Managing Revenue from Television Advertising Sales

Managing Revenue from Television Advertising ales Dana G. Popescu Department of Technology and Operations Management, INEAD ridhar eshadri Department of Information, Risk and Operations Management, University of Texas at Austin In the television industry, customers (advertisers) buy a variety of products (advertising plans) consisting of customized bundles of resources (airtime spots) sold in two markets: a forward market (upfront) and a spot market (scatter). Television networks must allocate resources to products at the time the plans are purchased. We study resource allocation and pricing strategies for this network revenue management problem with customized products. We analyze two categories of resource control mechanisms: (i) partitioned booking limits and (ii) first-come, first-served mechanisms. We find that the relative performance of these controls depends on the arrival process: the first control type yields better results on the upfront market but the second type performs better on the scatter market. Finally, we propose a value-based approach to the pricing of customized advertising plans by identifying customer segments with similar valuation for a given set of product attributes. Our method improves on the current industry practice of setting a price per spot (ratecard) and then making capacity allocation decisions based on spot prices. We use industry data to illustrate the main segmentation criteria and model fit. Key words: media, network revenue management, advertising, pricing, customized products 1. Introduction Non-premium television networks derive a significant part of their revenues from ad sales. 1 In 2010, the total spent on advertising in the United tates alone exceeded $131 billion, with more than $59 billion spent on television advertising (Bloomberg 2011). Networks sell airtime to advertisers during two periods, known as the upfront and scatter markets (Bollapragada et al. 2002, Phillips and Young 2010). Between 50% and 90% of a broadcast or cable network s inventory is sold upfront from six months to a year ahead of the ads air dates during a short period of one or two weeks. The remaining spots are sold throughout the year in the scatter market, sometimes the day of (or the day before) the airing date. In both these markets the advertisers buy advertising plans, which are bundles of spots in different shows, that satisfy a set of constraints. Advertisers use these spots to reach consumers 1 Ancillary sales (DVDs, merchandise licensing, etc.) bring in a much smaller part of a network s revenues. 1

2 Popescu and eshadri: Managing Revenue from TV Ad ales in the target demographic of the product or service they are promoting. The plans are built to order so that they match the advertiser s requirements, yet the television networks retain some flexibility in choosing the spots that go into the plan bundle. Given the high degree of customization, no two plans are the same. In this context, our paper addresses two main questions. First, how should networks allocate resources among different types of advertisers? econd, how should networks price these highly customized advertising plans? The media revenue management (MRM) problem studied in this paper is similar in many ways to the network revenue management (NRM) problem faced by airlines and hotels, but it involves the additional challenges of customized bundle design and pricing. In the classic NRM problem, a firm manages the allocation of m limited, flexible resources (hotel rooms or, as here, airtime spots) to n types of products (multi-day stays or, as here, ad campaigns), which are bundles of these resources sold at predetermined prices. 2 The RM decision is whether or not to accept a customer request at a given price, given the available capacity and estimated future demand. Unlike the NRM problem, where the product bundles are predetermined (e.g., a weekend stay involves staying on a Friday and aturday night), in the MRM problem the plans are unique and customized to the client s needs. Hence it involves an additional, product design stage (the plan-building step), which must be completed at the time the network receives the client request. Moreover, pricing is complicated by product customization and negotiations between advertisers and the network. Thus, the fundamental difference between the two is that the NRM problem studied in the literature applies to consumer markets, where a large number of buyers have similar wants and transactions are typically of small value, while the MRM problem described here applies to business markets, where there are fewer buyers, transactions are typically larger, buyers have a strong need for customized products and prices, and the usage of the product or service ultimately determines its value (see Narayndas 2005 for further discussion on the distinction between consumer and business markets). To sharpen our focus, we next describe in more detail the practical context of the ad selling process. This process involves four stages, as illustrated in Figure 1. First, the advertiser gives the network a plan proposal that specifies a target demographic, 3 together with the total number of impressions, 4 that the ad campaign should generate. The advertiser also tells the network how much she is willing to spend (i.e., the budget), which can be negotiated on. 2 A flexible resource is one that can be used in more than one product. 3 Each advertiser targets a particular demographic, which is most often defined in terms of an age bracket and gender (e.g., females of age 18 49, males of age 21 25). 4 The exposure of an ad is measured in terms of impressions or eyeballs i.e., the number of individuals who watch a particular ad. ince not every impression is valuable to a particular advertiser, her willingness to pay is only for impressions in the target demographic.

Popescu and eshadri: Managing Revenue from TV Ad ales 3 ÙÖ ½ Ë Ð ÈÖÓ ÓÖ ÍË Ì Ð Ú ÓÒ Æ ØÛÓÖ In order to reach a broader audience characterized by different work schedules, activity patterns, and viewing preferences, advertisers will require that ads air in different dayparts, 5 or times of the day. 6 Therefore, the plan proposal might also specify a daypart mix (i.e., the percentage of total impressions to air in each daypart) and a weekly mix (the percentage to air in each week). In the second stage, the network builds a plan that satisfies the advertiser s requirements, decides on an acceptable price for the plan, and then presents it to the advertiser for negotiation in the third stage. If negotiations lead to an agreement between the parties, then the deal is signed and the network is committed to delivering the total number of impressions in the particular demographic required by the advertiser, while satisfying the daypart and weekly mix constraints. The network must make decisions in three of the four stages illustrated in Figure 1. The first such decisions occur during the plan-building process, when the network must choose the shows in which to place the advertiser s ads while satisfying her specific requirements. Then, in the negotiation stage, the network must decide whether or not to sell the plan to the advertiser, given her willingness to pay for the plan and the estimated value of the plan for the network. Finally, the last stage involves decisions that concern the actual scheduling of ads within the shows and weeks already assigned to the advertiser (i.e., time of day, day of the week, location in a break, etc.). The ad scheduling problem has been thoroughly addressed in the literature (see, e.g., Bollapragada et al. 2004, Bollapragada and Garbiras 2004). This paper focuses instead on decisions pertaining to the selection of advertisers, the allocation of their ads to shows and the pricing of plans. In other words, our focus is on the second and third stages of the process just described in particular, on network problems associated with resource allocation and pricing. The current practices and challenges faced by the network in these two stages are reviewed next. 5 Dayparts are time segments that divide the day in a manner that generally coincides with a network s programming patterns. Each network has its own daypart division. The most common dayparts are early morning (5 a.m. to 9 a.m.), daytime (9 a.m. to 3 p.m.), early fringe (3 p.m. to 5 p.m.), early news (5 p.m. to 7 p.m.), prime access (7 p.m. to 8 p.m.), prime (8 p.m. to 11 p.m. Monday through aturday, 7 p.m. to 11 p.m. unday), late news (11 p.m. to 11:30 p.m.), late fringe (11:30 p.m. to 2 a.m.), and overnight (2 a.m. to 5 a.m.) (Television Bureau of Advertising, www.tvb.org). 6 This requirement somewhat contradicts the findings of Fisher et al. (1980) and Goettler (1999), which suggest that advertisers should prefer an ad on a single program with 2x viewers to an ad on two programs with x viewers each. The argument is that people might watch more than one program and so two programs reach fewer unique viewers.

4 Popescu and eshadri: Managing Revenue from TV Ad ales In terms of the resource allocation strategies used to generate a sales plan, current industry practice (as reflected in discussions with experienced network planners and by the media revenue management literature; see, e.g., Bollapragada et al. 2002) is to use available resources to build the least expensive plan that satisfies (to a reasonable degree) an advertiser s constraints and conditions. A first come, first served (FCF) policy is implemented both in the upfront and scatter markets, and booking limit controls are uncommon. The advertising plans are often developed manually by a team of experienced sales planners. electing the spots that go into a plan is a laborious process that typically takes several hours to a day. 7 The challenge is to find quick, intuitive, and easy-to-implement ways of allocating resources to products. In terms of pricing, the current industry practice (as reported in the extant literature on media revenue management; see Bollapragada et al. 2002, Goettler 1999, Wilbur 2008) is for networks to set a price per 30-second spot in a particular show. This base price is known as the ratecard. The (reference) price of a plan is then determined as the sum of the ratecards for the spots allocated to it, while the actual price might be lower and depends on the outcome of negotiations between network and advertiser. However, this pricing method does not fully take advantage of the fact that different advertisers derive different value from a given spot in a show, 8 which raises the second question addressed in this paper: how should a network set prices for advertising plans? pecifically, what are the relative benefits of a value-based approach (Hinterhuber 2008, Monroe 2002) and how can it be implemented? In order to answer these questions, we first model the MRM problem as a stochastic dynamic program and then compare several types of mechanisms motivated by industry practice and RM research. In particular, we focus on two types of resource control mechanisms: partitioned booking limits (PBL) and first-come, first-served mechanisms. Under PBL, inventory is reserved in advance for each class of advertisers. Once the booking limit for a specific class is reached, any incoming advertiser from that class will no longer be served. Under the FCF mechanism, all customers are served in the order of their arrival and no customer is turned away unless inventory has been exhausted. We focus on these two types of controls for tractability, but also for suitability, given the characteristics of arrival processes on the upfront and scatter markets. Our main findings are as follows. (1) For resource allocation, we show that a priority mechanism based on PBL dominates a FCF approach but only in the upfront market. Indeed, the 7 As explained in Bollapragada et al. (2002), planners generate more than 300 plans for the upfront market. This requires the sales force to work 12 16-hour days during a period of two or three weeks. 8 A beer marketer, for instance, might be willing to pay more than a jewelry marketer for a spot in a football game, not only because the predominantly male audience of football games matches the target demographic for beer (and not so for jewelry) but also because men are probably more receptive to brewers advertising messages when they are broadcast during sports programming. Thus, charging all advertisers the same price per spot, irrespective of their respective willingness to pay, might be suboptimal.

