), are independent of the price p when thi happen, the choice of advertier can be made (by chooing at random with probability proportional to a j ) without knowing the price. A before, we will be intereted in implementing optimal (i.e., maximally repreentative) allocation. We will, therefore, concentrate on characterizing allocation which can be implemented via a decentralized trategy. In the full verion of the paper [6] we how how to compute the optimal allocation in the preence of multiple contract. 3.1 Decentralization In thi ection, we examine what allocation can be implemented via a decentralized trategy. Note that it i not ufficient to imply ue a ditribution H j = 1 aj(p) a j(0) a in Propoition 1, ince thee contract compete amongt each 8 In general, not all of thee opportunitie might be uitable for every contract; we do not conider thi here for clarity of preentation. However the ame idea and method can be applied in that cae and the reult are qualitatively imilar.

11 other a well. Specifically, uing the ditribution 1 aj(p) a j(0) will lead to too few opportunitie being purchaed for contract j, ince thi ditribution i deigned to compete againt f alone, rather than 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 (p)/ of opportunitie being purchaed 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 implementation ue ditribution H j to bid for impreion.then, contract j win an impreion at price p with probability a j (p) = p H k (x) h j (x)dx, k j ince to win, the bid for contract j mut be larger than p and larger than the bid placed by each of the remaining m 1 contract. Since all the quantitie in the integrand are nonnegative, a j i non-increaing in p. Now aume that a j are differentiable a.e. and non-increaing. Let A(p) := j a j (p) be the total fraction of opportunitie at price p that the publiher need to acquire. Clearly, a j mut atify A(p), p. Let p := inf{p : A(p) < }. Let { R e p H j (p) := a j (x)/( A(x))dx p > p 0 ele (7) Then, H j (p) 0 and i continuou. Since a j (p) i non-increaing, H j(p) i monotone non-decreaing. Further, H( ) = 1 and H j (p ) = 0. Thu, H j i a ditribution function. We can verify that bidding according to H j will reult in the deired allocation (ee [6] for detail). Thu, we have contructed ditribution function H j (p) which implement the given non-increaing (and a.e. differentiable) allocation a j (p). If any a j i increaing at any point, the et of campaign cannot be decentralized. The following theorem generalize Propoition 1: Theorem 2. A et of allocation a j (p) can be implemented in an auction via a decentralized trategy iff each a j (p) i non-increaing in p, and j a j(p)/ 1. Having determined which allocation can be implemented by bidding in an auction in a decentralized fahion, we turn to the quetion of finding uitable allocation to implement. A in the ingle contract cae, we would like to implement allocation that are maximally repreentative, given the pend contraint. A we how in [6], the optimal allocation i decentralizable in two cae: 1. The target pend are uch that the olution decouple. In thi cae the allocation for each contract i independent of the other; we olve for the parameter of each allocation a in Section 2.1.

12 a j(p) a k (p) 2. The target pend are uch that, for all j, k, i independent of p. In thi cae we need to olve for the common lope and p min, and the contract pecific value p j max, which together determine the allocation. Thi can be done uing, for intance, Newton method. When the target pend are uch that the allocation i not decentralizable, the vector of target pend can be increaed to reach a decentralizable allocation. One way i to cale up the target pend uniformly until they are large enough to admit a eparable olution; thi ha the advantage of preerving the relative ratio of target pend. The minimum multiplier which render the allocation decentralizable can be found numerically, uing for intance binary earch. 4 Concluion Moving guaranteed contract into an exchange environment preent a variety of challenge for a publiher. Randomized bidding i a ueful compromie between minimizing the cot and maximizing the quality of guaranteed contract. It i akin to the mutual fund trategy common in the capital aet pricing model. We provide a readily computable olution for ynchronizing an arbitrary number of guaranteed campaign in an exchange environment. Moreover, the olution we detail appear table with real data. Reference [1] Mohe Babaioff, Jaon Hartline, and Robert Kleinberg. Selling banner ad: Online algorithm with buyback. In 4th Workhop on Ad Auction, [2] C. Boutilier, D. Parke, T. Sandholm, and W. Walh. Expreive banner ad auction and model-baed online optimization for clearing. In National Conference on Artificial Intelligence (AAAI), [3] Florin Contantin, Jon Feldman, S Muthukrihnan, and Martin Pal. Online ad lotting with cancellation. In 4th Workhop on Ad Auction, [4] I. Cizar. Why leat quare and maximum entropy? an axiomatic approach to interference for linear invere problem. Annal of Statitic, 19(4): , [5] Uriel Feige, Nicole Immorlica, Vahab S. Mirrokni, and Hamid Nazerzadeh. A combinatorial allocation mechanim with penaltie for banner advertiing. In Proceeding of ACM WWW, page , [6] Arpita Ghoh, Preton McAfee, Kihore Papineni, and Sergei Vailvitkii. Bidding for repreentative allocation for diplay advertiing. CoRR, ab/ , [7] R. Preton McAfee and John McMillan. Auction and bidding. Journal of Economic Literature, 25(2): , [8] Paul R. Milgrom. A convergence theorem for competitive bidding with differential information. Econometrica, 47(3): , May [9] David Parke and Tuoma Sandholm. Optimize-and-dipatch architecture for expreive ad auction. In 1t Workhop on Ad Auction, [10] Tuoma Sandholm. Expreive commerce and it application to ourcing: How we conducted $35 billion of generalized combinatorial auction. 28(3):45 58, [11] Robert Wilon. A bidding model of perfect competition. Rev. Econ. Stud., 44(3): , 1977.