Serving two masters? Optimizing mobile ad contracts with heterogeneous advertisers

Abstract

Our research question is motivated by a real problem faced by an existing demand aggregator. The aggregator represents multiple advertisers, each of whom signs one of two types of contracts with the aggregator for bidding on an RTB (real‐time bidding) platform. A quality contract occurs on a cost‐per‐impression (CPM, i.e., cost per thousand impressions) basis. The advertiser is promised a minimum number of impressions and pays a CPM that is a function of the targeting quality as measured by the realized click‐through rate (CTR). In a performance contract the advertiser pays on a CPC (cost‐per‐click) basis constrained by a budget. We develop and solve a generalized profit maximization problem that jointly optimizes the aggregator's bidding and allocation decisions. The allocation policy optimally assigns each arriving bidding opportunity to a specific advertiser. The bidding policy then computes a bid amount for that allocation based on the estimated click probability of the opportunity. Our solution has nice theoretical properties. First, neither policy depends on the memory carried in the system, that is, the sequence of previous states and decisions, making the solution easy to implement. Second, the allocation policy is shown to have a threshold structure. This enables the assignment of arriving opportunities into one of two distinct sets, each corresponding to a specific advertising contract type. The assessment of the impact of the change in various parameters on the solution is used to derive several interesting and important implications for the management of advertising contracts.

INTRODUCTION

The usage of mobile devices has increased exponentially during the past decade, contributing to 70% of all Internet traffic (Aleksandar, 2022). Not surprisingly, this has led to a shift in the digital advertising landscape with mobile advertising predicted to comprise 81% of the online advertising market in the United States by 2023 (Marketingcharts, 2019). The ability to target advertisements (henceforth ads) based on the location of mobile device users has further enhanced interest in mobile advertising. There exist multiple channels for advertising on mobile devices (mobile web, social media, etc.). We are interested in in‐app advertising where advertisements in the form of banner ads, videos, and so forth, are displayed when a user opens a mobile application (henceforth apps). This form of advertising is prevalent and comprises the largest fraction of mobile promotions. Planning the allocation of advertisement opportunities under advertiser‐enforced constraints is a growing stream of literature; for instance, Shen et al. (2021) propose a plan for displaying ads when advertisers have budget constraints. Similarly, Rui et al. (2017) develop an algorithm for the allocation of online display advertisements based on advertisers' ad valuations with respect to potential viewers. Our research adds to this current and important topic by considering similar concerns from the aggregator's perspective.

Background and problem description

We begin by describing the business background that is the backdrop for our study. The main entities involved are the advertisers, publishers, and intermediaries as shown in Figure 1. The advertisers constitute the demand side of this equation and wish to advertise their products and services on mobile devices. The publishers (the supply side) are mobile app owners that provide the real estate on which the ads are served. The presence of a large number of firms on either side of this relationship paves the way for intermediaries on both sides. Demand aggregators are the representatives on the demand side (for the advertisers), with supply‐side aggregators working on behalf of publishers (app‐owners). Contracts between the demand aggregator and the advertiser are primarily based on either the CPM (cost‐per‐impression) model or the CPC (cost‐per‐click) model. ¹ In mobile advertising the former model is preferred when the advertiser's interest lies in increasing brand awareness, whereas the CPC model is favored when the advertiser is interested in selling specific products and services via special promotions. Our focus is maximizing profits for the demand aggregator when both contracts are offered simultaneously.

FIGURE 1

Mobile advertisement ecosystem

A growing trend for such trading is via real‐time auctions (more commonly referred to as real‐time bidding or RTB). The RTB platform is a digital marketplace through which competing demand aggregators evaluate and bid in real time for ad‐spaces, buying them on behalf of their customers (advertisers). By recent estimates, RTB‐based ads comprise 90% of all digital advertising (Mitchell, 2022). The increasing use of RTB, together with the ability to target smaller geographical areas, has led to the growth of small business advertisers within this market. Our focus in this paper is on the demand side, that is, from the perspective of a demand aggregator who simultaneously contracts with one or more ad exchanges (using RTB platforms) and buys ad space on behalf of multiple small advertisers. This side of the market is highlighted in Figure 1 with a blue (left) outline.

An opportunity for an ad to be displayed arises each time a user opens an app on her mobile device. This opportunity results in the start of an auction on the RTB platform by the supply aggregator. Demand aggregators like Cidewalk ² are responsible for deciding the bid amount, placing the bid, and displaying the ad upon winning the bid. We use the term impression for an opportunity that has been won. Small business advertisers launch ad campaigns by displaying ads to customers within specific geographical areas over a short time horizon. The advertiser contracts with a demand aggregator for each ad campaign by opting either a CPM or CPC contract. In the CPM contract, the demand aggregator is paid per impression subject to the delivery of a minimum number of impressions. For the CPC contract, the demand aggregator is paid per click delivered while not exceeding the advertiser's budget constraint. For example, a CPM campaign might be designed to deliver 10,000 impressions within a specific zip code over a week with the demand aggregator receiving $50.00. On the other hand, under a CPC contract an aggregator might be paid $1.00 for every click delivered over the same time horizon with the total revenue not exceeding $100.00.

In recent times aggregators have begun considering a new kind of CPM‐based contract. In such a contract, the click‐through rate (CTR)

$(\frac{\text{Number}\text{of}\text{clicks}}{\text{Number}\text{of}\text{impressions}\text{delivered}})$

realized at the end of the campaign serves as a signal for the relevance of the ads to the audience. Thus, the CPM is a variable amount that increases with the realized CTR. We refer to this type of contract as a quality contract and the CPC contract described above as a performance contract. It should be noted that advertisers who opt for quality contracts (quality advertisers) display ads whose primary purpose is to increase brand awareness and use the realized CTR solely as a proxy for measuring the quality of targeting. On the other hand, advertisers opt for the performance contract (performance advertisers) when they wish to offer specific products, services, or promotions. Since the ads are designed with this intent, a click on such an ad is more likely to lead to a conversion. However, too many conversions can generate excess demand that may be detrimental to the advertiser. Having more customers than the advertiser is equipped to handle can lead to stock‐outs (demand that remains unfulfilled); such an occurrence is known to generate lost goodwill (Chopra & Meindl, 2007). Similar concerns have been studied in a related context by Liu and Mookerjee (2018) in which one consideration when optimizing profit is to avoid attracting too much traffic by overadvertising. Advertisers desirous of performance contracts therefore limit the number of clicks they want by imposing a budget amount. For example, a restaurant owner wishing to offer a special promotion for a new entree while also ensuring that he has the capacity to meet the increased demand would likely utilize such a contract. These contract types, quality and performance, are the focus of our paper.