Popescu and eshadri: Managing Revenue from TV Ad ales 5 converse holds: a FCF approach is preferable in the scatter market. The relative performance of these allocation mechanisms depends mainly on the length of the selling horizon relative to the time between customer arrivals. (2) For pricing, we propose a value-based mechanism whereby prices are set in terms of the requested demographic; we demonstrate that this mechanism dominates the current industry practice of ratecard pricing. Our results are verified both analytically and numerically. The rest of this paper is organized as follows. In ection 2 we position this work in the literature. In ection 3 we set up the model, and in ection 4 we describe and analyze different types of resource control mechanisms. In ections 5 and 6 we discuss the allocation mechanisms of (respectively) PBL and FCF. In ection 7 we analyze pricing strategy, and in ection 8 we illustrate numerically the willingness to pay estimation and customer segmentation as well as the performance of the heuristics presented in ections 5 and 6. We summarize our results and conclude in ection 9. 2. Literature Review As explained in the Introduction, the problem we study has much in common with the NRM problem, which has been widely addressed in the literature (see, e.g., Bertsimas and Popescu 2003, Talluri and van Ryzin 1998, 2004, van Ryzin and Vulcano 2008a, 2008b, Zhang 2011). The similarity is that in both cases the customers buy bundles of products. For instance, customers might reserve a hotel room for multiple nights or buy a plane ticket with an itinerary consisting of multiple stops. Also in both problems, the demand for products is random and there is a finite selling horizon after which inventory becomes perishable. The key difference is that, in the traditional network revenue management problem, the combination of resources that make up a bundle is predefined and the only decision left is whether or not to accept the customer s request; in the media revenue management problem, however, each product is unique and customized to meet the advertiser s constraints. Gallego and Phillips (2009) relax the assumption of predefined resource product allocation by modeling flexible products in a NRM framework. A flexible product is a menu of two or more substitute products. To customers who purchase a flexible product, the supplier may allocate any one of the alternatives near the end of the booking process. Thus, the allocation of resources to products is not predetermined and does not occur until the end of the selling horizon. In our case the products are also flexible, but there is an infinite number of resource combinations that can yield a substitute product and the allocation occurs at the time of sale. In Table 1 we summarize the main differences among traditional NRM, NRM with flexible products (NRM-FP), and the MRM model proposed here.

6 Ì Ð ½ ÓÑÔ Ö ÓÒ Ó ÆÊÅ ÆÊŹ È Ò ÅÊÅ ÅÓ Ð Popescu and eshadri: Managing Revenue from TV Ad ales NRM NRM-FP MRM Products pecific Flexible (two three options) Customized (infinite options) Resource-Product Mapping Predefined At the end of booking horizon At the time of sale Decisions Accept/Reject Accept/Reject; if Accept, then choose one resource allocation at the end of the booking horizon Accept/Reject; if Accept, then choose one resource allocation at the time of sale Different features of the media RM problem have been studied in the literature under various modeling assumptions. Araman and Popescu (2010) focus on the uncertainty in ratings, not on the variety of inventory. In their model, capacity is homogeneous. Ratings are uncertain during the upfront market, but information about ratings is revealed periodically during the scatter market. The network must decide how much capacity to sell on the upfront market (when ratings are uncertain) and how much capacity to reserve for the scatter market (when ratings are known). There is a penalty cost for underdelivery of impressions, and a newsvendor type of model is used to derive optimal static policies. The major differences between their model and ours is that they assume inventory is homogeneous in terms of audience ratings across different demographics whereas we assume that ratings are deterministic. In other words, we focus on the show selection problem while they solve the reservation problem. In this sense, the two models are complementary. In Bollapragada et al. (2002), the problem of show selection is formulated as a goalprogramming model. The network ranks all show week pairs based on their importance as perceived by management. The plans are built with the goal of minimizing the premium inventory used, where premium inventory is the inventory ranked above a certain level. This method is easy to implement. Yet because the ranking is based on management s subjective interpretation of a show s importance, there is plenty of room for errors when deciding the best allocation for a given plan. Their model ignores demand uncertainty. Moreover, since the same ranking is used irrespective of a plan s target demographic, their model does not fully exploit the fact that shows generate higher revenues when sold to advertisers seeking a particular demographic. Research on TV advertising revenue optimization has often considered the ad scheduling problem (Bollapragada et al. 2004, Bollapragada and Garbiras 2004, Crama et al. 2012). For example, Bollapragada and Garbiras formulate a goal-programming model to solve the scheduling problem. In their model, the emphasis is on satisfying as many product conflict and ad position constraints as possible by assigning an appropriate penalty to each violation of a constrain; the objective is to minimize the total penalty cost incurred. With all input being deterministic, the problem reduces to a mixed integer problem (MIP) that can be solved efficiently using various heuristics.

Popescu and eshadri: Managing Revenue from TV Ad ales 7 3. A Model for Media Resource Allocation Decisions Consider a television network with a fixed offering of shows over a yearly programming horizon. Faced with uncertain demand, the network wants to allocate airtime inventory to advertisers with the objective of maximizing expected revenue. In this section we describe the inventory, the demand, and the decision problem. 3.1. Inventory There are shows and M audience demographics. A spot in show i generates a ij impressions in demographic j. Let y t =(y1, t..., y t ) be the vector of spot inventory at time t, where y t i denotes the unsold number of spots in show i at time t. Then A t =(A t) ij,, j=1,m is the matrix of total impressions available at time t, where A t ij = a ij y t i. We denote by A0 the matrix of impressions available before the upfront market opens (at time 0). We assume A 0 to be deterministic and known by the network at time 0. ince A ij a ij, for convenience (and computational simplification) we will work with a continuous-space allocation rather than a discrete allocation (i.e., we will allocate percentages of a show, rather than spots i.e. demarcated 30-second intervals). The problem s complexity will thereby be reduced from integer, nonconvex optimization to continuous, convex optimization. The impact of this approach on network revenue is minimal, but the computational savings are significant. We define a continuous-space show allocation as an -dimensional vector α=(α i ),, where α i [0, 1] denotes the proportion of show i that is allocated to a given plan. Example 1. Let Y be a one-hour show that runs once per week, 50 weeks per year. The total spot inventory of Y is equal to 1,800 30-second commercial spots in a given year (18 minutes per show 50 times a year 2 = 1,800 30-second ads in a year). 9 Thus, allocating α Y = 0.0005 of show Y to a given plan would be equivalent to allocating one spot in the show. 3.2. Demand There are M segments (or classes; we use the two terms interchangeably) of advertisers that correspond to each potential target demographic (and possibly other segmentation criteria to be discussed in ection 8). Advertisers arrive in [0, T] according to a Poisson distribution with arrival rate λ. We discretize time in intervals of unit size, small enough to assume that no two arrivals occur at the same time. An advertiser arriving at time t from class j has demand d t j of impressions in demographic j. The probability that an arrival belongs to class j is β j and M j=1 β j = 1. The total demand of impressions in class j is denoted by D j, where D j = T t=1 dt j has density and distribution functions f j ( ) and F j ( ), respectively. 9 In the United tates, the Federal Communications Commission limits the combined duration of commercial breaks to 12 minutes per hour for children and 18 minutes for adults. The limit in the European Union is 12 minutes per hour; in the United Kingdom, the advertising airtime limit varies by time of day but averages 7 minutes per hour.

8 Popescu and eshadri: Managing Revenue from TV Ad ales Ì Ð ¾ Demand (millions) Ñ Ò Ò ÁÒÚ ÒØÓÖÝ Ö Ø Ö Ø Inventory (millions) Time (t) Type Quantity Type how 1 how 2 1 D 1 1 D 1 A 11 = 50 A 21 = 40 2 D 2 2 D 2 A 12 = 10 A 22 = 80. Example 2. If the arrivals in class j are Poisson(λ j ) (i.e., λ j = λβ j ) and if the individual demands of advertisers of class j are independent and identically distributed (i.i.d.) with mean µ j and standard deviation σ j, then D j is a random variable with mean D j = E(D j )=λ j Tµ j and variance V(D j )=λ j T(σ 2 j + µ 2 j). We assume that, for an advertiser of class j, the willingness to pay for a plan consisting of an allocation α=(α i ), is given by w j (α). The function w j (α) could depend on the audience level of the shows in the target demographic group and perhaps also on those shows characteristics, which could affect the effectiveness of the advertising message. 3.3. Decision Problem uppose an advertiser from class j arrives at time t. he has a demand of d t j impressions and a willingness to pay w j (α) for an allocation α C t j, where Ct j is the set of feasible allocations. 10 The network must decide whether to accept the advertiser s plan proposal; if the proposal is accepted, the network must choose the shows in which to place the advertiser s ads (i.e., determine the vector α). The network s revenue optimization problem can be expressed mathematically as max V(t, A t, d t ), (1) α Λ(A t ) where Λ(A t )= { α i [0, 1], α C t j α i A t ij d t j, P j(α) w j (α) } is the set of feasible allocations and P j (α) is the price paid by the advertiser for allocation α. The functional form of the value function V(t, A t, d t ) in (1) will depend on the control mechanism chosen by the network. Product Customization. We have emphasized that there are many plans capable of satisfying the advertiser s requirements (i.e., Λ(A t ) is potentially an infinite set). In order to solve the resource allocation problem (1), the network must know w j (α) for any allocation α Λ(A t ). Example 3. Consider the problem parameters described in Table 2. An advertiser of class D 1 (demographic 1) arrives at time t = 1, and her demand is d 1 1 = 1 million impressions. how 1 delivers 50 million impressions in D 1 whereas show 2 delivers 10 million impressions in D 1. We assume that there are no specific constraints in C 1 1 (e.g., no daypart constraints). Then the set Λ(A 1 ) of all feasible allocations for advertiser 1 of class D 1 is infinite. In this case the customer s requirements would be satisfied, for instance, by allocating either 0.02 of show 1 or 0.1 10 The typical constraints that affect Cj t are the daypart mix and the weekly mix (e.g., 20% of all impressions must be in prime time and 5% of all impressions must be in the first week of December ).

