{"title": "Dynamic Revenue Sharing", "book": "Advances in Neural Information Processing Systems", "page_first": 2681, "page_last": 2689, "abstract": "Many online platforms act as intermediaries between a seller and a set of buyers. Examples of such settings include online retailers (such as Ebay) selling items on behalf of sellers to buyers, or advertising exchanges (such as AdX) selling pageviews on behalf of publishers to advertisers. In such settings, revenue sharing is a central part of running such a marketplace for the intermediary, and fixed-percentage revenue sharing schemes are often used to split the revenue among the platform and the sellers. In particular, such revenue sharing schemes require the platform to (i) take at most a constant fraction \\alpha of the revenue from auctions and (ii) pay the seller at least the seller declared opportunity cost c for each item sold. A straightforward way to satisfy the constraints is to set a reserve price at c / (1 - \\alpha) for each item, but it is not the optimal solution on maximizing the profit of the intermediary. While previous studies (by Mirrokni and Gomes, and by Niazadeh et al) focused on revenue-sharing schemes in static double auctions, in this paper, we take advantage of the repeated nature of the auctions. In particular, we introduce dynamic revenue sharing schemes where we balance the two constraints over different auctions to achieve higher profit and seller revenue. This is directly motivated by the practice of advertising exchanges where the fixed-percentage revenue-share should be met across all auctions and not in each auction. In this paper, we characterize the optimal revenue sharing scheme that satisfies both constraints in expectation. Finally, we empirically evaluate our revenue sharing scheme on real data.", "full_text": "Dynamic Revenue Sharing\u2217\n\nSantiago Balseiro\nColumbia University\nNew York City, NY\n\nsrb2155@columbia.edu\n\nMax Lin\nGoogle\n\nNew York City, NY\nwhlin@google.com\n\nVahab Mirrokni\n\nGoogle\n\nNew York City, NY\n\nmirrokni@google.com\n\nRenato Paes Leme\n\nGoogle\n\nNew York City, NY\n\nrenatoppl@google.com\n\nSong Zuo\u2020\n\nTsinghua University\n\nBeijing, China\n\nsongzuo.z@gmail.com\n\nAbstract\n\nMany online platforms act as intermediaries between a seller and a set of buyers.\nExamples of such settings include online retailers (such as Ebay) selling items\non behalf of sellers to buyers, or advertising exchanges (such as AdX) selling\npageviews on behalf of publishers to advertisers. In such settings, revenue sharing\nis a central part of running such a marketplace for the intermediary, and \ufb01xed-\npercentage revenue sharing schemes are often used to split the revenue among the\nplatform and the sellers. In particular, such revenue sharing schemes require the\nplatform to (i) take at most a constant fraction \u03b1 of the revenue from auctions and\n(ii) pay the seller at least the seller declared opportunity cost c for each item sold.\nA straightforward way to satisfy the constraints is to set a reserve price at c/(1\u2212 \u03b1)\nfor each item, but it is not the optimal solution on maximizing the pro\ufb01t of the\nintermediary.\nWhile previous studies (by Mirrokni and Gomes, and by Niazadeh et al) focused on\nrevenue-sharing schemes in static double auctions, in this paper, we take advantage\nof the repeated nature of the auctions. In particular, we introduce dynamic revenue\nsharing schemes where we balance the two constraints over different auctions\nto achieve higher pro\ufb01t and seller revenue. This is directly motivated by the\npractice of advertising exchanges where the \ufb01xed-percentage revenue-share should\nbe met across all auctions and not in each auction. In this paper, we characterize\nthe optimal revenue sharing scheme that satis\ufb01es both constraints in expectation.\nFinally, we empirically evaluate our revenue sharing scheme on real data.\n\n1\n\nIntroduction\n\nThe space of internet advertising can be divided in two large areas: search ads and display ads. While\nsimilar at \ufb01rst glance, they are different both in terms of business constraints in the market as well as\nalgorithmic challenges. A notable difference is that in search ads the auctioneer and the seller are the\nsame party, as the same platform owns the search page and operates the auction. Thus search ads are\na one-sided market: the only agents outside the control of the auctioneer are buyers. In display ads,\non the other hand, the platform operates the auction but, in most cases, it does not own the pages in\n\u2217We thank Jim Giles, Nitish Korula, Martin P\u00e1l, Rita Ren and Balu Sivan for the fruitful discussion and their\ncomments on early versions of this paper. We also thank the anonymous reviewers for their helpful comments.\nA full version of this paper can be found at https://ssrn.com/abstract=2956715.\n\u2020The work was done when this author was an intern at Google. This author was supported by the National\nBasic Research Program of China Grant 2011CBA00300, 2011CBA00301, the Natural Science Foundation of\nChina Grant 61033001, 61361136003, 61303077, 61561146398, a Tsinghua Initiative Scienti\ufb01c Research Grant\nand a China Youth 1000-talent program.\n31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.\n\n\fwhich the ads are displayed, making the main problem the design of a two-sided market, referred to\nas ad exchanges.