In an RTB environment ad space is marketed on a CPM basis. Regardless of the contract type, the cost to the aggregator is dependent on the acquisition cost of the impressions purchased. It should be clear that the bidding strategy employed by the aggregator plays a crucial role in the acquisition cost. Bidding too high will increase costs and thereby reduce profits. On the other hand, bidding too low could mean lost impressions, potentially resulting in the aggregator either violating the terms of the quality contract or missing revenue opportunities under the performance contract. It should also be noted that the number of available opportunities (the supply) is an exogenous and fixed value over which the aggregator has no control.

Contributions

Our objective in this paper is to determine optimal bidding and allocation strategies in order to maximize profits for a demand aggregator who simultaneously contracts with multiple advertisers where each advertiser opts for one of the two contract types. We contribute to the extant literature on advertising by being the first to (i) present and solve the quality contract problem where the pricing mechanism is a function of the “quality” of the impressions displayed; (ii) simultaneously consider two different contract types, each chosen by multiple advertisers; and (iii) study this profit maximization problem from the perspective of the demand aggregator as intermediary. Note that while we use the context of mobile advertising to showcase our results, they are applicable to the field of digital advertising as a whole.

We develop and solve a generalized profit maximization problem that jointly optimizes the aggregator's bidding and allocation policies. The bidding policy computes an optimal bid amount based on each arriving opportunity, and the allocation policy optimally assigns each opportunity to a specific advertiser. The optimal policies have nice theoretical properties. First, neither of these policies depend on the memory (the sequence of previous states and decisions) carried in the system. Thus, these policies are easy to implement in practice. Second, the allocation policy is shown to have a threshold structure. This implies that the space of arriving click probabilities can be partitioned into two sets, each corresponding to a specific type of advertising contract.

We have several interesting insights. We find that, independent of the optimal bidding strategy employed for the quality advertiser, it is infeasible or suboptimal in some cases to utilize all of the performance advertiser's budget. Similarly, it may be suboptimal to purchase exactly the minimum number of impressions required by the quality advertiser; again, this is independent of the bidding decision for the performance advertiser. Our solution furnishes clear recommendations regarding actions the aggregator can take in order to achieve a desired position in the market; for example, the aggregator may prefer to be in a state where exactly the number of impressions contracted for by the quality advertisers are purchased.

The problem we solve is inherently difficult due to the dynamic and stochastic nature of the state of the system, described by the number of impressions that have been displayed at any point in time and the number of clicks generated. This state likely changes with each arriving opportunity, and a priori we would expect this state to play a significant role in the bid decision for every new opportunity. We employ an elegant approach to solving this complex problem. We first solve a problem with one quality and one performance advertiser. Then, we demonstrate that the problem with multiple advertisers of each of these two types can be condensed to a problem with two virtual advertisers, one of each type. We use Lagrangian relaxation in order to find the optimal solution, and demonstrate that the optimal bidding and allocation policies are both independent of the previous sequence of states and decisions. This independence and the closed‐form nature of our solutions have important ramifications for the aggregator since they enable implementation of optimal RTB and allocation decisions.

After presenting a brief survey of existing literature in Section 2, we provide a detailed description of the aggregator's problem in Section 3. Section 4 details the solution process and the comprehensive sensitivity analysis conducted in order to further enhance and validate our results. Managerial implications are discussed in Section 5. We conclude the paper with a summary of results and a discussion on future directions.

LITERATURE REVIEW

There is a vast amount of literature in the space of digital advertising. In this section, we provide a summary of the extant research that is closely related to our problem. We categorize this research into the following streams: (1) optimizing digital advertising campaigns, (2) optimal ad auction bidding strategies, (3) digital advertising pricing mechanisms, and (4) advertising economics.

The first stream of literature focuses on displaying ads on different platforms (e.g., websites, smart‐phones, tablets, and Internet‐enabled game consoles) while optimizing a given objective (e.g., revenue and clicks) over a planning horizon. Apart from the characteristics of the ads to be shown (e.g., size and location) and the issues associated with fitting a set of ads into the available display space, advertiser constraints based on ad saturation and competition are also considered (e.g., Dawande et al., 2005; Kumar et al., 2007). In order to maximize a publisher's (owner of the website where the ads are displayed) benefit, Najafi‐Asadolahi and Fridgeirsdottir (2014) study how to optimize the pricing of display ads under the CPC model. Balseiro et al. (2014) study the revenue optimization problem faced by a publisher who simultaneously sells impressions through two channels: via an RTB exchange and also directly to the advertisers. Mookerjee et al. (2017) study an optimization problem that jointly considers both the publisher as well as the ad network. The goal of the ad network is to maximize revenue while also meeting a CTR constraint specified by the publisher. Aseri et al. (2018) offer near‐optimal policies for mobile‐promotion platforms purchasing mobile app impressions for advertisers to meet their location and volume requirements. Additionally, several patents exist for methods measuring the effectiveness of an Internet ad campaign (Gerken, 2008; Harvey et al., 2010; Lindsay et al., 2010; Srinivasan & Shamos, 2010). Shen et al. (2021) propose a planning approach that is different from the event‐based auction mechanism for a publisher to optimize its digital advertising. In this planning approach, the publisher first applies an arbitrary‐point–inflated Poisson regression model to forecast the number of clicks, and then solves a mixed‐integer nonlinear optimization problem that maximizes its ad revenue. Lejeune and Turner (2019) study an advertisement allocation problem under resource constraints when advertisers want ads spread out across various market segments. Our paper contributes to this stream by studying a profit optimization problem faced by an aggregator who purchases impressions through RTB for multiple advertisers with different contract types. We study a new type of allocation problem where these advertisers are one of two types (performance or quality). This type of allocation problem among heterogeneous advertisers has not been addressed in the past.