Popescu and eshadri: Managing Revenue from TV Ad ales 9 of show 2 or by allocating a combination of both show 1 and show 2 (e.g., 0.01 of show 1 and 0.05 of show 2). If we are considering continuous allocations α [0, 1] then there are infinite number of possibilities, of which the network must choose one. To make the best decision, the network would have to know advertiser 1 s willingness to pay for each of the (infinite) potential combinations of shows. As in ection 3.2, we assume that the willingness of an advertiser with demand d j to pay for a particular show allocation depends on the audience level of the shows in the target demographic group and possibly also on the shows characteristics, which could have an impact on the advertising message s effectiveness. Furthermore, we conjecture that advertising customers can be segmented in such a way that the willingness to pay for the plan of a customer from a particular segment depends only on the number of impressions that the plan generates in the advertiser s target demographic. 11 Thus we assume that the willingness to pay of customers in a segment j has a linear expression, 12 w j (α)=v j α i A ij. (2) This equation simply states that the willingness to pay of each advertiser from class j is equal to v j per impression in demographic j. Throughout ections 4-7 we assume for simplicity of exposition that the segmentation is done on the basis of demographics only. In ection 8 we use industry data to identify the main segments and estimate the willingness to pay per impression in each segment. We also explain how to account for other segmentation criteria such as market (upfront/scatter) and quantity (low/medium/high volume), without changing the model setup. 4. Types of Resource Controls In this section we first formulate the network s problem as a stochastic dynamic program and then justify the subsequent focus of this paper on a particular subset of control mechanisms. Assume that a customer of type j arrives at time t and has a demand of d t j impressions. We can define an allocation policy by solving the following stochastic dynamic program: { V(t, A t, d t j )=max v j α C α i A t t ij + E[V(t+ 1, I (1 α)a t, d t+1 } )] α i A t ij dt j ; 0 α i 1, j where I (1 α) R is a diagonal matrix with(i (1 α) ) ii = 1 α i for,. o each time a customer arrives, we find the allocation vector α that maximizes the combined revenue from the current 11 Our analysis of the plans sold during the period 2006 2008 by several major U cable networks supports this belief. pecifically, we observe a linear relationship between the logarithm of the price of the advertising plans for a given demographic in a given market and the logarithm of the number of impressions in those plans. Performing a simple linear regression of the logarithm of the price of the advertising plan (as the dependent variable) on the number of impressions in the plan (as the independent variable) yields an extremely high R 2 in the range 0.92 0.95 (see ection 8). 12 Industry data show that quantity discounts are prevalent, but these can be accounted for by further segmenting advertisers according to size (e.g., small, medium, and large advertisers). We can then assume that the willingness to pay for a plan within each segment is a linear function of the number of impressions (see ection 8). (3)

10 Popescu and eshadri: Managing Revenue from TV Ad ales customer with the expected value of the remaining inventory. However, problem (3) is for all practical purposes intractable. 13 ince practicality requires that the resource allocation method be easy to implement, in this paper we focus on two types of control mechanisms: partitioned booking limits and first-come, first-served (a.k.a. no-holdback) mechanisms. Partitioned booking limits. Under PBL, available resources are divided into separate buckets one for each class of advertisers that can be sold only to the designated class. At the start of the selling period (t = 0), the network computes a static resource allocation among the M classes of advertisers by taking into account the demand distribution in each class. The booking limit control has the following structure: allocate proportion α ij of show i to class j for i = 1, and j = 1, M, where 0 α ij 1. In other words, accept advertisers from class j as long as demand in class j does not exceed α ija ij. This paper considers only static (as opposed to dynamic) booking limits, computed only once at the beginning of the booking horizon, so that the model will be amenable to analytic treatment. First-come, first served. Under FCF resource control, all requests are accepted provided that resources are available. Customers are served in the order of their arrival, and a customer is rejected only if there are no more resources left. Therefore, the network s sole concern is to find the resource allocation that satisfies the current offer and maximizes total expected revenues. In other words, the network chooses α such that: { V FC (t, A t, d t j )= max v j α C α i A t t ij + E[V FC(t+1, I (1 α) A t, d t+1 )] j [0,1] α i A t ij = min ( d t j, A t ij) }. (6) The FCF mechanism is equivalent to a static bid-price control where the bid price is a price per impression, which is equal to a type-j advertiser s willingness to pay (v j ). Depending on the arrival process, one type of control will work better than the other. We illustrate the performance of the two types of controls with the following simplified example. 13 One can improve the tractability of this program by replacing the expectation of the value function with the expectation of the wait-and-see (W) value, which is the value of the following linear program (LP): } W(t, A t )=max. (4) α { M v j α ij A t ij α ij A t M ij Dt j j=1, M; α ij 1 i= 1, ; α ij 0 i= 1,, j=1, M j=1 j=1 Here α ij is the fraction of show i allocated to customers of type j, and W(t, A t ) is the maximum revenue that can be obtained from time t onward. Program (4) assumes that the TV network waits until time T, observes the total demand in each demographic, and then builds all the advertising plans at once. Because the D t j = T k=t dk j are random variables of known distribution, we can compute the expected value E[W(t, A t )]. An allocation policy can be found by solving the following stochastic LP: { Ṽ(t, A t, d t j )=max v j α α i A t ij + E[W(t+1, I (1 α) At )] } α i A t ij dt j ; 0 α i 1 i= 1,. (5) The wait-and-see value replaces the actual value function in the stochastic program, but solving (5) is still nontrivial. Even approximate solutions based on Monte Carlo simulation are both hard to implement and computationally expensive. For more approximation methods, see Birge and Louveaux (1997).

Popescu and eshadri: Managing Revenue from TV Ad ales 11 10000 dλ=2000 10000 dλ=1500 8000 dλ=1000 Expected Revenue 8000 6000 4000 FCF PBL 2000 0 50 100 λ ÙÖ ¾ Expected Revenue 8000 6000 4000 FCF PBL 2000 0 50 100 λ Expected Revenue 7000 6000 5000 4000 FCF PBL 3000 0 50 100 150 λ È Ä Ú Ö Ù Ë A 1 = A 2 = 1000 β 1 = β 2 = 0.5 v 1 = 10 v 2 = 6 ½¼¼¼ ÖÙÒ µ Example 4. uppose there are two classes of customers arriving in period[0, 1] (i.e., T = 1) and buying resources for delivery (airing) in period [1, 2]. There is only one show, which generates A 1 impressions in demographic 1 and A 2 impressions in demographic 2. Let λ be the arrival rate of customers and let β and 1 β be the respective probabilities that a customer is categorized as class 1 or 2. Each customer has a constant demand for d impressions in a particular demographic (demographic 1 for customers of class 1 and demographic 2 for customers of class 2). As before, a class-i customer s willingness to pay for impressions in demographic i is denoted v i. While keeping a constant total demand (i.e., λ d), we vary the arrival rate λ. We then plot the network s expected revenue under PBL and under FCF. The network chooses one of these control mechanisms at time t = 0. If booking limits are used, then the limits are computed at t=0 and remain fixed throughout period [0, 1]. Figure 2 displays the results. It is clear that for large values of λ, PBL outperforms FCF control, unless the expected demand is equal or less than the available inventory. As the arrival rate decreases, the FCF control outperforms PBL. This suggests that if arrivals are highly concentrated within a short period and if the time between the ads sales date and airing date is long enough, then booking limits perform better than the FCF control. In contrast, if arrivals are scattered over a long time horizon and if there is often more time between successive arrivals than between sales and delivery (airing date), then FCF control is preferred. This is not surprising, for in the limits that is, when there are infinite arrivals within a period 14 or when there is at most one arrival per period these two types of control are (respectively) optimal. This straightforward observation is stated in Proposition 1 below. Proposition 1. uppose the individual demands of advertisers of class j are i.i.d. and the total expected demand exceeds the available inventory (i.e., M j=1 y j > 0, where y is a solution to (7)). v j α ij A 0 ij α ij A 0 M ij + y j= D j j=1, M; α ij 1 i= 1, ; α ij 0, y j 0 j=1, M, i= 1, { M max α,y j=1 j=1 } (7). 14 Note that infinity is not required, as rapid convergence is achieved with arrival rates of 100+ per period. uch rates are common during the upfront market.

12 Popescu and eshadri: Managing Revenue from TV Ad ales By varying λ while holding the total expected demand in each demographic (i.e., D j ) as well as the partition of the arrivals ((i.e., β j ) fixed, we make the following observations: (i) if the arrival rate λ in [0, T] is infinite, then a PBL control is optimal; (ii) if λ is small enough such that the probability of two or more arrivals in [0, T] goes to zero, then FCF control is optimal. While this is a stylized example, it motivates our focus on these two types of controls that we will study next. In the graphs depicted in Figure 2, the arrival processes at the left and right end of the spectrum mimic the arrival processes in the upfront and scatter market, respectively. As noted in the Introduction, 50% 90% of a TV network s inventory is sold upfront during a short period of one or two weeks. The bulk of advertiser arrivals occur during this period. Given that the plan-building process takes anywhere from several hours to several days, hundreds of plans are constructed at the same time by decentralized sales teams. The network s objective is to maximize expected revenue by satisfying part of the demand in the upfront market while reserving enough inventory for the scatter market where prices are higher, demand is more volatile, and inventory is perishable. Thus, it is reasonable to conclude that partitioned booking limits are more suitable for the upfront market. On the scatter marker, however, arrivals are distributed throughout the entire year. The plans sold on the scatter market are small in size and may be sold just a day before (or even the same day as) airing. The time between customer arrivals in the scatter market is often longer than the time between sale and delivery (airing) date. Almost any forecast of scatter demand will be difficult to obtain and highly inaccurate. The network s main concern in the scatter market is to avoid being left with unsold spots. This explains why networks are less inclined to use booking limits when allocating inventory in the scatter market: such limits entail rejecting some customers in expectation of higher revenues from future arrivals. For this reason, we propose FCF controls for the scatter market. 5. Partitioned Booking Limits (Upfront Market) In this section we study partitioned booking limits for the media revenue management problem. o that we may better understand the structure of the optimal policy, we first derive closedform solutions for the two-class problem and then generalize the results to M classes. For now, we ignore the advertiser constraint sets Cj t ; the problem is fully separable (for the main type of constraints that go into Cj t ) and reducible to solving several smaller problems without advertiser constraints. 15 We denote by A the matrix of available inventory at time t=0 (i.e., we omit the 15 Each show belongs to exactly one daypart and so, if we assume that demand in each daypart is independent of demand in other dayparts, then the resource allocation problem is separable into independent problems corresponding to each daypart. Thus it is sufficient to focus on characterizing the optimal policy for a single daypart, and we henceforth assume that there is only one daypart. Our neglect of the weekly mix constraints can be similarly justified.