\nThe problem of designing an ad exchange can be decomposed in two parts: the \ufb01rst is to design\nan auction, which will specify how an ad impression will be allocated among different prospective\nbuyers (advertisers) and how they will be charged from it. The second component is a revenue sharing\nscheme, which speci\ufb01es how the revenue collected from buyers will be split between the seller (the\npublisher) and the platform. Traditionally the problems of designing an auction and designing a\nrevenue sharing scheme have been merged in a single one called double auction design. This was the\ntraditional approach taken by [4], [3] and more recently in the algorithmic work of [2, 5]. The goals\nin those approaches have been to maximize ef\ufb01ciency in the market, maximize pro\ufb01t of the platform\nand to characterize when the pro\ufb01t maximizing policy is a simple one.\nThose objectives however, do not entirely correspond to actual problem faced by advertising ex-\nchanges. Take platform-pro\ufb01t-maximization, for example. The ad-exchange business is a highly\ncompetitive environment. A web publisher (seller) can send their ad impressions to a dozen of\ndifferent exchanges. If an exchange tries to extract all the surplus in the form of pro\ufb01t, web publishers\nwill surely migrate to a less greedy platform. In order to retain their inventory, exchanges must align\ntheir incentives with the incentives of those of web publishers.\nA good practical solution, which has been adopted by multiple real world platforms, is to declare a\n\ufb01xed revenue sharing scheme. The exchange promises it will keep at most an \u03b1-fraction of pro\ufb01ts,\nwhere the constant \u03b1 is typically the outcome of a business negotiation between the exchange and the\nweb publisher. After the fraction is agreed, the objective of the seller and the exchange are aligned.\nThe exchange maximizes pro\ufb01ts by maximizing the seller\u2019s revenue.\nIf revenue sharing was the only constraint, the exchange could simply ignore sellers and run an\noptimal auction among buyers. In practice, however, web-publishers have outside options, typically\nin the form of reservation contracts, which should be taken into account by the exchange. Reservation\ncontracts are a very traditional form of selling display ads that predates ad exchanges, where buyers\nand sellers make agreements of\ufb02ine specifying a volume of impressions to be transacted, a price per\nimpression and a penalty for not satisfying the contract. Those agreements are entered in a system (for\nexample Google\u2019s Doubleclick for Publishers) that manages reservations on behalf of the publisher.\nThis reservation system determines for each arriving impression the best matching of\ufb02ine contract\nthat impression could be allocated to as well as the cost of not allocating that impression. The cost of\nnot allocating an impression takes into account the potential revenue from allocating to a contract and\nthe probability of paying a penalty for not satisfying the contract.\nFrom our perspective, it is irrelevant how a cost is computed by reservation systems. It is suf\ufb01cient\nto assume that for each impression, the publisher has an opportunity cost and it is only willing to\nsell that particular impression in the exchange if its payout for that impression exceeds the cost.\nExchanges therefore, allow the publisher to submit a cost and only sell that impression if they are\nable to pay the publisher at least the cost per that impression.\nWe design the following simple auction and revenue sharing scheme that we call the na\u00efve policy:\n\n\u2022 seller sends to the exchange an ad impression with cost c.\n\u2022 exchange runs a second price auction with reserve r \u2265 c/(1 \u2212 \u03b1).\n\u2022 if the item is sold the exchange keeps an \u03b1 fraction of the revenue and sends the remaining\n\n1 \u2212 \u03b1 fraction to the seller.\n\nThis scheme is pretty simple and intuitive for each participant in the market. It guarantees that if the\nimpression is sold, the revenue will be at least c/(1 \u2212 \u03b1) and therefore the seller\u2019s payout will be at\nleast c. So both the minimum payout and revenue sharing constraints are satis\ufb01ed with probability\n1. This scheme has also the advantage of decoupling the auction and the revenue sharing problem.\nThe platform is free to use any auction among the buyers as long as it guarantees that whenever the\nimpression is matched, the revenue extracted from buyers is at least c/(1 \u2212 \u03b1).\nDespite being simple, practical and allowing the exchange to experiment with the auction without\nworrying about revenue sharing, this mechanism is sub-optimal both in terms of platform pro\ufb01t and\npublisher payout. The exchange might be willing to accept a revenue share lower than \u03b1 if this grants\nmore freedom in optimizing the auction and extracting more revenue.\nMore generally, the exchange might exploit the repeated nature of the auction to improve revenue\neven further by adjusting the revenue share dynamically based on the bids and the cost. In this setting,\n\n2\n\n\fwe can think of the revenue share constraints to be enforced on average, i.e., over a sequence of\nauctions the platform is required to bound by \u03b1 the ratio of the aggregate pro\ufb01t and the aggregate\nrevenue collected from buyers. This allows the platform to increase the revenue share on certain\nqueries and reduce in others.\nIn the repeated auctions setting, the exchange is also allowed to treat the minimum cost constraint on\naggregate: the payout for the seller needs to be at least as large as the sum of costs of the impressions\nmatched. The exchange can implement this in practice by always paying the seller at least his cost\neven if the revenue collected from buyers is less than the cost. This would cause the exchange to\noperate at a loss for some impressions. But this can be advantageous for the exchange on aggregate if\nit is able to offset these losses by leveraging other queries with larger pro\ufb01t margins.