Our paper is also related to the literature on optimal bidding strategies in online ad auctions from the perspective of advertisers (see Choi et al., 2020, for a comprehensive review). Given the auction mechanisms, different constraints faced by the advertisers, such as the available budget and the number of impressions required, are considered in the bid calculations in previous studies. For instance, Ghosh et al. (2009) propose a bidding agent to purchase a given number of impressions with respect to a budget constraint, while Balseiro and Gur (2019) study advertisers' bidding strategies in repeated auctions with budgets. This literature is related to the topic of attracting potential buyers to the website. In this space, Liu and Mookerjee (2018) study how advertisers compete with two decision processes under the CPC pricing model in a duopoly setting. These two decision processes are spending based (beginning with the spending budget and determining the traffic accordingly) versus traffic based (beginning with a target level of traffic and then determining the appropriate spending budget). They show that traffic‐based competition generates higher profits for both firms. When the decision process itself is a choice, then spending‐based advertising is chosen by both firms in equilibrium. We contribute to the literature by developing an optimal bidding and allocation policy from the perspective of an aggregator who assigns impressions to its advertisers in order to meet their corresponding CPM or CPC contracts.

Additionally, our paper is related to the studies on pricing mechanisms in online advertising. Asdemir et al. (2012) apply the principal–agent setting in order to study factors that affect the preference of the CPM model versus the CPC model. Hu et al. (2016) study the incentive issues of the advertiser and publisher for performance advertising pricing models, that is, CPC and cost‐per‐action (CPA) models. Liu and Viswanathan (2012) identify conditions when a publisher should apply the strategy of pay‐per‐impression, pay‐for‐performance, or a combination of both. Balseiro and Candogan (2017) explore a contract design problem for delivering a fixed number of impressions when advertisers' budgets and targeting criteria are private and provide an optimal bidding policy, which is stationary. In our study, we extend the CPM model studied previously to a pricing mechanism in which the CPM is a function of the “quality” of the impressions delivered. Such a pricing system has not yet been studied in previous research in this domain.

Finally, there are several papers in the domain of two‐sided markets that consider the issue of advertising for mobile apps. Ji et al. (2019) study the problem of joint advertising investment and in‐app advertising adoption decisions by platform owners and app developers on a mobile platform using a differential game theoretical model. They show that a platform owner may delay or even not offer an in‐app advertising program when revenue is low. Guo et al. (2019) conduct an economic analysis of reward advertising. Reward advertising is a monetization mechanism for app developers in which consumers choose to view ads in exchange for premium app content. Their study shows that it is often optimal to offer reward ads jointly with the direct selling of premium content. Hao et al. (2017) develop a two‐sided market model of in‐app advertising, with advertisers and app users as the two sides of the market, to study a mobile platform owner's optimal advertising revenue‐sharing contract. On the mobile platform, app developers set the price for the apps while the platform owner sets the price for the ads. Their study identifies a between‐agent subsidization strategy for the platform owner in which it is optimal for the owner to subsidize the developer via the advertising channel, leading to greater profits for both. These papers are different from our research context in which we consider the problem of a demand aggregator; none of the studies cited above have examined that side of the advertising economy.

THE AGGREGATOR'S PROBLEM

We begin this section by highlighting aspects of the problem common to the two contracts that we study. We then introduce the model notation and present the complete problem faced by the demand aggregator.

Model preliminaries

There are two critical and important components that are common to the two contracts; the first is the prediction of the click probability associated with an opportunity, and the second is the bid function. We briefly describe both components.

Click‐probability prediction

The ad‐delivery process is illustrated in Figure 2. When a user opens an app, this event is relayed to an RTB platform where an auction ensues to win the right to show an ad for the resultant opportunity. The exchange provides data pertaining to the opportunity, such as the app opened, time, user location, user device type, to all demand aggregators who have contracted with this ad exchange. ³ Each demand aggregator then uses these data to independently evaluate its interest in this opportunity and decide an appropriate bid amount.

FIGURE 2

Advertisement delivery process

We assume that a demand aggregator (henceforth, referred to as aggregator for conciseness) can predict the click probability,

$c_i$

, associated with each arriving opportunity, i. This can be done using a logistic regression model developed based on the past opportunity data provided by the RTB platform together with the corresponding click outcomes. These historical data also enable us to fit a distribution

$f(c)$

for the click probabilities. We use the term mean click probability to refer to the mean of

$f(c)$

. It is important to note that the ad itself is usually not considered for the click‐probability prediction. There are two reasons for this. First, the aggregator does not know the exact ad that will be displayed, but only has access to a link pointing to the ad location. This arrangement gives advertisers the flexibility to display different ads depending on their needs. Second, and more importantly, the duration of the campaigns managed by an aggregator is relatively short (e.g., 1 week), and the campaign may not repeat over time. It is therefore difficult to obtain reliable data to predict the click probability associated with an opportunity based on the ad being displayed. For these reasons the click probability of an opportunity is predicted in real time, using only the information available in the bid request from the ad exchange.

The bid function

The other aspect requiring elaboration is the bid function faced by the aggregator at the ad exchange. The bid function

$b(x)$

represents the relationship between the bid amount for an opportunity and the corresponding probability (x) of winning that opportunity. Using data collected from Cidewalk, we estimate the bid amount needed as a linear function of the win probability (

$b(x)=\alpha x$

). The data consist of 13,520,613 opportunities that arrived (from an RTB) over a week, of which the aggregator won 1,050,102. The lowest bid amount was $0 and the highest was $3.96. We first bin the bid amounts into 396 categories in increments of one cent. ⁴ For each bin we determine the win probability by dividing the number of won bids with the total number of opportunities within that bin. As expected, higher bid amounts result in higher probabilities of winning the opportunity. Next we fit a regression model to predict the bid amount for a given win probability. The adjusted R‐squared for this model is 0.9823 demonstrating an excellent fit on the data, and the slope of the line is significant at 99.99% confidence interval. We therefore conclude that a linear bid function

$b(x)=\alpha x$

is appropriate for computing the bid amount given a desired win probability. It follows that the expected cost of a bid is

$b(x)\times x=\alpha x^2$

, that is, the expected bid cost function is convex in the win probability.