Popescu and eshadri: Managing Revenue from TV Ad ales 13 superscript 0). At time t = 0, the television network computes the static partitioned booking limits and sets a price P j = v j per impression in demographic j. Thus we assume that the network can extract all surplus from the advertisers. 5.1. Two-Class Problem During the sales stage, the network s objective is allocate resources to advertisers such that the expected revenue is maximized. The resource allocation problem for two classes of advertisers can be formulated as follows: { [ ( max P 1 E min D 1, α [ ( α i A i1 )]+ P 2 E min D 2, (1 α i )A i2 )]} s.t. 0 α i 1,, j=1, M. (8) Proposition 2 describes the optimal resource allocation policy for this problem. Proposition 2. Assume that A 11 A 12 A 21 A 22 optimal static booking limits satisfy α i = A 1 A 2. Then there exists an index m such that the { 1 if i= 1, 2,..., m 1, 0 if i=m + 1,...,. Also, α m is a solution to P 1 A m 1 Pr ( D 1 α ia i1 ) = P2 A m 2 Pr ( D 2 (1 α i )A i2 ). Not surprisingly, the optimal booking policy balances the opportunity cost of reserving one more unit of inventory for class 1 with the opportunity cost for reserving one more unit of inventory for class 2. The ratio P 1 A i1 denotes the rate of displacement between two advertiser classes in other words, P 2 A i2 the rate at which the network will take away one spot in show i from an advertiser of class 2, and sell it to an advertiser of class 1, without a change in revenue. For example, if P 1 A i1 P 2 A i2 = 2, then the network generates the same revenue from selling two spots in show i to a class-2 advertiser as it would from selling one spot in show i to a class-1 advertiser. Proposition 2 states that shows will be allocated to class-1 advertisers in decreasing order of displacement rate. The ratio A ij A kj denotes the rate of substitution between shows i and k for advertisers of class j that is, the rate at which advertisers of class j will give up a spot in show k for a spot in show i while maintaining their same level of utility. For example, if A ij A kj = 2 then a class-j advertiser will give up two spots in show k for one spot in show i. If show i (resp., show k) is sold exclusively to class-1 (resp., class-2) advertisers, then the rate of substitution between shows i and k is higher for advertisers of class 1 than for those of class 2. Bundle Design. Once the shows have been partitioned among the different classes of advertisers, the network will still need to solve a resource allocation problem for each advertiser within the inventory allocated for its class. uppose a class 1 advertiser with demand d arrives at some (9)

14 Popescu and eshadri: Managing Revenue from TV Ad ales point during the upfront market. Then the network will compute a feasible allocation of the form: allocate proportion α 1 i of show i such that m α1 i d and α 1 i α i for all i = 1,. At this stage, the network could incorporate additional constraints into the resource allocation problem, such as show diversity (i.e., spread the advertiser s ads over a number of different shows). 5.2. M-Class Problem In reality there are more than two advertiser classes. The resource allocation problem for M classes of advertisers can be formulated as follows: [ ( M max min D j, α ij A α ij)] α ij 1, ; α ij 0 i= 1,, j=1, M j=1 { M P j E j=1 }. (10) Establishing the PBL for the M-class case is a convex optimization problem with separable objective function and linear constraints. For this type of problem, polynomial-time algorithms (see, e.g., Hochbaum and eshadri 1993, Hochbaum and hanthikumar 1990) can be used. The following statement gives the intuition behind the structure of the optimal policy. Proposition 3. (i) The M-class static booking problem for media resource allocation is a convex optimization problem with separable objective function. (ii) The optimal allocation strategy has the following form: there exist at most M 1 shows such that α ij / {0, 1}; for all other shows, α ij {0, 1}. For example, if there are five classes of advertisers and the network airs 20 shows, then only 4 of those 20 shows can be allocated to more than one class of advertisers. The rest of the shows will be reserved exclusively for one of the five advertiser classes. If there is unlimited demand for each class of advertisers then each show will be sold exclusively to one class the one for which the target demographic matches the shows s audience. We say there is a match when the displacement ratio corresponding to a particular show is highest for that class of advertisers. Deterministic Linear Programming Heuristic. Although problem (10) is solvable in polynomial time, it assumes that the distribution of demand is known and has a tractable analytical expression (or can be approximated by one). Yet often the network can only estimate the moments of the demand distribution. In such cases, a simple heuristic to find the static booking limits for the M-class problem is to solve the deterministic linear program associated with (10). Let D j be the expected demand for demographic j. Then we can write the deterministic LP as follows: v=max α { M P j α ij A ij α ij A ij D j j=1, M; j=1 M } α ij 1 i= 1, ; α ij 0 j=1, M, i= 1,.(11) j=1 The expected revenue from implementing this policy is M j=1 P j E [ min { α ij A ij, D j }]. It is easy to see that v is an upper bound on the expected profit from implementing the deterministic

Popescu and eshadri: Managing Revenue from TV Ad ales 15 linear program (DLP). As shown in Cooper (2002) and Talluri and van Ryzin (2004), if we scale up capacity and demand by a factor of θ (a positive integer) then the solution of the DLP converges to the optimal solution. In other words, if we consider the sequence of problems of type (11) with capacities θa ij and demand vectors D θ j such that E(D θ j )=θ D j and 1 θ Dθ j 6. FCF Control (catter Market) D D j, then 1 θ Π(Dθ ) D v. As in the preceding case of partitioned booking limits, for simplicity of exposition we will first derive the optimal policy for the two-class problem and then discuss its implications for the M-class problem. 6.1. Two-Class Problem If there are only two classes of advertisers, then the optimal resource allocation policy under the FCF strategy is given by the following proposition. Proposition 4. Let (A t i1, At i2), be the vector of available impressions at time t and let d t j be the demand in period t for class j. Denote by α the -dimensional allocation vector, where α i is the proportion of show i allocated to satisfy demand d t j (0 α i 1 for all i= 1, ). W.l.g., we assume that At 11 At A 12 t 21 A22 t At 1. A t 2 (i) If j = 1 and A t ij > d t j, then there exists an index m such that the optimal allocation vector satisfies: 1, i= 1, 2,..., m 1; d α i = t j m 1 A t ij A, i= m ; m j 0, i= m + 1,...,. (ii) If j = 2 and At ij > dj t, then there exists an index m such that the optimal allocation vector satisfies: (iii) If A t ij d j t, then α i = 1 for all,. 0, i= 1, 2,..., m 1; d α i = t j i=m +1 At ij A, i=m ; m j 1, i=m + 1,...,. Thus, the optimal policy is to allocate shows to advertisers in a way that minimizes the total number of free impressions. That is, a class-j advertiser pays only for impressions of type j but a spot in a show generates impressions in both demographics 1 and 2; hence the advertiser receives free impressions in the nontarget demographic. Of course, a revenue-maximizing network tries to give away a minimum number of free impressions (a.k.a. lost impressions). It is important to recognize that the solution for the two-class problem is pathwise optimal; in other words, the solution is optimal with probability 1 irrespective of the sample path. This is a very strong solution concept, especially since we make no assumption on the distribution of

16 Popescu and eshadri: Managing Revenue from TV Ad ales demand. o regardless of the actual demand distribution and sample path, this FCF policy maximizes the network s revenue provided that the network prefers not to reject customers when inventory is still available. The FCF allocation policy is appropriate when the selling horizon is short and rationing resources carries a high risk of leftover inventory. It is also recommended when the probability of excess demand is low or when the network is more averse to unsold inventory than to selling inventory at suboptimal prices. Finally, the FCF policy is useful in cases of little or no information about demand, since it does not require any knowledge about the demand distribution. 6.2. M-Class Problem When we generalize the problem to M classes of advertisers, there is no longer a pathwise optimal solution under the FCF allocation strategy. We shall characterize the scenarios in which a pathwise optimal solution does exist. For all other scenarios, we provide heuristics to solve the general problem (which is otherwise difficult to solve). ome additional notation will be employed as follows. We rewrite A ij = A i1 β ij, so A i = (A i1, A i1 β i2,..., A i1 β im ) is the vector of impressions in show i. Let δ=(0,..., d,..., 0) be an M-dimensional demand vector with δ j = d and δ k = 0 for all k = j, k= 1, M. Let α be the -dimensional allocation vector. We say that α is a feasible allocation vector if αa δ. We say that α is a pathwise optimal allocation vector if α is feasible and α is a least element of Λ={αA 0 α i 1, ; αa δ}. Therefore, irrespective of which class of customer arrives later, we cannot improve the allocation because the demand δ has been satisfied by minimizing the number of free impressions in all other demographics. Lemma 1. Let α be a pathwise optimal solution for a customer with demand δ. If there exist i, k {1, 2,..., } such that 0<α i < 1 and 0<α k < 1, then there exists a θ> 0 such that A i = θa k. This lemma states that a pathwise optimal solution requires all ads to be placed in one show. Ads would not be placed in more than one show unless one show is not enough to satisfy all demand or all shows generate proportional number of impressions in each demographic. Next we show that, for a pathwise optimal solution to exist at any arrival epoch, the vectors {A i }, must satisfy strict conditions. Proposition 5. A pathwise optimal allocation exists at any arrival epoch if and only if, for any A i and A k, either β ij β kj = 1 or β ij β kj = s ik for s ik a positive number. In contrast to the two-class case, with M classes a pathwise optimal solution does not exist for all inventory matrices A. Because the conditions imposed on A are strong, it is unlikely that a pathwise optimal solution exists in any real-life setting. olving the dynamic allocation problem