\nIn this paper, we attempt to characterize the optimal scheme for repeated auctions and measure on\ndata the improvement with respect to the simple revenue sharing scheme discussed above.\nFinally, while we discuss the main application of our results in the context of advertising exchanges,\nour model and results apply to the broad space of platforms that serve as intermediaries between\nbuyers and sellers, and help run many repeated auctions over time. The issue of dynamic revenue\nsharing also arises when Amazon or eBay act as a platform and splits revenues from a sale with\nthe sellers, or when ride-sharing services such as Uber or Lyft split the fare paid by the passenger\nbetween the driver and the platform. Uber for example mentions in their website3 that: \u201cDrivers\nusing the partner app are charged an Uber Fee as a percentage of each trip fare. The Uber Fee varies\nby city and vehicle type and helps Uber cover costs such as technology, marketing and development\nof new features within the app.\u201d\n\n1.1 Our Results and Techniques\n\nWe propose different designs of auctions and revenue sharing policies in exchanges and analyze\nthem both theoretically and empirically on data from a major ad exchange. We compare against the\nna\u00efve policy described above. We compare policies in terms of seller payout, exchange pro\ufb01t and\nmatch-rate (number of impressions sold). We note that match-rate is an important metric in practice,\nsince it represents the volume of inventory transacted in the exchange and it is a proxy for the volume\nof the ad market this particular exchange is able to capture.\nFor the auction, we restrict our attention to second price auctions with reserve prices, since we aim at\nusing theory as a guide to inform decisions about practical designs that can be implemented in real\nad-exchanges. To be implementable in practice the designs need to follow the industry practice of\nrunning second-price auctions with reserves. This design will be automatically incentive compatible\nfor buyers. On the seller side, instead of enforcing incentive compatibility, we will assume that\nimpression costs are reported truthfully. Note that the revenue sharing contract guarantees, at least\npartially, when the constraint binds (which always happens in practice), the goals of the seller and the\nplatform are partially aligned: maximizing pro\ufb01t is the same as maximizing revenue. Thus, sellers\nhave little incentive to misreport their costs. In fact, this is one of the main reason that so many\nreal-world platforms such as Uber adopt \ufb01xed revenue sharing contracts. In the ads market, moreover,\nsellers are also typically viewed as less strategic and reactive agents. Thus, we believe that the latter\nassumption is not too restrictive in practice.4\nWe will also assume Bayesian priors on buyer\u2019s valuations and on seller\u2019s costs. For the sake of\nsimplicity, we will start with the assumption that seller costs are constant and show in the full version\nhow to extend our results to the case where costs are sampled from a distribution.\nWe will focus on the exchange pro\ufb01t as our main objective function. While this paper will take the\nperspective of the exchange, the policies proposed will also improve seller\u2019s payout with respect\nto the na\u00efve policy. The reason is simple: the na\u00efve policy keeps exactly \u03b1 fraction of the revenue\nextracted from buyers as pro\ufb01t. Any policy that keeps at most \u03b1 and improves pro\ufb01t, should improve\nrevenue extracted from buyers at least at the same rate and hence improve seller\u2019s payout.\nSingle Period Revenue Sharing. We \ufb01rst study the case where exchange is required to satisfy\nthe revenue sharing constraint in each period, i.e., for each impression at most an \u03b1-fraction of the\n\n3See https://www.uber.com/info/how-much-do-drivers-with-uber-make/\n4While in this paper we focus on the dynamic optimization of revenue sharing schemes when agents report\ntruthfully, it is still an interesting avenue of research to study the broader market design question of designing\ndynamic revenue sharing schemes while taking into account agents\u2019 incentives.\n\n3\n\n\frevenue can be retained as pro\ufb01t. We characterize the optimal policy. We \ufb01rst show that the optimal\npolicy always sets the reserve price above the seller\u2019s cost, but not necessarily above c/(1 \u2212 \u03b1). The\nexchange might voluntarily want to decrease its revenue share if this grants freedom to set lower\nreserve prices and extract more revenue from buyers.\nWhen the opportunity cost of the seller is low, the optimal policy for the exchange ignores the seller\u2019s\ncost and prices according to the optimal reserve price. When the opportunity cost is high, pricing\naccording to c/(1 \u2212 \u03b1) is again not optimal because demand is inelastic at that price. The exchange\ninternalizes the opportunity cost, prices between c and c/(1 \u2212 \u03b1), and reduces its revenue share if\nnecessary. For intermediate values of the opportunity cost, the exchange is better off employing the\nna\u00efve policy and pricing according to c/(1 \u2212 \u03b1).