The decisions

The goal of the problem is to simultaneously arrive at two optimal decisions: (1) the allocation policy and (2) the bidding policy. The allocation policy determines which type of advertiser should be assigned the opportunity, depending on the click probability,

$c_i$

, of that opportunity. The bidding policy then computes a win probability based both on the advertiser type as well as the click probability.

Problem formulation

We now present the aggregator's problem. Table 1 contains all model notation and facilitates easier understanding of the discussion that follows. Recall that small business advertisers launch campaigns in order to advertise to customers within specific geographical areas over short time horizons (usually a week). The aggregator simultaneously contracts with N such advertisers in order to manage their campaigns. Some of these are quality advertisers that opt for the quality contracts—we represent this set of advertisers as

${\mathrm{\Theta }}_Q$

. The remaining are performance advertisers who opt for the performance contracts where

${\mathrm{\Theta }}_P$

represents the set of these advertisers. Per the quality contract, each quality advertiser

$q\in {\mathrm{\Theta }}_Q$

is guaranteed a delivery of

$M_q$

impressions (won opportunities) over the time horizon and in the specified location.

The aggregator is paid

$m({\eta }_q)$

for each impression delivered. The revenue function,

$m({\eta }_q)$

, is a function of the realized CTR (

${\eta }_q$

) for the q ^th advertiser. Under the performance contract, the revenue from each performance advertiser

$p\in {\mathrm{\Theta }}_P$

is on a CPC basis with the total revenue from an advertiser not exceeding their budget,

$A_p$

. We assume that all campaigns begin at the same time and the durations are same. Thus, at the initial state of the system, the number of impressions shown, and the number of clicks generated, are both equal to zero. Based on the aggregator's knowledge of the market and the arrival rate of opportunities, the aggregator is able to specify the number of opportunities (K) that is available over the campaign duration. It should also be noted that K is an exogenous value over which the aggregator has no control and is independent of the number of advertisers.

TABLE 1

Model notation (parameters, decision variables, and functions)

Variable	Description
Market
K	Total number of opportunities available during a campaign period
$b(x_i^{\pi }(·))=\alpha x_i^{\pi }(·)$	The bid function where α is the cost parameter
Aggregator
$\mathbb{Z}$	Profit for the demand aggregator
r	Revenue per click
η	Realized click‐through rate (CTR)
$m(\eta )={\gamma }_0+{\gamma }_1\eta$	Revenue per impression where γ0 and γ1 are the revenue parameters
π	The bidding and allocation policy
$x_i^{\pi }(·)$	The target probability for the i th arrival under policy π
Advertiser
${\mathrm{\Theta }}_Q$	Set of quality advertisers
${\mathrm{\Theta }}_P$	Set of performance advertisers
$N=|{\mathrm{\Theta }}_Q|+|{\mathrm{\Theta }}_P|$	Total number of advertisers who have contracted with the aggregator during one campaign period
$M_q$	The minimum number of impressions required for the q th quality advertiser
$A_p$	The maximum budget available for the p th performance advertiser
Click probability
$c_i$	The predicted click probability of the i th arrival
$\stackrel{\sim }{c}$	A random variable from the click probability distribution
$f(c)$ , $F(c)$	The $pdf$ and the $cdf$ of the click probability distribution
${\mu }_c$	The mean of the click probability distribution
${\sigma }_c$	The variance of the click probability distribution
$\overline{c}$	The click probability threshold
${\mu }_q$	The mean of the truncated click probability distribution allocated to quality advertisers
${\sigma }_q$	The variance of the truncated click probability distribution allocated to quality advertisers
${\mu }_p$	The mean of the truncated click probability distribution allocated to performance advertisers
${\sigma }_p$	The variance of the truncated click probability distribution allocated to performance advertisers
Allocation
${\mathcal{Y}}_i^{\pi }(·)$	Probability of allocating the i th opportunity to a quality advertiser
$y_{q,i}^{\pi }(·)$	Probability of allocating the i th opportunity to the q th quality advertiser
$y_{p,i}^{\pi }(·)$	Probability of allocating the i th opportunity to the p th performance advertiser
States
${\stackrel{\sim }{u}}_i$	Uniformly distributed random variable corresponding to the realized value of an impression
${\stackrel{\sim }{v}}_i$	Uniformly distributed random variable corresponding to the realized value of a click
${\mathbb{I}}_{x_i}(·)$	Indicator variable: equal to 1 when an impression is won, and 0 otherwise
${\mathbb{I}}_{c_i}(·)$	Indicator variable: equal to 1 when a click is generated, and 0 otherwise
$U_{q,i},U_{p,i}$	The set of realized values of ${\stackrel{\sim }{u}}_i$ for i opportunities that are associated with quality advertiser q or performance advertiser p
$V_{q,i},V_{p,i}$	The set of realized values of ${\stackrel{\sim }{v}}_i$ for i opportunities that are associated with quality advertiser q or performance advertiser p
$U_i$ ( $V_i$ )	The set of realized values of ${\stackrel{\sim }{u}}_i$ ( ${\stackrel{\sim }{v}}_i$ ) for i opportunities for all advertisers
${\stackrel{\sim }{U}}_i$ $({\stackrel{\sim }{V}}_i)$	The random variable associated with $U_i$ $(V_i)$
$j_{q,i}^{\pi }(U_{q,i-1}),j_{p,i}^{\pi }(U_{p,i-1})$	The number of opportunities won for the quality advertiser q (or performance advertiser p) under policy π just prior to the i th opportunity given the realized history $U_{q,i}$ (or $U_{p,i}$ ).
$j_i^{\pi }(U_{i-1})$	The total number of impressions won across all advertisers before opportunity i $=\sum _{q\in {\mathrm{\Theta }}_Q}{j}_{q,i}^{\pi }(U_{q,i-1})+\sum _{p\in {\mathrm{\Theta }}_P}{j}_{p,i}^{\pi }(U_{p,i-1})$
$h_{q,i}^{\pi }(V_{q,i-1}),h_{p,i}^{\pi }(V_{p,i-1})$	The number of clicks generated for quality advertiser q (or performance advertiser p) under policy π just prior to the i th opportunity given the realized history $V_{q,i}$ (or $V_{p,i}$ ).
$h_i^{\pi }(V_{i-1})$	The total number of clicks generated across all advertisers before opportunity i $=\sum _{q\in {\mathrm{\Theta }}_Q}{h}_{q,i}^{\pi }(V_{q,i-1})+\sum _{p\in {\mathrm{\Theta }}_P}{h}_{p,i}^{\pi }(V_{p,i-1})$