Popescu and eshadri: Managing Revenue from TV Ad ales 17 for the M classes (i.e., problem (6)) may not be practically feasible owing to the overall complexity of the problem (and sometimes the unknown distribution of demand on the scatter market). For this reason, we propose two heuristics for resource allocation that seek to minimize this opportunity cost. Both heuristics have the advantages of being easy to implement and requiring no assumptions on the distribution function of the demand. 6.2.1. Minimum Lost Impressions Heuristic The concept of pathwise optimality is the starting point for this heuristic. A pathwise optimal solution requires that, at each moment in time, the allocation gives away the minimum number of so-called free impressions in each demographic. However, a television network might be interested in minimizing these lost impressions only for a relatively small subset of demographics. (This is a reasonable assumption when one considers that 70% 90% of ad sales occur within three or four demographic categories.) 16 A pathwise optimal allocation exists only under highly restrictive assumptions. Relaxing the pathwise optimality requirement allows us to look for allocations under which the network can minimize either the total number of free (lost) impressions or the value of those impressions. uppose a customer of class j arrives at time t and requires d t j impressions in demographic j. Let J be the subset of demographics that are of greatest interest for the network. The network can satisfy customer requirements while minimizing free impressions for demographics in J if it solves the following allocation problem: { min α α i A t } ik α i A t ij dt j ; 0 α i 1,. (12) k J j Remark 1. If a pathwise optimal solution exists, then the solution to (12) is pathwise optimal. If (12) is infeasible, then α i = 1 for all i = 1, and all available inventory will be sold to the customer who arrived at time t. To account for impressions in different demographics having different value for the network, we can weight the lost impressions by their price. Here is the allocation for which the value of lost impressions is minimized: { min α P k α i A t ik α i A t ij dt j ; 0 α i 1, k J j 6.2.2. Minimum Price Heuristic A similar heuristic to what is currently used in practice (cf. Bollapragada et al. 2002) is that of building the least expensive (or least valuable) plan, where a plan s value is the sum of the prices (ratecards) of the spots allocated to that plan. We can determine the ratecard values by solving the dual of the deterministic linear program (11). The dual values γ i should give us an approximate value for a show: { M min ξ,γ j=1 D j ξ j + } γ i ξ j A 0 ij + γ i P j A 0 ij j=1, M,, ; ξ j 0 j= 1, M; γ i 0 i= 1,. 16 For the television networks in our data set, more than 70% of the sales are concentrated in 3 to 5 demographics. }.

18 Popescu and eshadri: Managing Revenue from TV Ad ales Once we have the dual values, we can find an allocation for an advertiser of type j and with demand d t j by solving { min α γ i α i } α i A t ij dt j j=1, M; α i 1 i= 1, ; α i 0 i= 1,. In the numerical example given in ection 8 we show that, for the scenarios considered, the minimum lost impressions heuristic always outperforms the minimum price heuristic. The explanation lies in the suboptimality of setting a price per spot and making resource allocation decisions based on spot prices, as opposed to setting a price per impression and using those prices in the resource allocation model. In the next section we discuss the pricing strategy in more detail. 7. Pricing trategy During the pre-sales stage, one of the most important tasks for a TV network is to set prices. The current pre-sales process for television networks is similar to the one described in Bollapragada et al. (2002) in connection with the NBC network. The main steps are summarized in Figure 3. ÙÖ ÈÖ ¹Ë Ð Ø Ú Ø ÓÐÐ ÔÖ Ø Ðº ¾¼¼¾µ The figure shows that, once the programming schedule is determined, television networks estimate demand and then set ratecards for each show. Ratecards consist of base prices for 30- second spots on an airing of a particular show. 17 The ratecards are used as reference prices during the sales-planning stage. When building a plan for an advertiser, the network attempts to ensure that the value of the plan (i.e., the sum of the ratecards of all spots included the plan) is less than the advertiser s budget. Their objective is to build the cheapest plan, in ratecard amounts, that satisfies all the advertiser s requirements. We show that the networks pre-sales process (Figure 2) is suboptimal and that their pricing strategy leads to suboptimal resource allocation. Rather than pricing resources (i.e., the spots in a show) before making resource allocation decisions, the networks should be pricing impressions in each demographic. In marketing terminology, this distinction corresponds to that between a pure components strategy and a pure bundling strategy (see, e.g., Banciu et al. 2010). 17 For example, Wilbur (2008) proposes a model for rate-card estimation. In his model, the aggregate inverse demand for advertising in a given program is p s = q s λ q + V s λ V + d s λ d + x s λ x + Φ s, where p s is the price of an ad during show s, q s is the show s ad level, V s is the number of viewers watching show s, d s is a vector of viewer demographics, and x s represents program characteristics that affect advertising effectiveness; the λ are advertiser preference parameters and Φ s is an error term.

Popescu and eshadri: Managing Revenue from TV Ad ales 19 The following proposition characterizes the optimal pricing strategy conditional on the capacity control mechanisms of PBL and FCF. Proposition 6. Let a class-j advertiser s willingness to pay for a plan consisting of show allocation α be given by w j (α)=v j α i A ij. If the network uses either a PBL or a FCF control, then it earns more revenue by charging a price P j = v j per impression in demographic j than by charging any fixed price per spot. Proposition 6 states that there is inefficiency in pricing spots. Therefore, networks should instead price impressions and the pre-sale process should be changed to the one portrayed in Figure 4. ÙÖ ÈÖ ¹Ë Ð Ø Ú Ø ÙÒ Ö Ê Ú ÈÖ Ò ËØÖ Ø Ý The spot-pricing inefficiency arises mainly because an advertiser s willingness to pay for a plan is a function of the number of impressions it is likely to generate. Moreover, advertisers of different classes are more willing or less willing to pay for a spot depending on the extent of the show s viewership that matches the advertiser s target demographic. A network that simply charges a single price per spot reduces its power to employ price discrimination and hence will be unable to extract all surplus from advertisers. Another reason that pricing impressions generates more revenue than pricing spots is that networks have flexibility in choosing the shows in which to place the ads. Even if an advertiser requires some percentage of impressions to come from a given show, absent complete show selection (i.e., a complete specification of the vector α) the remaining network flexibility will mean that Proposition 6 still holds. Example 5 (Value of Flexibility). As a simple illustrative example, suppose there are only two shows ( 1 and 2 ) and two demographics (D 1 and D 2 ). The annual audience size of each show for each demographic is given in Table 3. This table reports, for instance, that 1 generates 5 million impressions in demographic D 1 and 8 million impressions in demographic D 2. The demand for impressions in each demographic is equal to 10 million. Advertiser j s willingness to pay for an impression in demographic j is v j, where v 1 = 1 and v 2 = 2 (column 4). The network s price for show i is r i. Ì Ð Ñ Ò Ö Ø Ö Ø Demo Imps. 1 (million) Imps. 2 (million) Valuation (per imp.) Demand (million) D1 5 10 1 10 D2 8 4 2 10

20 Popescu and eshadri: Managing Revenue from TV Ad ales Consider two scenarios. In the first, a network charges a price per impression and must determine what allocation of resources (shows) will maximize its own revenue. This allocation problem can be written as v = max α ij {( 5α 11 + 10α 21 )+2(8α 12 + 4α 22 )} s.t. 5α 11 + 10α 21 10; 8α 12 + 4α 22 10; α 11 + α 12 1; α 21 + α 22 1; α 11, α 12, α 21, α 22 0. In the second scenario, the network charges a constant price for a (fraction of) show and so must solve this pricing allocation problem: d = max r j {(α 11 + α 12 )r 1 +(α 21 + α 22 )r 2 } s.t. 5α 11 + 105α 21 10; 8α 12 + 4α 22 10; α 11 + α 12 1; α 21 + α 22 1; (C1) α 11 r 1 + α 21 r 2 5α 11 + 10α 21 ; (C2) α 12 r 1 + α 22 r 2 2(8α 12 + 4α 22 ); α ij 0, r i 0 (, 2, j=1, 2). Adding constraints (C1) and (C2) results in the inequality (α 11 + α 12 )r 1 +(α 21 + α 22 )r 2 (5α 11 + 10α 21 )+2(8α 12 + 4α 22 ). Thus, the revenue obtained by pricing fractions of shows is less than the revenue obtained by pricing impressions in each demographic. For this particular example we have r 1 = 0, r 2 = 20.92, v = 25 million, and d = 20.92 million; the loss in revenue is 4.08 million (or 16.3%). 8. Numerical Analysis In this section we present two numerical examples. In ection 8.1 we use industry data to illustrate some of the model s main assumptions; in ection 8.2, we test the performance of the heuristics presented in ections 5 and 6. 8.1. Example 1: Advertiser egmentation and Network Estimates of Willingness to Pay In ection 3.3 we assumed that advertisers can be segmented in such a way that a class-j advertiser s willingness to pay for a plan (consisting of a resource allocation α) is a linear function of the number of impressions promised by the plan: w j (α)=v j α ia ij. In this section, we justify that assumption empirically. We identify the main advertiser segmentation criteria and then show that the plan s price for advertisers belonging to the same segment is well approximated by a linear function of the number of impressions in the plan. We also estimate the price per impression in each of the identified advertiser segments. Our example is based on data from two large U.. cable network. The data set contains all plans sold by the network on the upfront and scatter markets during 2006 2008 (two years). Each observation includes advertiser information, plan requirements, show selection and characteristics, and the final negotiated price for a particular plan. 18 18 Values are perturbed to preserve confidentiality. Also for this reason we identify neither the network nor its specific demographic categories (Demo1 Demo19).