\nMulti Period Revenue Sharing. We then study the case where the revenue share constraint is\nimposed over the aggregate buyers\u2019 payments. We provide intuition on the structure of the optimal\npolicy by \ufb01rst solving a Lagrangian relaxation and then constructing an asymptotically optimal heuris-\ntic policy (satisfying the original constraints) based on the optimal relaxation solution. In particular,\nwe introduce a Lagrange multiplier for the revenue sharing constraint to get the optimal solution\nto the Lagrangian relaxation. The optimal revenue sharing policy obtained from the Lagrangian\nrelaxation pays the publisher a convex combination between his cost c and a fraction (1 \u2212 \u03b1) of the\nrevenue obtained from buyers. Depending on the value of the multiplier, the reserve price could be\nbelow c, exposing the platform to the possibility of operating at a loss in some auctions.\nThe policy obtained from the Lagrangian relaxation, while intuitive, only satis\ufb01es the revenue sharing\nand cost constraints in expectation. Because this is not feasible for the platform, we discuss heuristic\npolicies that approximate that policy in the limit, but satisfy the constraints surely in aggregate over\nthe T periods. Then we discuss an even stronger policy that satis\ufb01es the aggregate constraints for any\npre\ufb01x, i.e., at any given time t, the constraints are satis\ufb01ed in aggregate from time 1 to t.\nComparative Statics. We compare the structure of the single period and multi period policies. The\n\ufb01rst insight is that the optimal multi-period policy uses lower reserve prices therefore matching more\nqueries. The key insight we obtain from the comparison is that multi-period revenue sharing policies\nare particularly effective when markets are thick, i.e. when a second highest bid is above a rescaled\nversion of the cost often and cost are not too high.\nEmpirical Insights. To complement our theoretical results, we conduct an empirical study simu-\nlating our revenue sharing policies on real world data from a major ad exchange. The data comes\nfrom bids in a second price auction with reserves (for a single-slot), which is truthful. Our study\ncon\ufb01rms the effectiveness of the multi period revenue sharing policies and single period revenue\nsharing policies over the na\u00efve policy. The results are consistent for different values of \u03b1: the pro\ufb01t\nlifts of single period revenue sharing policies are +1.23% \u223c +1.64% and the lifts of multi period\nrevenue sharing policies are roughly 5.5 to 7 times larger (+8.53% \u223c +9.55%).\nWe do an extended overview in Section 7, but leave the further details to the full version. We omit the\nrelated work here, which can be can be found in the full version.\n\n2 Preliminaries\n\nSetting. We study a discrete-time \ufb01nite horizon setting in which items arrive sequentially to an\nintermediary. We index the sequence of items by t = 1, . . . , T . There are multiple buyers bidding in\nthe intermediary (the exchange) and the intermediary determines the winning bidder via a second\nprice auction. We assume that the bids from the buyers are drawn independently and identically\ndistributed across auctions, but potentially correlated across buyers for a given auction.\nWe will assume that the pro\ufb01t function of the joint distribution of bids is quasi-concave. The expected\npro\ufb01t function corresponds to the expected revenue of a second price auction with reserve price r and\nopportunity cost c:\n\n\u03a0(r, c) = E(cid:2)1{bf \u2265 r} (max(r, bs) \u2212 c)(cid:3) .\n\nt and bs\n\nt are the highest- and second-highest bid at time t. Our assumption on the bid\n\nwhere bf\ndistribution will be as follows:\nAssumption 2.1. The expected pro\ufb01t function \u03a0(r, c) is quasi-concave in r for each c.\n\n4\n\n\fThe previous assumption is satis\ufb01ed, for example, if bids are independent and identically distributed\naccording to a distribution with increasing hazard rates (see, e.g., (author?) [1]).\nMechanism. The seller submitting the items sets an opportunity cost of c \u2265 0 for the items. The\npro\ufb01t of the intermediary is the difference between the revenue collected from the buyers and the\npayments made to the seller. The intermediary has agreed to a revenue sharing scheme that limits the\npro\ufb01t of the intermediary to at most \u03b1 \u2208 (0, 1) of the total revenue collected from the buyers.\nThe intermediary implements a non-anticipative adaptive policy \u03c0 that maps the history at time t\n: R+ \u2192 R+\nt \u2208 R+ for the second price auction and a payment function p\u03c0\nto a reserve price r\u03c0\nthat determines the amount to be paid to the seller as a function of the buyers\u2019 payments. That is,\nt \u2265 r\u03c0\nthe item is sold whenever the highest bid is above the reserve price, or equivalently bf\nt . The\nintermediary\u2019s revenue is equal to the buyers\u2019 payments of max(r\u03c0\nt) and the seller\u2019s revenue\nt)). The intermediary\u2019s pro\ufb01t is given by the difference of the buyers\u2019\nis given by p\u03c0\npayments and the payments to the seller, i.e., max(r\u03c0\nt)). From the perspective\nof the buyers, the mechanism implemented by the intermediary is a second price auction with\n(potentially dynamic) reserve price r\u03c0\nt . The intermediary\u2019s problem amounts to maximizing pro\ufb01ts\nsubject to the revenue sharing constraint. The revenue sharing constraint can be imposed at every\nsingle period or over multiple periods. We discuss each model at a time.\nNa\u00efve revenue sharing scheme. The most straightforward revenue sharing scheme is the one that\nsets a reserve above c/(1 \u2212 \u03b1) and pay the sellers a (1 \u2212 \u03b1)-fraction of the revenue:\n\nt) \u2212 p\u03c0\n\nt (max(r\u03c0\n\nt (max(r\u03c0\n\nt , bs\n\nt , bs\n\nt , bs\n\nt , bs\n\nt\n\nt \u2265 c\nr\u03c0\n\n1\u2212\u03b1 ,\n\nt (x) = (1 \u2212 \u03b1)x.