Given multiple simultaneously operating contracts, the aggregator evaluates each arriving opportunity and decides the appropriate bid amount. Upon winning, the bid is allocated to one of the

$N(=|{\mathrm{\Theta }}_Q|+|{\mathrm{\Theta }}_P|)$

advertisers and an ad from that advertiser is displayed.

It should be clear that the choice of bid amount is a crucial decision for the aggregator. Bidding too low could result in lost impressions, lowering revenue, while bidding too high could lead to lower profits due to the higher cost of acquiring impressions. The arriving opportunities are heterogeneous in terms of the characteristics discussed in Section 3.1.1. This heterogeneity is condensed and captured (through the predictive model) by the click‐probability,

$c_i$

, which is therefore unique to each arrival. Aggregators assess the profit‐maximizing potential of each arrival based on its click probability and the current state (number of impressions displayed so far and number of clicks generated) of the campaign. This assessment enables them to arrive at a measure of the attractiveness of the new opportunity – the targeted win probability of the bid. The aggregator targets a higher win probability for a more attractive opportunity. The bid function relates the win probability to an appropriate bid amount. Henceforth, we will refer to this targeted win probability as the target probability. Therefore, given the following:

• total number of bidding opportunities in a campaign,

• number of impressions won before the i ^th arrival,

• number of clicks generated before the i ^th arrival,

• predicted click probability and its associated distribution,

the policy π defines two rules that constitute the optimal decision for each arriving opportunity: (i) the allocation policy regarding which advertiser should be allocated a particular opportunity, and (ii) a bidding policy that determines the corresponding target probability,

$x_i^{\pi }$

. Based on the value of target probability, the bidding amount is calculated using the bid function

$b(x_i^{\pi })$

.

The event of winning a bid and that of clicking an advertisement are each random processes and we model them accordingly. In the absence of any specific priors of the probability distributions of these events we model them as random draws from a uniform distribution. If the randomly drawn value is less than a threshold, we consider the event to have occurred and to have not occurred otherwise. For a given opportunity, the threshold for winning an impression is the win probability, and the predicted click probability provides the threshold for clicking on an impression.

In the case of bidding, let

$\{\stackrel{\sim }{u_i}:i\in \{1,2,\text{…},K\}\}$

denote a set of i.i.d. uniformly distributed random variables, each with support [0,1]. We say that we win opportunity i under policy π if and only if

$\stackrel{\sim }{u_i}\le x_i^{\pi }(j_i^{\pi }(U_{i-1}),h_i^{\pi }(V_{i-1}),c_i)$

. A vector

$U_i$

is the set of realized values of the random variables (corresponding to the impressions that are won) for i opportunities, and

${\stackrel{\sim }{U}}_i$

represents the random variable associated with this vector. Let

$j_i^{\pi }(U_{i-1})$

be the number of opportunities won under policy π just prior to the i ^th opportunity given the realized history

$U_{i-1}$

. Let

$j_{q,i}^{\pi }(U_{q,i-1})$

be the number of opportunities won for quality advertiser q under policy π just prior to the i ^th opportunity given the realized history

$U_{q,i-1}$

. Similarly, let

$j_{p,i}^{\pi }(U_{p,i-1})$

be the number of opportunities won for performance advertiser p under policy π. The total number of impressions won across all advertisers before opportunity i is represented by

$j_i^{\pi }(U_{i-1})={\sum }_{q\in {\mathrm{\Theta }}_Q}{j}_{q,i}^{\pi }(U_{q,i-1})+{\sum }_{p\in {\mathrm{\Theta }}_P}{j}_{p,i}^{\pi }(U_{p,i-1})$

.

Similarly, let

$\{\stackrel{\sim }{v_i}:i\in \{1,2,\text{…},K\}\}$

denote a set of i.i.d. uniformly distributed random variables, each with support [0,1]. We say that a click is generated if and only if

$\stackrel{\sim }{v_i}\le c_i$

. Following notation analogous to that used for tracking the impressions, the total number of clicks generated across all advertisers before opportunity i is represented by

$h_i^{\pi }(V_{i-1})={\sum }_{q\in {\mathrm{\Theta }}_Q}{h}_{q,i}^{\pi }(V_{q,i-1})+{\sum }_{p\in {\mathrm{\Theta }}_P}{h}_{p,i}^{\pi }(V_{p,i-1})$

.

Although in general the win probability x does not have to depend upon anything, but in a dynamic process such as ours, the win probability, which is a decision, should ideally incorporate the history of winning and clicking. We accordingly set the win probability x to be a function of the numbers of wins (

$j_i^{\pi }$

) and number of clicks (

$h_i^{\pi }$

) occurring up to the time period i. Likewise, the win probability should also be allowed to depend upon the likelihood of clicking the ad; if the likelihood is high, the win probability can be different from when the likelihood is low. The win probability is accordingly denoted by

$x_i^{\pi }(j_i^{\pi }(U_{i-1}),h_i^{\pi }(V_{i-1}),c_i)$

. The bid amount that will result in this probability of winning the bid is

$b(x_i^{\pi }(j_i^{\pi }(U_{i-1}),h_i^{\pi }(V_{i-1}),c_i))$

. For example,

$x_4^{\pi }(j_4^{\pi }(U_3),h_4^{\pi }(V_3),c_4)$

represents the win probability targeted by policy π for opportunity

$i=4$

, given that

$j_4^{\pi }(U_3)$

and

$h_4^{\pi }(V_3)$

impressions and clicks, respectively, of the previous three impressions have been won and that the click probability for this (the fourth) impression is c ₄. The bid amount corresponding to this target probability is

$b(x_4^{\pi }(j_4^{\pi }(U_3),h_4^{\pi }(V_3),c_4))$

. The probability of allocating an impression i (an opportunity that has been won) to a quality advertiser q is denoted by

${\mathcal{y}}_{q,i}^{\pi }(j_{q,i}^{\pi }(U_{q,i-1}),h_{q,i}^{\pi }(V_{q,i-1}),c_i)$

; likewise the probability of allocating an impression i to a performance advertiser p is denoted by

${\mathcal{y}}_{p,i}^{\pi }(j_{p,i}^{\pi }(U_{p,i-1}),h_{p,i}^{\pi }(V_{p,i-1}),c_i)$

.