Popescu and eshadri: Managing Revenue from TV Ad ales 21 8.1.1. Dependent Variable The dependent variable is the price of a plan (i.e., the amount paid by the advertiser for the whole plan). In light of the ad sales process summarized in ection 1 and discussions with experienced network planners, we expect that plan prices will reflect the plan size (total number of impressions), the target audience, the daypart mix, and the market on which the plan was sold (upfront versus scatter). Initially we ran a regression with CPM as the dependent variable and a large number of independent variables. The independent variables that we considered were unit length (i.e., number of 30-second ads and 15-second ads), market, daypart mix, demographic, plan size, plan date, program type. Many of these variables proved to be nonsignificant. We do not report those results here, preferring to focus on models that provide a good fit and incorporate a sensible number of variables. 19 8.1.2. Model and Results We assume that the price of any plan has this functional form: Plan_Price= v j (Imps j ) ϕ, (13) where v j is a class-j advertiser s willingness to pay (per impression) in demographic j. o a plan s price is given by the unit price of an impression in demographic j multiplied by the number of impressions in the plan (Imps j ) and then adjusted by ϕ, the size of the quantity discount. 20 Now taking the logarithm on both sides yields ln(plan_price)=ln(v j )+ ϕ ln(imps j ). (14) To control for time variation of prices during the two-year period, we introduce a dummy variable (Y) for the year, which is set equal to 1 if the plan was sold in year 2006 2007 (and set to 0 otherwise). Note that by year we refer to the scheduling year (eptember August) and not the calendar year. The aggregated data indicate that upfront prices differ from scatter prices; the latter are, on average, higher than the former. Therefore, we control for the type of market with the indicator dummy U (Upfront). Finally, we conjecture that the demographic of a plan affects its price. Hence we introduce a dummy variable (D i ) for each demographic in the data set. Denoting V j = ln(v j ), we obtain this model: ln(plan_price) = M j=1 V j D j + ϕ ln(imps j )+γy+µu. (15) 19 Other papers in the literature on factors affecting ad prices have focused on the shows characteristics. Thus, as shown in Brown and Cavazos (2003) and Wilbur (2008), comedies and reality shows tend to earn much more than other shows; action dramas, science fiction shows, and news tend to earn much less than the average. Wilbur analyzes a two-sided model of the market. He estimates a model of consumer demand for programs on the one side and, on the other side, a model of advertiser demand for audiences. These estimates reveal that premiums for various program characteristics reflect the viewing preferences of the target demographics. Our analysis is related to their finding that different types of shows attract different audiences and therefore have different prices. However, our analysis differs in allowing the price of a given ad spot to vary depending on the advertiser s demographic. 20 Note that the underlying assumption is that following negotiations, the network extracts the willingness to pay from the advertiser adjusted by a quantity discount, which reflects the advertiser s bargaining power.

22 Popescu and eshadri: Managing Revenue from TV Ad ales We estimate (15) using ordinary least squares. We only consider those demographics for which at least 15 observations are available. The results are summarized in Table 4. The values for Y, D i and Constant are linearly scaled due to confidentiality of data. Variable Network 1: Coef. [td. Err.] Ì Ð Network 2: Coef. [td. Err.] Ê Ö ÓÒ Ê ÙÐØ Variable (contd.) Network 1: Coef. [td. Err.] Network 2: Coef. [td. Err.] Y.033[.023] 0.144 [.011] D11 0.116* [.051] U 0.229*** 0.077* D12 0.514*** [0.039] [0.025] [0.099] D1 0.373*** D13 0.954*** [0.110] [0.099] D2 0.381*** [0.090] D14 0.797*** [0.086] 0.426*** [0.047] D3 0.648*** [0.088] D15 0.656*** [0.085] 0.294*** [0.046] D4 1.592*** [0.090] 0.787*** [0.053] D16 0.844*** [0.086] 0.632*** [0.045] D5 0.288* [0.114] D17 0.034 [0.055] D6 0.609*** [0.103] 0.223*** [0.052] D18 0.464*** [0.049] D7 0.126 [.102] D19 0.464*** [0.046] D8 0.319*** [0.090] 0.047 [0.045] ln(imps) 0.838*** [0.012] 0.921*** [0.007] D9 0.344*** [0.051] Constant 4.025*** [0.111] 2.817*** [0.074] D10 0.269*** [0.061] No. of Obs. 940 1227 R-quared 0.9203 0.9578 *=ignificant at the 5% level; ***=ignificant at the 1% level. The R 2 values are high, which suggests that demographic alone explains an overwhelming part of a plan s price. It is noteworthy that some demographics exhibit a premium (e.g., demographics D7 for network 1 and D8 for network 2) while others exhibit a high discount (e.g., demographics D4 and D16). In general, the broader the demographic, the lower the CPM. Moreover, we note that Network 1 offers a significant discount for plans sold in the upfront market compared to the ones sold in the scatter market. This is not true for Network 2, however, whose prices are slightly lower in the scatter market than in the upfront market. The quantity discount is higher for Network 1 than for Network 2. For the latter, the price of the plan can be well approximated by a linear function of the number of impressions. For the former, additional segmentation based on order quantity might be required. 8.1.3. ummary of Data Analysis o far in this section we have analyzed the factors that affect the price of a plan. We found that, for a given demographic, the number of impressions

Popescu and eshadri: Managing Revenue from TV Ad ales 23 Ì Ð ÌÓØ Ð Ê Ú ÒÙ Ý Ë Ò Ö Ó Ò ÐÐÓ Ø ÓÒ ÈÓÐ Ý ÓÖÖ Ð Ø Ñ Ò µ cenario DLP MLI MP Max 1 561,640 505,751 493,332 591,126 2 561,640 442,826 434,328 591,126 3 561,640 587,967 586,060 591,126 4 561,640 540,066 539,430 591,126 5 561,640 535,882 535,882 591,126 6 561,640 453,561 453,561 591,126 in a plan explains more than 90% of the variability in plan prices. Based on this analysis, we conclude that the main segmentation criteria are demographic, market, and (potentially) size. This means that two advertisers interested in buying the same number of impressions in the same demographic on the same market have equal willingness to pay per impression. Moreover, the willingness to pay for the plan of advertisers within each segment can be well approximated by a linear function of the number of impressions in the plan. Although the analysis in ections 4 6 considered only one segmentation criterion (demographic), the same model can also account for the other two criteria. For instance, to account for the different willingness to pay on the two different markets, we can separate class j into classes j s (the scatter type) and j u (the upfront type), where A ij s = A ij u and v j s > v j u. To model small- and large-budget advertisers, we can separate class-j advertisers into j (denoting a small-budget advertiser of class j) and j ( a large-budget advertiser), where A ij = A ij and v j > v j. 8.2. Example 2: Heuristic Performance Analysis In the rest of this section we test the performance of the three heuristics proposed in ections 5 and 6. We present two cases to test the effectiveness of the proposed methods. In the first we assume that the demands in each demographic are perfectly correlated; in the second case, we assume that such demands are independent. We compute the revenues from each of the following policies: deterministic linear program (DLP), minimum lost impressions (MLI) heuristic, and minimum price (MP) heuristic. We also compute the maximum revenue (Max) the network could have obtained by waiting until time T and build all plans at once. We simulate ten runs of these experiments and report the averages. 8.2.1. Case 1: Perfectly Correlated Demand We assume that there are three demographics. The demand in each demographic is D j = β j D, where the β j are fixed and given (respectively) by 0.5, 0.3, and 0.2. This corresponds to a setting where the demands in each demographic are perfectly correlated. The price per impression in each demographic is (respectively) 30, 50, and 25. The network has the following inventory: A 1 =(5000, 3200, 4000), A 2 =(1000, 800, 200), A 3 =(2000, 500, 300), A 4 =(1000, 800, 1200), A 5 =(700, 200, 1000), A 6 =(3000, 5000, 1000), A 7 =(800, 200, 100), A 8 =(1000, 800, 1000).

24 Popescu and eshadri: Managing Revenue from TV Ad ales Ì Ð ÌÓØ Ð Ê Ú ÒÙ Ý Ë Ò Ö Ó Ò ÐÐÓ Ø ÓÒ ÈÓÐ Ý ÁÒ Ô Ò ÒØ Ñ Ò µ cenario DLP MLI MP Max 1 626,500 603,939 603,939 632,668 2 592,486 508,913 444,445 617,996 3 626,500 595,414 587,264 639,000 4 463,274 550,731 521,347 561,936 5 533,459 587,905 536,284 587,905 Average 568,444 569,380 538,656 607,901 Here the first (resp., second and third) values in each 3-tuple is the total number of impressions in demographic 1 (resp., 2 and 3) available at the start of the year. This inventory is a smallscale representation of an actual network s inventory within a daypart, yet it has enough variety to capture the differences in number of impressions across demographics that we encounter in actual television shows. For simplicity, we assume that the total demand D has a uniform distribution in [12,000, 36,000]. We shall investigate six scenarios for the arrival process. In the first scenario, all class-1 advertisers arrive in [0, t 1 ], all class-2 advertisers arrive in [t 1, t 2 ], and all class-3 advertisers arrive in [t 2, T], where an advertiser s class is given by the demographic of her target audience. In the second scenario, arrivals are ordered as follows: class 1, then class 3, then class 2 (third scenario: class 2, then class 1, then class 3; fourth scenario: class 2, then class 3, then class 1; fifth scenario: class 3, then class 2, then class 1; sixth scenario: class 3, then class 1, then class 2). We explore these stylized arrival scenarios to illustrate the worst and best performances of the four policies. (Random arrival scenarios are discussed in Case 2.) We simulate ten values for the demand averaging D=23, 359 and compute the revenues under each of the six arrival scenarios. The average revenues obtained by the network for each scenario policy pair are summarized in Table 5. The MLI and MP heuristics perform worse than the DLP for scenarios in which the low-revenue customers arrive ahead of the high-revenue customers (scenarios 1, 2, 4, 5, and 6). This result is intuitive: since both heuristics serve all customers in the order of their arrival and no customer is rejected, these policies will perform poorly when more-valuable customers arrive late and there is not enough inventory left to satisfy their demand. cenario 3 is the only one in which the MLI and MP outperform the DLP. In this scenario, the high-revenue (class-2) customers arrive first, followed by class-1 customers, and the low-revenue customers (class 3) arrive last. In each instance, the MLI outperforms the MP policy. This observation holds also for the random arrival scenario described next (as Case 2). 8.2.2. Case 2: Independent Demand For our second example we let the mix of demand be random (i.e., the demands in each demographic are independent). We assume that there are ten advertisers who arrive in a random sequence. At each arrival epoch, a random number x [0, 1] is generated. If x 0.5 then the advertiser is of class 1, and if x (0.5, 0.8] then she is of class 2;