\np\u03c0\n\nr\u2217 = arg maxr\u2265c/(1\u2212\u03b1) \u03a0(r, 0) .\n\n(1)\nt). Thus, the\n\nSince the revenue sharing is \ufb01xed, the intermediary\u2019s pro\ufb01t is given by \u03b1 max(r\u03c0\nintermediary optimizes pro\ufb01ts by optimizing revenues, and the optimal reserve price is given by:\n\nt , bs\n\nThe na\u00efve revenue sharing scheme sets a reserve above c/(1 \u2212 \u03b1) and pays the seller (1 \u2212 \u03b1) of\nthe buyers\u2019 payments. This guarantees that the payment to the seller is always no less than c, by\nconstruction, because the payment of the buyers is at least the reserve price. Since the intermediary\u2019s\npro\ufb01t is a fraction \u03b1 of the buyers\u2019 payment, the seller\u2019s cost does not appear in the objective, and the\nobjective of the seller is \u03b1\u03a0(r, 0). Note, however, that the seller\u2019s cost does appear as a constraint in\nthe intermediary\u2019s optimization problem: the reserve price should be at least c/(1 \u2212 \u03b1).\nThis is the baseline that we will use to compare the proposed policies with in the experiment section.\nThis policy is suboptimal for various reasons. Consider for example the extreme case where the\nbuyers alway bid more than c and less than c/(1 \u2212 \u03b1). In this case, the pro\ufb01t from the na\u00efve revenue\nsharing scheme is zero. However, the intermediary can still obtain a non-zero pro\ufb01t by setting the\nreserve somewhere between c and c/(1 \u2212 \u03b1), which results in a revenue share less than \u03b1. If the\nrevenue sharing constraint is imposed over multiple periods instead of each single period, we are able\nto dynamically balance out the de\ufb01cit and surplus of the revenue sharing constraint over time.\n\n3 Single Period Revenue Sharing Scheme\n\n(cid:80)T\nt \u2265 r\u03c0\n\u03c0\nt (x) \u2265 (1 \u2212 \u03b1)x ,\ns.t. p\u03c0\nt (x) \u2265 c ,\n\u2200x .\np\u03c0\n\nt=1\n\n\u2200x\n\nIn this case the revenue sharing scheme imposes that in every single period the pro\ufb01t of the interme-\ndiary is at most \u03b1 of the buyers\u2019 payment. We start by formulating the pro\ufb01t maximization problem\nfaced by the intermediary as a mathematical program with optimal value J S.\nt (max(r\u03c0\n\nE(cid:2)1{bf\n\nt } (max(r\u03c0\n\nt)))(cid:3)\n\nJ S (cid:44) max\n\nt) \u2212 p\u03c0\n\nt , bs\n\nt , bs\n\n(2a)\n\n(2b)\n(2c)\nThe objective (2a) gives the pro\ufb01t of the intermediary as the difference between the payments collected\nfrom the buyers and the payments made to the seller. The revenue sharing constraint (2b) imposes that\nintermediary\u2019s pro\ufb01t is at most a fraction \u03b1 of the total revenue, or equivalently (x \u2212 p\u03c0\nt (x))/x \u2264 \u03b1\nwhere x is the payment from the buyers. The \ufb02oor constraint (2c) imposes that the seller is paid at\nleast c. These constraints are imposed at every auction.\nWe next characterize the optimal decisions of the seller in the single period model. Some de\ufb01nitions\nare in order. Let r\u2217(c) be an optimal reserve price in the second price auction if the seller\u2019s cost is c:\n\nr\u2217(c) = arg maxr\u22650 \u03a0(r, c).\n\n5\n\n\fTo avoid trivialities we assume that the optimal reserve price is unique. Because the pro\ufb01t function\n\u03a0(r, c) has increasing differences in (r, c) then the optimal reserve price is non-decreasing with the\ncost, that is, r\u2217(c) \u2265 r\u2217(c(cid:48)) for c \u2265 c(cid:48).\nOur main result in this section characterizes the optimal decision of the intermediary in this model.\nt (x) = max(c, (1 \u2212 \u03b1)x) and\nTheorem 3.1. The optimal decision of the intermediary is to set p\u03c0\nt = max{min{\u00afc, r\u2217(c)}, r\u2217(0)} where \u00afc = c/(1 \u2212 \u03b1).\nr\u03c0\nThe reserve price \u00afc = c/(1 \u2212 \u03b1) in the above theorem is the na\u00efve reserve price that satis\ufb01es the\nrevenue sharing scheme by in\ufb02ating the opportunity cost by 1/(1 \u2212 \u03b1). When the opportunity cost\nc is very low (\u00afc \u2264 r\u2217(0)), pricing according to \u00afc is not optimal because demand is elastic at \u00afc and\nthe intermediary can improve pro\ufb01ts by increasing the reserve price. Here the intermediary ignores\nt = r\u2217(0) and pays the seller according to\nthe opportunity cost, prices optimally according to r\u03c0\nt (x) = (1 \u2212 \u03b1)x. When the opportunity cost c is very high (\u00afc \u2265 r\u2217(c)), pricing according to \u00afc\np\u03c0\nis again not optimal because demand is inelastic at \u00afc and the intermediary can improve pro\ufb01ts by\ndecreasing the reserve price. Here the intermediary internalizes the opportunity cost, prices optimally\naccording to r\u03c0\n\nt = r\u2217(c) and pays the seller according to p\u03c0\n\nt (x) = max(c, (1 \u2212 \u03b1)x).\n\n4 Multi Period Revenue Sharing Scheme\n\nt ))(cid:3)\n\nE(cid:2)1{bf\n\n(cid:80)T\ns.t. (cid:80)T\n(cid:80)T\nt=1 1{bf\nt=1 1{bf\n\nt=1\n\nIn this case the revenue sharing scheme imposes that the aggregate pro\ufb01t of the intermediary is at most\n\u03b1 of the buyers\u2019 aggregate payment. Additionally, in this model the opportunity costs are satis\ufb01ed on\nan aggregate fashion over all actions, that is, the payments to the seller need to be at least the \ufb02oor\nprice times the number of items sold. The intermediary decision\u2019s problem can be characterized by\nthe following mathematical program with optimal value J M , where x\u03c0\n\nt = max(r\u03c0\n\nt , bs\nt)\n\nt (x\u03c0\n\nt ) \u2265 0 ,\n\nt \u2265 r\u03c0\nt } (p\u03c0\nt } (p\u03c0\n\nt } (x\u03c0\nt (x\u03c0\nt (x\u03c0\n\nt \u2265 r\u03c0\nt \u2265 r\u03c0\n\nJ M (cid:44) max\u03c0\n\nt \u2212 p\u03c0\nt ) \u2212 (1 \u2212 \u03b1)x\u03c0\nt ) \u2212 c) \u2265 0 , .