For ease of exposition, we define:

1

In order to illustrate the evolution of the states we use a simple example with

$K=8$

, and show in Table 2 how the state transition variables evolve with each incoming opportunity i. The first column denotes the opportunity and the second column is the realization of the random variable

${\stackrel{\sim }{u}}_i$

, denoted by

$u_i$

. The third column is the targeted win probability. If the opportunity is won (

$u_i<x_i^{\pi }(·)$

), the opportunity is allocated to either a quality advertiser (q) or a performance advertiser (p) as depicted in column 4. A “−” in that column means that the opportunity was not won. The fifth, sixth, and seventh columns show the number of impressions just before the arrival of the i ^th opportunity. Therefore,

$j_{q,2}^{\pi }$

is 1 because only one impression has been allocated to a quality advertiser prior to the arrival of opportunity

$i=2$

. Column 8 stores the vectors of the realized values of the random variables

${\stackrel{\sim }{u}}_i$

for all i where the impression was won. The ninth column with

$v_i$

shows the realized values of the random variables

${\stackrel{\sim }{v}}_i$

for all i where the impression was clicked. The tenth column is the predicted click probability for the corresponding opportunity. Accordingly, variables in the remaining columns are updated.

TABLE 2

Illustrative example demonstrating state evolution

				Impressions						Clicks
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)	(14)
i	$u_i$	$x_i^{\pi }(·)$	$p/q$	$j_{p,i}^{\pi }$	$j_{q,i}^{\pi }$	$j_i^{\pi }$	$U_i$	$v_i$	$c_i$	$h_{q,i}^{\pi }$	$h_{p,i}^{\pi }$	$h_i^{\pi }$	$V_i$
1	0.2	0.6	q	0	0	0	$\{0.2\}$	0.7	0.2	0	0	0	$\{0\}$
2	0.7	0.3	−	0	1	1	$U_1\bigcup \{0\}$	0.3	0.1	0	0	0	$V_1\bigcup \{0\}$
3	0.4	0.5	p	0	1	1	$U_2\bigcup \{0.4\}$	0.4	0.6	0	0	0	$V_2\bigcup \{0\}$
4	0.6	0.7	p	1	1	2	$U_3\bigcup \{0.6\}$	0.5	0.7	0	1	1	$V_3\bigcup \{0.5\}$
5	0.1	0.3	p	2	1	3	$U_4\bigcup \{0.1\}$	0.3	0.5	0	2	2	$V_4\bigcup \{0.3\}$
6	0.6	0.5	−	3	1	4	$U_5\bigcup \{0\}$	0.8	0.1	0	2	2	$V_5\bigcup \{0\}$
7	0.4	0.1	−	3	1	4	$U_6\bigcup \{0\}$	0.2	0.3	0	2	2	$V_6\bigcup \{0\}$
8	0.4	0.6	q	3	1	4	$U_7\bigcup \{0.4\}$	0.4	0.6	0	2	2	$V_7\bigcup \{0\}$

The constrained optimization problem, Multiple Advertisers‐Quality Contract‐Performance Contract ( MA‐QC‐PC ) faced by the aggregator can then be formally expressed as follows:

MA‐QC‐PC

subject to:

2

3

4

5

State transition functions for quality advertisers

6

7

8

9

10

11

12

13

14

Since each quality advertiser requires a minimum number of impressions to be purchased and displayed on its behalf, the set of impression constraints for every quality advertisers in

${\mathrm{\Theta }}_Q$

is represented by Constraint (2). ⁵ Similarly, Constraint (3) represents the set of budget constraints for the performance advertisers in

${\mathrm{\Theta }}_P$

stating that the total revenue that the aggregator will extract from each performance advertiser will not exceed that advertiser's budget. Next, (4) is an equation defining the realized CTR for each quality advertiser. Each of the constraints (one for each opportunity) in (5) ensures that the sum of the probabilities for allocating the opportunity to advertisers is one. Equations (6) to (9) represent the transition functions for the quality advertisers. Equation (6) tracks the expected number of successful impressions so far from the quality advertisers. Equation (7) initiates the count for impressions won. Equation (8) tracks the expected number of clicks for the performance advertisers. Equation (9) initiates the expected count for clicks. Similarly, (10) to (13) represent the transition functions for the performance advertisers.

In the above formulation, we do not enforce a budget constraint for the quality advertiser. Since quality advertisers are interested in spreading brand awareness, having more consumers see (and potentially click on) their ads is not harmful. This is unlike the case of performance advertisers who require a budget constraint because generating too many clicks could exceed their capacity to serve demand. In practice since the impressions won by the aggregator are distributed across many advertisers, a particular quality advertiser will not receive too many impressions in excess of her desired number. Next, we present the solution to the aggregator's problem MA‐QC‐PC in the following section.