Popescu and eshadri: Managing Revenue from TV Ad ales 25 otherwise, the advertiser is of class 3. The total demand is uniformly drawn from [12,000, 36,000]. The demand of each advertiser is constant and is given by D/10. As in Case 1, we compute the revenues from each of the following policies: DLP, MLI, and MP heuristics. We also compute the maximum revenue the network could have obtained had it waited until time T and built all plans at once. For this experiment we simulate five runs, and the values are given in Table 6. The same observations as in the case of deterministic arrivals hold also for random arrivals. Both the DLP and the MLI perform well, and once again the MLI performs better than the MP. These results are in line with our observation in ection 7 as regards the suboptimality of ratecard pricing. Because the MP heuristic first computes a ratecard and then determines the resource allocation, we are not surprised that it is outperformed by the MLI heuristic in which the inventory is valued in terms of impressions rather than prices per spot. 9. Conclusions In this paper we proposed a new model for optimizing revenue from advertising sales in the television industry. We described the ad sales process and showed that the price of an advertising plan in the upfront or scatter market can be well approximated by a function of the CPM for a demographic and the number of impressions in the plan. Our model explains more than 90% of the volatility in plan prices. This finding enabled us to model the capacity allocation problem by segmenting advertisers in such a way that (i) the price per impression is fixed for the target demographic in each segment and (ii) the willingness to pay of advertisers within a segment is a linear function of the number of impressions that the plan generates in their target demographic(s). The segmentation criteria identified by our analysis were demographic, market and sometimes size (quantity). We presented several methods for optimizing resource allocation among various segments of advertisers. The solution concepts are summarized in Table 7. We proposed several static allocation policies that are easy to implement. The first such policy is partitioned booking limits. For the two-class problem we characterized the optimal solution. We showed that the M-class problem is a convex optimization problem with separable objective for which polynomial-time algorithms exist in the literature. A simple heuristic for computing the PBL is to solve the deterministic linear program, which the literature has shown to be asymptotically optimal. We also discussed first-come, first-served allocation methods. For the two-class problem, the optimal FCF control is always pathwise optimal; for the M-class problem, conditions for the existence of a pathwise optimal allocation are extremely restrictive and the optimal policy is difficult to find. That being said, we used insights gained from the pathwise method to derive two heuristics that are also easy to implement. The first of these, the minimum price heuristic,

26 Ì Ð Long elling Horizon (upfront Market) Plans are sold 6 months to 1 year before airing date; Concentrated Arrivals; hort elling Horizon (catter Market) Plans are sold much closer to airing date; cattered Arrivals; Optimal ËÙÑÑ ÖÝ Ó Ê ÙÐØ ËÓÐÙØ ÓÒ ÓÒ ÔØ tatic Partitioned Booking Limits Convex optimization problem with separable objective (polynomial-time algorithm available); First-Come, First-erved Pathwise optimal solution for two-class problem always exists; Pathwise optimal solution for the n-class problem exists only under strict conditions; Otherwise, optimal solution hard to derive; Popescu and eshadri: Managing Revenue from TV Ad ales Heuristics Deterministic LP Asymptotically optimal. Minimum Lost Impressions Pathwise optimal for two-class problem; Pathwise optimal for n- class problem if such a solution exists. Minimum Price Plan imilar to the heuristic currently used in practice (i.e., build the cheapest plan), but ratecards are computed by solving the dual of the DLP. is closely related to what TV networks currently use: first estimate the price of a spot in a show and then build the least valuable plan that satisfies the advertiser s constraints. However, in our method the shows ratecards are computed by solving the dual of the deterministic linear program. The minimum price heuristic is consistently outperformed by our second heuristic, minimum lost impressions. This second heuristic follows directly from the concept of pathwise optimality and is guaranteed to produce a pathwise optimal solution if one exists. Even if such a solution does not exist, the heuristic provides a reasonable criterion by which to allocate capacity: minimize the number of lost impressions in the main demographics. Last of all, we showed that the current industry practice namely, first setting a price per spot in a show and then allocating resources based on those prices is suboptimal. More revenue would be generated by employing a value-based approach in which the network first estimates a price per impression in each demographic of each segment and then uses that price to derive resource allocation strategies, since this approach enables networks to extract all the advertisers surplus. Appendix PROPOITION 2: Proof : This is a convex optimization problem with concave objective function and affine constraints. To see this, let φ= α i A i1 and θ = (1 α i )A i2. Then the problem can be written as: { max Π(φ, θ)= P1 E[min(D 1, φ)]+ P 2 E[min(D 2, θ)] } α,θ,φ s.t. 0 α i 1,, j= 1, M; φ= α i A i1 ; θ= (1 α i )A i2. (16) Also, 2 Π(φ,θ) = P φ 2 1 f 1 (φ), 2 Π(φ,θ) = P θ 2 2 f 2 (θ), and 2 Π(φ,θ) = 2 Π(φ,θ) = 0. Recall that f φ θ θ φ 1 ( ) and f 2 ( ) are the respective continuous density functions of D 1 and D 2.

Popescu and eshadri: Managing Revenue from TV Ad ales 27 Because (16) is a convex optimization problem, we know that a maximizer α exists. Moreover, the Karush Kuhn Tucker (KKT) conditions given as (17) (19) are necessary and sufficient conditions for problem formulation (8) that the maximizer α must satisfy. For all i = 1,, we have the following system: α i (P 1 A i1 F C 1 ( ) ( ) α i A i1 P 2 A i2 F2 C α i )A i2 ) λ i = 0; (17) (1 Let (α, λ ) be a solution to this system. ( ) ( ) (i) If P 1 A i1 F1 C α i A i1 P2 A i2 F2 C (1 α i)a i2 < 0, then: (1) A i1 A i2 < P 2F C 2( (1 α i )A i2) P 1 F 1( C α i A ; (2) α i = 0. ( α i A i1) P2 A i2 F2 C ( (1 α i)a i2 ) > 0, then: (ii) If P 1 A i1 F1 C (1) A i1 A i2 > P 2F2( C (1 α i )A i2) P 1 F 1( C α i A i1) ; (2) λ i > 0 and α i = 1. λ i (1 α i ) = 0; (18) Therefore, m is the highest index, such that A i1 A i2 = P 2F2( C (1 α i )A i2) P 1 F 1( C α i A i1). Remark 2. The argument above ((ii) (2)) implicitly assumes that if P 1 A i1 F1 C λ i 0. (19) ( α i A i1) ( ) P 2 A i2 F2 C (1 α i)a i2 > 0, then we cannot have α i = λ i = 0. To see that this is indeed true, note that α maximizes the Lagrangian for λ=λ, i.e., α = arg max α [0,1] L(α, λ ), where L(α, λ)= P 1 E[min(D 1, α i A i1 )]+P 2 E[min(D 2, (1 α i )A i2 )]+ λ i (1 α i ). (20) If λ i = 0 and (P 1 E[min(D 1, α i A i1)]+p 2 E[min(D 2, (1 α i)a i2 )]) α i α > 0, then by slightly increasing α i above zero we can increase L(α, λ ), so α is not a maximizer of the Lagrangian for λ=λ. That would be a contradiction, thus it must be that λ i > 0 and α i = 1. PROPOITION 3: Proof : (i) Let y j = α ij A ij. Then the problem can be rewritten as { M } max P j E[min(ξ j (T), y j )] α ij j=1 s.t. M j=1 α ij 1, ; y j α ij A ij = 0 j=1, M; α ij 0,, j= 1, M. (21) It is easy to see the objective function is concave and separable in the variables y 1,..., y M. This, together with the linearity of constraints, implies that the problem is one of convex optimization with separable objective. (ii) Denote the Lagrangian of (21) as [ ( L(α, λ)= M j=1 P j E min D j, α ij A ij )]+ λ i (1 M j=1 ) α ij. (22)

28 Popescu and eshadri: Managing Revenue from TV Ad ales Let y j = α ij A ij and αj y j = A j. Then E[min(D j, y j )] = y j 0 x df j (x) + F c j(y j )y j and αij E[min(D j, y j )]= A ij F c j(y j ). The KKT optimality conditions can be written as follows: (A ij P j F c j (y j) λ i )α ij = 0 i= 1,, j=1, M; (23) λ i (1 M j=1 α ij ) = 0 i= 1, ; (24) α ij 0, λ i 0 i= 1,, j=1, M. (25) By (23), if demographics j and k have a nonzero share in show i (i.e., if α ij > 0 and α ik > 0), then A ij P j F c j (y j)= A ik P k F c k (y k). (26) Next we show that you cannot have both α ij > 0 and α pj > 0 or both α ik > 0 and α pk > 0. In other words, it is suboptimal to have two shows allocated to the same two demographics unless the shows generate proportional number of impressions in each of those demographics that is, unless A ij A ik = A pj A pk. By way of contradiction, assume that such an allocation is not suboptimal. Then, by (26), we have A ij P j F c j (y j)= A ik P k F c k (y k) and A pj P j F c j (y j)= A pk P k F c k (y k). This implies A ij A ik = A pj A pk. In that case, an equivalent solution with at most one show allocated to the two demographics would be: if α ij A ij A pj α pk (i.e., α ij A ik α pk A pk ), then α ij = 0, α ik = α ik+ α ij, α pj = α pj+ α ij A ij A pj, α pk = α pk α ij A ij A pj ; (27) if α ij A ij A pj > α pk (i.e., α ij A ij > α pk A pj ), then α ij = α ij α pk A pj A ij, α ik = α A pj ik+ α pk, α pj A = α pj+ α pk, α pk = 0. (28) ij Thus, there will always exist an optimal solution in which there are no two shows with positive allocations to the same two demographics. imilarly we demonstrate that you cannot have three shows (i, p, s) with positive allocations to j and k, to j and r, or to r and k (respectively). That is: one cannot have α ij > 0 and α ik > 0 or α pj > 0 and α pr > 0 or α sr > 0 and α sk > 0. (29) If the preceding statement holds then it follows that A ij P j F c j (y j)= A ik P k F c k (y k), A pj P j F c j (y j)= A pr P r F c r(y r ), A sr P r F c r(y r )= A sk P k F c k (y k), (30) which implies A pr A sk A ij A pj A sr = 1. As before, in this case the solution can be transformed into an A ik equivalent solution in which there are only two shows containing three demographics and one A show containing exclusively one of the three demographics. Thus, assuming α pr α ij ij A pj α sk A sk α ij A ik, this equivalent solution would be: α ij = 0; α ik = α ik+ α ij ; α pj = α pj+ α ij A ij A pj ; α pr = α pr α ij A ij A pj ; α sr= α sr + α ij A ij A pj A pr A sr ; α sk = α sk α ij A ij A pj A pr A sr. and