\n\n(3a)\n(3b)\n(3c)\nThe objective (3a) gives the pro\ufb01t of the intermediary as the difference between the payments\ncollected from the buyers and the payments made to the seller. The revenue sharing constraint (3b)\nimposes that intermediary\u2019s pro\ufb01t is at most a fraction \u03b1 of the total revenue. The \ufb02oor constraint (3c)\nimposes that the seller is paid at least c. These constraints are imposed over the whole horizon.\nThe stochastic decision problem (3) can be solved via Dynamic Programming. To provide some\nintuition of the structure of the optimal solution we solve a Lagrangian relaxation of the problem\nwhere we introduce a dual variable \u03bb \u2265 0 for the \ufb02oor constraint (3c) and a dual variable \u00b5 \u2265 0 for\nthe revenue sharing constraint (3b). Lagrangian relaxations provide upper bounds on the optimal\nobjective value and introduce heuristic policies of provably good performance in many settings (e.g.,\nsee [7]). Moreover, we shall see the optimal policy derived from the Lagrangian relaxation is optimal\nfor problem (3) if constraints (3c) and (3b) are imposed in expectation instead of almost surely:\nTheorem 4.1. Let \u00b5\u2217 \u2208 arg min0\u2264\u00b5\u22641\nt (x) = (1 \u2212 \u00b5\u2217)c + \u00b5\u2217(1 \u2212 \u03b1)x and\nt = r\u2217(c(\u00b5\u2217)) is optimal for problem (3) when constraints (3c) and (3b) are imposed in expectation\nr\u03c0\ninstead of almost surely, where\n\n\u02c6\u03c6(\u00b5). The policy p\u03c0\n\n\u02c6\u03c6(\u00b5) (cid:44) T(cid:0)1 \u2212 \u00b5(1 \u2212 \u03b1)(cid:1) supr \u03a0(cid:0)r,\n\n(cid:1).\n\n(1\u2212\u00b5)c\n1\u2212\u00b5(1\u2212\u03b1)\n\nRemark 4.2. Although the multi period policy proposed is not a solution to the original program (3),\nwe emphasize that it naturally induces heuristic policies (e.g., see Algorithm 1) that are asymptotically\noptimal solutions to the original multi period problem (3) without relaxation (see Theorem 6.1).\n\n5 Comparative Analysis\n\nWe \ufb01rst compare the optimal reserve price of the single period and multi period model.\nProposition 5.1. Let rS (cid:44) max{min{\u00afc, r\u2217(c)}, r\u2217(0)} be the optimal reserve price of the single\nperiod constrained model and rM (cid:44) r\u2217(c(\u00b5\u2217)) be the optimal reserve price of the multi period\nconstrained model. Then rS \u2265 rM .\n\n6\n\n\fThe previous result shows that the reserve price of the single-period constrained model is larger or\nequal than the one of the multi-period constrained model. As a consequence, in the multi-period\nconstrained model items are allocated more frequently and the social welfare is larger.\nWe next compare the intermediary\u2019s optimal pro\ufb01t under the single period and multi period model.\nThis result quanti\ufb01es the bene\ufb01ts of dynamic revenue sharing and provides insight into when dynamic\nrevenue sharing is pro\ufb01table for the intermediary.\nProposition 5.2. Let \u00b5S \u2208 [0, 1] be such that r\u2217(c(\u00b5S)) = rS. Then\n\nJ S \u2264 J M \u2264 J S + (1 \u2212 \u00b5S)T E [(1 \u2212 \u03b1)bs \u2212 c]+ .\n\nThe previous result shows that the bene\ufb01t of dynamic revenue sharing is driven, to a large extent, by\nthe second-highest bid and the opportunity cost c. If the market is thin and the second-highest bid bs\nis low, then the truncated expectation E (cid:44) E [(1 \u2212 \u03b1)bs \u2212 c]+ is low and the bene\ufb01t from dynamic\nrevenue sharing is small, that is, J S \u223c J M . If the market is thick and the second-highest bid bs is\nhigh, then the bene\ufb01t of dynamic revenue sharing depends on the opportunity cost c. If the \ufb02oor\nprice c is very low, then rS = r\u2217(0) and \u00b5S = 1, implying that the coef\ufb01cient in front of E is zero,\nand there is no bene\ufb01t of dynamic revenue sharing J S = J M . If the \ufb02oor price c is very high, then\nrS = r\u2217(c) and \u00b5S = 0, implying that the coef\ufb01cient in front of E is 1. However, in this case the\ntruncated expectation E is small and again there is little bene\ufb01t of dynamic revenue sharing, that is,\nJ S \u223c J M . Thus the sweet spot for dynamic revenue sharing is when the second-highest bid is high\nand the opportunity cost is neither too high nor too low.\n\n6 Heuristic Revenue Sharing Schemes\n\nSo far we focused on the theory of revenue sharing schemes. We now switch our focus to applying\ninsights derived from theory to the practical implementation of revenue sharing schemes. First we\nnote that while the policy in the statement of Theorem 4.1 is only guaranteed to satisfy constraints in\nexpectations, a feasible policy of the stochastic decision problems should satisfy the constraints in an\nalmost sure sense.\nWe start then by providing two transformations that convert a given policy satisfying constraints in\nexpectation to another policy satisfying the constraints in every sample path.\n\n6.1 Multi-period Refund Policy\n\nOur \ufb01rst transformation will keep track of how much each constraint is violated and will issue a\nrefund to the seller in the last period (see Algorithm 1).