MODEL SETUP AND SOLUTION

Two heterogeneous advertisers

We first solve this problem with just two heterogeneous advertisers. The formulation (QC‐PC) of maximizing the aggregator's profit with one or more quality advertisers and one or more performance advertisers is provided in Section E of the Supporting Information. The decision variable

${\mathcal{Y}}_i^{\pi }(·)={\mathcal{Y}}_i^{\pi }(j_i^{\pi }(U_{i-1}),h_i^{\pi }(V_{i-1}),c_i)$

is the probability of allocating impression i to the quality advertiser. The variable

${\eta }_q$

represents the realized CTR achieved by the quality advertiser. The original problem reduces to a form described in Section C of the Supporting Information. We rewrite that model and present the final formulation by executing the following steps:

• (i)

  substitute the revenue function for

  $m({\eta }_q)$
  in the objective function,

• (ii)

  substitute

  $b(x_i^{\pi }(·))$
  with
  $\alpha x_i^{\pi }(·)$
  ,

• (iii)

  substitute the expression for

  ${\eta }_q$
  as defined above,

• (iv)

  substitute

  $\mathbb{E}[{\mathbb{I}}_{x_i}(·){\mathbb{I}}_{c_i}(·)]$
  with
  $\mathbb{E}[{\mathbb{I}}_{x_i}(·)]\mathbb{E}[{\mathbb{I}}_{c_i}(·)]$
  —this substitution is valid because a click event is independent of the impression for any opportunity i,

• (v)

  replace

  $\mathbb{E}[{\mathbb{I}}_{x_i}(·)]$
  with
  $\mathbb{E}[x_i^{\pi }(·)]$
  —this substitution is valid since
  $\mathbb{E}[{\mathbb{I}}_{x_i}(·)]=\mathbb{E}[\mathbb{I}({\stackrel{\sim }{u}}_i\le x_i^{\pi }(·))]$
  and the random variable
  ${\stackrel{\sim }{u}}_i$
  follows a uniform distribution, and

• (vi)

  replace

  $\mathbb{E}[{\mathbb{I}}_{c_i}(·)]$
  with
  $c_i$
  .

15

We solve this constrained optimization problem by applying Lagragian relaxation and obtain the optimal target win probabilities for both advertiser types. A noteworthy aspect of the solution is that the optimal target probabilities (

$x_{q,i}^{*}$

and

$x_{p,i}^{\ast }$

) are independent of the state (number of impressions won and number of clicks generated so far) of the system and depend only on the click probability of the arrival i. This has important implications for implementation since it ensures that the aggregator does not need to keep track of the state of a campaign. This is particularly beneficial when simultaneously managing multiple advertisers (addressed in the next subsection). A similar solution approach and solution characteristics are sought in other related papers as well. For example, Balseiro and Candogan (2017) study the optimal contract offered by an intermediary in a setting where advertisers' budgets and targeting criteria are private. They characterize the problem using the Lagrangian dual of the original problem (which is dynamic to begin with) and derive an optimal stationary bidding policy.

Solution structure

Before analyzing the solution, we briefly discuss the solution structure under our optimal policy

${\pi }^{\ast }$

. For each arriving impression, the aggregator is faced with two decisions: How much to bid and how to allocate each impression? Our solution provides an optimal target probability and therefore an optimal bid amount, conditioned on which type of advertiser will be awarded the opportunity. We, therefore, first answer the latter question. Note that we do not consider the case where

$M>K$

since this would be an infeasible contract; for the rest of the analysis, we assume

$M\le K$

. ⁶

We begin by demonstrating that the optimal allocation decision is a threshold policy; the proof is provided in Supporting Information in the discussion following (A14). Our solution provides an optimum threshold value

${\overline{c}\phantom{|}}^{\ast }$

against which the click probability

$c_i$

of each arriving impression is compared. If

$c_i$

is below the threshold, then that impression is allocated to the quality advertiser, otherwise the impression is allocated to the performance advertiser. The threshold therefore serves to split the click probability distribution into two separate regions, each with its own mean and variance. This division has interesting as well as important managerial implications resulting in two opportunity sets that are distinctly different from each other with respect to click probability. The click probability of every opportunity in the first segment is lower than the click probability of every opportunity in the second. Further, the optimal policy assigns all opportunities with lower click probabilities to the quality advertiser and those with higher click probabilities to the performance advertiser. The rationale for such an allocation can be explained as follows. The revenue from a quality advertiser is guaranteed even for a click probability of zero (γ₀ is guaranteed so long as an impression is delivered), whereas revenue from a performance advertiser is realized only when a click occurs. Additionally, although higher click probabilities and therefore higher CTRs increase the revenue per impression, the expected revenue from an impression with a higher click probability is significantly larger from the performance contract than the expected revenue for the same impression from the quality contract. Asdemir et al. (2012) find an analogous result from the publisher's perspective. They show that the publisher prefers a CPC contract over CPM for opportunities with higher click probabilities.

Let

$F(\overline{c})$

denote the cumulative distribution function for the click probability. Then, given K opportunities,

$K_q=KF({\overline{c}\phantom{|}}^{\ast })$

represents the number of opportunities assigned to the quality advertiser, while

$K_p=K(1-F({\overline{c}\phantom{|}}^{\ast }))$

is the number of opportunities assigned to the performance advertiser. The mean and variance of the click probability distribution in the region where

$c_i<{\overline{c}\phantom{|}}^{\ast }$

are denoted by

${\mu }_q$

and

${\sigma }_q$

, respectively, while

${\mu }_p$

and

${\sigma }_p$

are the corresponding values when

$c_i\ge {\overline{c}\phantom{|}}^{\ast }$

. Here,

${\overline{c}\phantom{|}}^{\ast }$

is

$\overline{c}$

computed on optimal values of the decision variables.

Suppose

$\widehat{M}$

is the number of unconstrained optimal impressions, which is a lower bound to the number of impressions that the aggregator will purchase on behalf of the single quality advertiser. Similarly

$\widehat{A}$

is the unconstrained optimal budget, which is an upper bound to the budget that the aggregator will extract from the single performance advertiser. The aggregator's complete problem includes a required minimum number of impressions (M) that must be purchased for the quality advertiser, as well as a maximum available budget (A) for the performance advertiser. Therefore, in the optimal solution to this constrained problem, the aggregator always allocates the larger of M and

$\widehat{M}$

to the quality advertiser. If

$M\ge \widehat{M}$

then the impression constraint becomes tight, and remains slack otherwise. Similarly, the aggregator will use the smaller of A and

$\widehat{A}$

for the performance advertiser, resulting in a tight constraint if all of A is used, and a slack constraint otherwise.