Popescu and eshadri: Managing Revenue from TV Ad ales 29 The argument can be extended to prove that, for M classes of advertisers, there will be at most M 1 shows allocated to more than one class. PROPOITION 4: Proof : This follows from the proof of Proposition 5. LEMMA 1: Proof : Let r i = 1 A A i and r k ij = 1 A A k. If r i = r k then obviously there exists a θ > 0 such that kj A i = θa k. Assume r i = r k, and suppose that r il r kl for all l = 1, M with at least one strict inequality. Then, after setting z i = min ( 1, α i A ij +α k A kj A ij ), zp = α p for all p {1,..., } {i, k}, and ), we have p=1 z p A pl p=1 α p A pl for all l= 1, M with at least one strict z k = max ( 0, d M l=1,l =k z l A lj A kj inequality. Then α is not pathwise optimal. A similar argument shows that assuming r il r kl for all l = 1, M also leads to a contradiction. It then follows that r il < r kl for some l and that r il > r kl for some l. As a result, the same vector z constructed previously will yield fewer lost impressions than the vector α in demographic l contradicting the pathwise optimality of α (just assume a path in which all future customers are of type l ). We have thus shown that the only case in which the pathwise optimality of α is not contradicted is r i = r k. Hence there exists a θ> 0 such that A i = θa k. PROPOITION 5: Proof : We first prove that the condition is necessary and then we demonstrate sufficiency. Define the following demand vectors: d 1 =(A i1 ǫ, 0,..., 0),..., d l =(0, 0,..., A i1 β il ǫ,..., 0),..., d M = (0, 0,..., A i1 β im ǫ). We assume ǫ is small enough that each demand can be fully satisfied by a fraction of show i or of show k. For every l = 1, M, let u li be an allocation vector that satisfies demand d l by placing the minimum number of ads exclusively in show i (i.e., u li = (u li 1,..., u li ) with u li p = 0 for all p = i and u li i = ǫ); the vector u lk is defined similarly (i.e., u lk =(u1 lk,..., ulk ) with u lk p = 0 for all p = k and u lk = A i1 β il ǫ k ). Now observe that, for each d A k1 β l, one of the two vectors u li or kl u lk must produce a better allocation along all M dimensions. uppose (to derive a contradiction) that this is not the case, and consider the following path: only customers with demand d l arrive up until inventory in all shows except for shows i and k is depleted. If another customer with demand d l arrives when the only available inventory is in show i and/or k, then no pathwise optimal solution exists because neither u li A u lk A nor u li A u lk A. This result contradicts our assumption that a pathwise optimal solution exists at every arrival epoch. u 1k k W.l.g. we can assume that u 1i u 1k, which is shorthand for u 1i A u 1k A, where u 1i i (since β i1 = β k1 = 1). Then the following inequality holds: = A i1 ǫ A k1 which is equivalent to = ǫ and A i1 β il ǫ A k1 β kl A i1 ǫ A k1 l= 2, M, (31)

30 Popescu and eshadri: Managing Revenue from TV Ad ales If u hi u hk for some h>1, then This, in turn, is equivalent to A i1 ǫ A k1 A i1 β ih ǫ A k1 β kh, β kh β ih, β il β kl l= 2, M. (32) But from (32) and (34) it follows that, if u hi u hk for some h>1, then A i1 β il ǫ A k1 β kl A i1 β ih ǫ A k1 β kh l= 2, M. (33) β il β kl β ih β kh l= 2, M. (34) β ih = β kh, β il β jl 1 l= 2, M; (35) if u hk u hi for some h>1 then the reverse is true: β ih β kh, β il β kl β ih β kh l= 2, M. (36) o if there is an h 1 > 1 and an h 2 > 1 such that u h 1 k u h 1 i and u h 2k u h 2i, then by (36) we have β ih1 β kh1 = β ih 2 β kh2 = s ik, (37) where s ik is some positive number less than 1 (by (32)). Let D i ={d l : u li u lk, l > 1} and D k = {d l : u lk u li, l> 1}. Then we can summarize our necessary conditions as follows: 1. if d l D i then β il = β kl by (35); 2. if d l D k then β il β kl = s ik by (37). To prove sufficiency, we let two shows i and k be such that A i1 A k1. If β il β kl = 1 and s ik 1 (resp., s ik 1), then u lk u li (resp., u lk u li ). To see this, note that u li A = (A i1 ǫ, A i1 β i2 ǫ,..., A i1 β im ǫ) and u lk A = (A i1 ǫ, A i1 β k2 ǫ,..., A i1 β km ǫ). If s ik 1 then u li A u lk A, which implies u li u lk ; similarly, if s ik 1 then u li A u lk A, which implies u lk u li. β If il = s β ik and s ik 1 (resp., s ik 1), then u lk u li (resp., u lk u li ). To see this, note kl that u li A = (A i1 ǫ, A i1 β i2 ǫ,..., A i1 β im ǫ) and u lk A = ( A A i1 β il ǫ A k1, A A k1 β k1 β i1 β il ǫ A k2,..., A kl A k1 β k1 β i1 β il ǫ) km kl A k1 β = kl (A i1 ǫs ik, A i1 ǫs ik β k2,..., A i1 ǫs ik β km ). Then u lk A u li A, which implies that u lk u li. Thus, for any two shows i and k and demand vector d l, we have either u li u lk or u lk u li. Hence there exists an r {1,..., } such that u lr u lk for all l = 1, M. Therefore, u lr is pathwise optimal. PROPOITION 6: Proof : If the network uses partition booking limits as a resource control mechanism and if it sets the price per impression in a given demographic graphic equal to a type-j advertiser s willingness to pay (i.e., P j = v j, j = 1, M), then the revenue maximization problem can be formulated as { M [ ( v = max P j E min α j=1 M D j (T), ij A ij)] α ij 1 i= 1, ; α ij 0,, j=1, M α j=1 }. (38)

Popescu and eshadri: Managing Revenue from TV Ad ales 31 If the network charges a price per show (r i, i = 1, ), then the joint pricing-allocation (revenue maximization) problem can be formulated as follows: Here { M d = max r,α j=1 EΠ(α, r, D j (T)) { M } Π(α, r, D j (T))=max r i β β ij j=1 s.t. 0 β ij α ij i= 1,, j=1, M; M j=1 } α ij 1 i= 1, ; α ij 0,, j=1, M. (39) A ij β ij D j (T) j=1, M; β ij r i P j A ij β ij j=1, M. (40) Let α and r be the optimal solutions to (39) and let α be the optimal solution to (38). Then α is feasible for (38). By (40) we have Π(α, r, D j (T)) P j min ( D j (T), α ij A ij ) for any realization of D j (T). Then M j=1 E Dj (T)[Π(α, r, D j (T)] P j E Dj (T)[ min ( Dj (T), α ij A ij )] P j E Dj (T)[ min ( Dj (T), α ij A ij )]. Example 5 in ection 7 shows that the inequality is strict in at least some cases. imilar arguments can be used to show that when a FCF control is used then ratecard pricing yields a lower expected revenue than value-based pricing. References [1] Araman, V. F., I. Popescu. 2010. Media Revenue Management with Audience Uncertainty: Balancing Upfront and pot Market ales. M&OM. 12, 190-212. [2] Banciu, M., E. Gal-Or, P. Mirchandani. 2010. Bundling trategies When Products Are Vertically Differentiated and Capacities Are Limited. Management cience. 56, 2207-2223. [3] Bertsimas, D., I. Popescu. 2003. Revenue Management in a Dynamic Network Environment. Transportation cience. 37, 257-277. [4] Birge, J. R., F. Louveaux. 1997. Introduction to tochastic Programming. pringer eries in Operations Research, NY. [5] Bloomberg BusinessWeek. 2011. U.. Advertising Expenditures Rose 6.5% in 2010. [6] Bollapragada,., M.R. Bussieck,. Mallik. 2004. cheduling Commercial Videotapes in Broadcast Television. Operations Research. 52, 679-689. [7] Bollapragada,., H. Cheng, M. Phillips, M. Garbiras, M. choles, T. Gibbs, M. Humphreville. 2002. NBC s Optimization ystems Increase Revenues and Productivity. Interfaces. 32, 47-60. [8] Bollapragada,., M. Garbiras. 2004. cheduling Commercials on Broadcast Television. Operations Research. 52, 337-345. [9] Brown, K.., R. J. Cavazos. 2003. Empirical Aspects of Advertiser Preferences and Program Content of Network Television. FCC, Media Bureau taff Research Paper. [10] Cooper, W.L. 2002. Asymptotic Behavior of an Allocation Policy for Revenue Management. Operations Research. 50, 720-727.

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