\n\nALGORITHM 1: Heuristic Refund Policy from Lagrangian Relaxation\n1: Determine the optimal dual variable \u00b5\u2217 \u2208 arg min0\u2264\u00b5\u22641 \u02c6\u03c6(\u00b5)\n2: for t = 1, . . . , T do\n3:\n4:\n5:\n6:\nend if\n7:\n8: end for\n\nt = r\u2217(c(\u00b5\u2217))\nt \u2265 r\u03c0\nthen\nt = max(r\u03c0\nt ) = (1 \u2212 \u00b5\u2217)c + \u00b5\u2217(1 \u2212 \u03b1)x\u03c0\n\nCollect the buyers\u2019 payment x\u03c0\nPay the seller p\u03c0\n\nSet the reserve price r\u03c0\nif item is sold, that is, bf\n\nt , bs\nt)\n\nt (x\u03c0\n\nt\n\nt\n\n9: Let DF =(cid:80)T\n10: Let DR =(cid:80)T\n\nt } (p\u03c0\nt } (p\u03c0\n11: Pay the seller \u2212 min{DF , DR, 0}\n\nt=1 1{bf\nt=1 1{bf\n\nt \u2265 r\u03c0\nt \u2265 r\u03c0\n\nt ) \u2212 c) be the \ufb02oor de\ufb01cit.\nt ) \u2212 (1 \u2212 \u03b1)x\u03c0\n\nt (x\u03c0\nt (x\u03c0\n\nt ) be the revenue sharing de\ufb01cit.\n\nThe following result analyzes the performance of the heuristic policy. We omit the proof as this is a\nstandard result in the revenue management literature.\nTheorem 6.1 (Theorem 1, [7]). Let J H be the expected performance of the heuristic policy. Then\n\n\u221a\nJ H \u2264 J M \u2264 J H + O(\n\nT ).\n\nThe previous result shows that the heuristic policy given by Algorithm 1 is asymptotically optimal\nfor the multi-period constrained model, that is, it implies that J H /J M \u2192 1 as T \u2192 \u221e. When the\n\n7\n\n\fnumber of auctions is large, by the Law of Large Numbers, stochastic quantities tend to concentrate\naround their means. So the \ufb02oor and revenue sharing de\ufb01cits incurred by violations of the respective\nconstraints are small relative to the platform\u2019s pro\ufb01t and the policy becomes asymptotically optimal.\nPre\ufb01x and Hybrid Revenue Sharing Policies. We also propose several other policies satisfying\neven more stringent business constraints: revenue sharing constraints can be satis\ufb01ed in aggregate\nover all past auctions at every point in time. Construction details could be found in the full version.\n\n7 Overview of Empirical Evaluation\n\nIn this section, we use anonymized real bid data from a major ad exchange to evaluate the policies\ndiscussed in previous sections. Our goal will be to validate our insights on data. In the theoretical\npart of this paper we made simplifying assumptions, that not necessarily hold on data. For example,\nwe assume quasi-concavity of the expected pro\ufb01t function \u03a0(r, c). Even though this function is not\nconcave, we can still estimate it from data and optimize using linear search. Our theoretical results\nalso assume we have access to distributions of buyers\u2019 bids. We build such distributions from past\ndata. Finally, in our real data set bids are not necessarily stationary and identically distributed over\ntime. Even though there might be inaccuracies from bids changing from one day to another, our\nrevenue sharing policies are also robust to such non-stationarity.\nData Sets The data set is a collection of auction records, where each record corresponds to a real\ntime auction for an impression and consists of: (i) a seller (publisher) id, (ii) the seller declared\nopportunity cost, and (iii) a set of bid records. The maximum revenue share \u03b1 that the intermediary\ncould take is set to be a constant. To show that our results do not rely on the selection of this constant,\nwe run the simulation for different values of \u03b1 (\u03b1 = 0.15, 0.2, 0.25), while due to the limit of space,\nwe only present the numbers for \u03b1 = 0.25 and refer the readers to the full version for more details.\nOur data set will consist of a random sample of auctions from 20 large publishers over the period of 2\ndays. We will partition the data set in a training set consisting of data for the \ufb01rst day and a testing\nset consisting of data for the second day.\nPreprocessing Steps Before running the simulation, we need to do some preprocessing of the data\nset. The goal of the preprocessing is to learn the parameters required by the policies we introduced\nfor each seller, in particular, the optimal reserve function r\u2217 and the optimal Lagrange multiplier \u00b5\u2217.\nWe will do this estimation using the training set, i.e., the data from the \ufb01rst day.\nThe \ufb01rst problem is to estimate \u03a0(r, c) and r\u2217(c). To estimate \u03a0(r, c) for a given impression we\nlook at all impressions in the training set with the same seller and obtain a list of (bf , bs) pairs. We\nbuild the empirical distribution where each of those pairs is picked with equal probability. This allows\nus to evaluate and optimize \u03a0(r, c) with a single pass over the data using the technique described\nin [6]. For each seller, to estimate \u00b5\u2217, we enumerate different \u00b5\u2019s from the discretization of [0, 1]\n(denoted by D) and evaluate the pro\ufb01ts of these policies on the training set. Then the estimation (\u02c6\u00b5\u2217)\nof \u00b5\u2217 is the \u00b5 that yields the maximum pro\ufb01t on the training set, i.e., \u02c6\u00b5\u2217 (cid:44) arg max\u00b5\u2208D\n\u02c6pro\ufb01t(\u00b5)\n\n7.1 Evaluating Revenue Sharing Policies\n\nWe will evaluate the different policies discussed in the paper on testing set (day 2 of the data set) using\nthe parameters \u02c6r\u2217(c) and \u02c6\u00b5\u2217 learned from the training set during preprocessing. For each revenue\nsharing policy we evaluate, we will be concerned with the following metrics: pro\ufb01t of the exchange,\npayout to the sellers, match rate which corresponds the number of impressions allocated, revenue\nextracted from buyers and buyers values which is the sum of highest bids over allocated impressions\n(we assume that buyers report their values truthfully in the second-price auction). In addition, the\naverage intermediary\u2019s revenue share will be calculated.\nThe policies evaluated will be the following: NAIVE: na\u00efve policy (Section 2), SINGLE: single\nperiod policy (Section 3), REFUND: multi period refund policy (Algorithm 1), PREFIX and HYBRID.5\nIn Table 1, we report the results of the policies described above or \u03b1 = 0.25 (see the full version\nfor more values of \u03b1). The metrics are reported with respect to the NAIVE policy. In other words,\nthe cell in the table corresponding to revenue of policy P is the revenue lift of P with respect to\n\n5The details of policy PREFIX and HYBRID are omitted here, see the full version for further details.\n\n8\n\n\fpro\ufb01t\n0.00%\n\nrevenue\npayout match rate\npolicy\nNAIVE\n0.00%\n0.00%\n0.00%\nSINGLE +1.64% +2.97%\n+1.07% +2.64%\nREFUND +9.55% +9.57% +10.71% +9.56%\nPREFIX \u22121.00% +2.16% \u221218.51% +1.37%\nHYBRID +4.61% +6.90%\n+6.74% +6.33%\n\nbuyers values\n0.00%\n+1.39%\n+9.64%\n\u22122.90%\n+4.55%\n\nrev. share\n25.00%\n24.76%\n25.00%\n24.41%\n24.60%\n\nTable 1: Performance of the policies for \u03b1 = 0.25.\n\nNAIVE: revenue lift(P) = revenue(P)/revenue(NAIVE) \u2212 1. The only metric that is not reported as a\npercentage lift is the revenue share in the last column: rev share(P) = pro\ufb01t(P)/revenue(P).\nInterpreting Simulation Results What conclusions can we draw from the lift numbers? The \ufb01rst\nconclusion is that even though the theoretical model deviates from practice in a number of different\nways (concavity of \u03a0(r, c), precise distribution estimates, stationarity of bids), we are still able to\nimprove over the na\u00efve policy. Notice that the na\u00efve policy implements the optimal reserve price\nsubject to a \ufb01xed revenue sharing policy. So all the gains from reserve price optimization are already\naccounted for in our baseline.\nWe start by observing that even for SINGLE, which is a simple policy, we are able to considerably\nimprove over NAIVE across all performance metrics. This highlights that the observation that \u201cpro\ufb01t\nand revenue can be improved by reducing the share taken by the exchange\u201d is not only a theoretical\npossibility, but a reality on real-world data.\nNext we compare the lifts of SINGLE, which enforces revenue sharing constraints per impression,\nversus REFUND, which enforces constraints in aggregate. We can see that the lift is 5.8 times larger\nfor REFUND compared to SINGLE. For \u03b1 = 0.25, the lift6 for SINGLE is +1.64% while REFUND is\n+9.55%. This shows the importance of optimizing revenue shares across all auctions instead of\nper auction. Additionally, we observe that the match rate and buyers values of REFUND are higher\nthan those of SINGLE. This is in agreement with Proposition 5.1: because the reserve price of the\nsingle-period constrained model is typically larger than the one of the multi-period constrained model,\nwe expect REFUND to clear more auctions, which in turns leads to higher buyer values.\nFinally, we bire\ufb02y analyze the performance of PREFIX and HYBRID policies. While PREFIX is\nproposed to guarantee more stringent constraints, it fails to have a positive impact on pro\ufb01t. Instead,\nwith some slight modi\ufb01cations, HYBRID is able to overcome these shortcomings by granting the\nintermediary more freedom in picking reserve prices. As a result, we obtain a policy that is consistently\nbetter than SINGLE. Even though not as good as REFUND in terms of revenue lift, HYBRID satis\ufb01ed\nthe more stringent constraints that are not necessarily satis\ufb01ed by REFUND. To sum up, the policies\ncan be ranked as follows in terms of performance:\n\nREFUND (cid:31) HYBRID (cid:31) SINGLE (cid:31) NAIVE \u223c PREFIX.\n\nReferences\n[1] Santiago R. Balseiro, Jon Feldman, Vahab Mirrokni, and S. Muthukrishnan. Yield optimization of display\n\nadvertising with ad exchange. Management Science, 60(12):2886\u20132907, 2014.\n\n[2] Renato Gomes and Vahab S. Mirrokni. Optimal revenue-sharing double auctions with applications to ad\n\nexchanges. In 23rd International World Wide Web Conference, WWW \u201914, 2014, pages 19\u201328, 2014.\n\n[3] R Preston McAfee and John McMillan. Auctions and bidding. Journal of economic literature, 25(2):699\u2013\n\n738, 1987.\n\n[4] R. Myerson and M. Satterthwaite. Ef\ufb01cient mechanisms for bilateral trading. Journal of Economics Theory\n\n(JET), 29:265\u2013281, 1983.\n\n[5] Rad Niazadeh, Yang Yuan, and Robert D. Kleinberg. Simple and near-optimal mechanisms for market\n\nintermediation. In Web and Internet Economics, WINE 2014. Proceedings, pages 386\u2013399, 2014.\n\n[6] Renato Paes Leme, Martin P\u00e1l, and Sergei Vassilvitskii. A \ufb01eld guide to personalized reserve prices. In\n\nProceedings of WWW, pages 1093\u20131102, 2016.\n\n[7] Kalyan Talluri and Garrett van Ryzin. An analysis of bid-price controls for network revenue management.\n\nManagement Science, 44(11):1577\u20131593, 1998.\n6The reader might ask how to interpret lift numbers. The annual revenue of display advertising exchanges is\non the order of billions of dollars. At that scale, 1% lift corresponds to tens of millions of dollars in incremental\nannual revenue. We emphasize that this lift is in addition to that obtained by reserve price optimization.\n\n9\n\n\f", "award": [], "sourceid": 1530, "authors": [{"given_name": "Santiago", "family_name": "Balseiro", "institution": "Duke University"}, {"given_name": "Max", "family_name": "Lin", "institution": "Google"}, {"given_name": "Vahab", "family_name": "Mirrokni", "institution": "Google Research NYC"}, {"given_name": "Renato", "family_name": "Leme", "institution": "Google Research"}, {"given_name": "IIIS", "family_name": "Song Zuo", "institution": "IIIS, Tsinghua University"}]}