In effect, the Problem QC‐PC is a combination of two problems: one of the quality advertiser alone and the other of the performance advertiser alone. The resultant solution consists of four scenarios with both (budget and impression) constraints being tight, both constraints being slack, or just one being tight. The fulfillment or violation of each of the following two conditions ⁷ determines which specific scenario is applicable for the current pair of advertisers:

TightImpConstraint

TightBudgetConstraint

Given two contracts with two heterogeneous advertisers, it is first necessary to identify the applicable scenario by evaluating the two conditions listed above. However, the two conditions are dependent on the optimal threshold

${\overline{c}\phantom{|}}^{\ast }$

, which is different for each one of the four scenarios. All four

${\overline{c}\phantom{|}}^{\ast }$

values are therefore computed first. The two conditions are then evaluated using each of the four

${\overline{c}\phantom{|}}^{\ast }$

values. Since the conditions are evaluated using different

${\overline{c}\phantom{|}}^{\ast }$

values for each scenario, the scenarios may not be mutually exclusive. If multiple scenarios prove to be relevant, then the one with the highest profit will be applicable. On the other hand, if we are unable to find any

${\overline{c}\phantom{|}}^{\ast }$

such that the conditions for the corresponding scenario are satisfied, then the optimal solution is for the aggregator to offer the contract type with the higher profit. It should now be clear that the applicable scenario together with the corresponding

${\overline{c}\phantom{|}}^{\ast }$

is determined before the onset of a campaign. Once the optimal threshold is computed, the allocation decision is straight forward. If

$c_i<{\overline{c}\phantom{|}}^{\ast }$

, then

${\mathcal{Y}}_i^{\pi *}$

is set to 1 and the impression is allocated to the quality advertiser; otherwise,

${\mathcal{Y}}_i^{\pi *}$

is set to 0 and the impression is allocated to the performance advertiser.

We are now in a position to answer the first of the two questions posed at the beginning of this subsection, that is, how much to bid. We first determine the optimal target probability based on which advertiser will be allocated the impression. The conditions (16) and (17) presented below ensure that the target probability remains less than 1, which is the case in practice as well. We therefore assume that they always hold:

16

17

To summarize, the following decisions are made at the start of a campaign with two heterogeneous advertisers: the applicable scenario and the optimal threshold

${\overline{c}}^{\ast }$

value. The campaign then begins and for each arriving opportunity: (1) The click probability is compared with

${\overline{c}}^{\ast }$

and the allocation decision is made and (2) the appropriate optimal target probability is then computed. The overall decision process is depicted in Figure 3 and the four scenarios are summarized in Table 3.

TABLE 3

Four scenarios based on boundary conditions

	$A>\widehat{A}$	$A\le \widehat{A}$
$M\ge \widehat{M}$	Scenario I	Scenario III
$M<\widehat{M}$	Scenario IV	Scenario II
	$\widehat{M}=\frac{m({\mu }_q)\mathrm{KF}({\overline{c}\phantom{|}}^{*})}{2\alpha }$ , $\widehat{A}=\frac{K(1-F({\overline{c}\phantom{|}}^{*}))r^2({\sigma }_p+{\mu }_p^2)}{2\alpha }$

FIGURE 3

Allocating and bidding process

Solution

We now present the solution together with a detailed analysis. Our problem comprises of input parameters, which can be grouped into three different categories: (i) market‐determined parameters: K and α; (ii) aggregator‐determined revenue parameters: γ₀, γ₁, and r; and (iii) advertiser‐determined parameters: M and A. We have four outcome variables: the optimal target probability for the quality advertiser (

$x_{q,i}^{\ast }$

), the optimal target probability for the performance advertiser (

$x_{p,i}^{\ast }$

), the optimal threshold policy (

${\overline{c}}^{\ast }$

), and the expected total profit (

${\mathbb{Z}}^{\ast }$

). The

$x_{q,i}^{\ast }$

and

$x_{p,i}^{\ast }$

, in all cases, are interior solutions and depend on the click probability (

$c_i$

) of each arrival.

The optimal bidding and allocation policy under each of these four scenarios are given in Propositions 1 to 4. In the discussion that follows each proposition, we analyze the impact of changing these values for each of the input parameters on the outcome variables. We substitute

$c_i$

(a random variable) with

${\mu }_q$

or

${\mu }_p$

in order to study the general impact of changing input parameters on the target probabilities. This enables us to also understand the overall change in the bid cost for each advertiser type. We present a total of 112 results, of which 92 were derived analytically and the remaining 20 were determined numerically. Details of these numerical experiments are provided in Section A of the Supporting Information. Proposition 1

Scenario I: The following conditions hold:

TightImpConstraint

$$\begin{array}{c}\hfill A>\frac{K(1-F({\overline{c}\phantom{|}}^{\ast }))r^2({\sigma }_p+{\mu }_p^2)}{2\alpha }.\end{array}$$

SlackBudgetConstraint

The solution is given by

$x_{q,i}^{\ast }=\frac{M}{KF({\overline{c}\phantom{|}}^{\ast })}+\frac{{\gamma }_1(c_i-{\mu }_q)}{2\alpha }$

and

$x_{p,i}^{\ast }=\frac{r}{2\alpha }{c}_i$

where

${\overline{c}\phantom{|}}^{\ast }$

is a solution of

${\overline{c}\phantom{|}}^{\ast }=\frac{2M\alpha -{\gamma }_1{\mu }_{q}KF({\overline{c}\phantom{|}}^{\ast })}{KF({\overline{c}\phantom{|}}^{\ast })(r-{\gamma }_1)}$

. The profit is given by

${\mathbb{Z}}^{\ast }=m({\mu }_q)M+KF({\overline{c}\phantom{|}}^{\ast })\frac{{\gamma }_1^2{\sigma }_q}{4\alpha }-\frac{\alpha M^2}{KF({\overline{c}\phantom{|}}^{\ast })}+\frac{K(1-F({\overline{c}\phantom{|}}^{\ast }))({\mu }_p^2+{\sigma }_p)r^2}{4\alpha }$

.

TABLE 4

Summary of comparative statics results—Scenario I

	${\mathbf{M}}_{\mathbf{q}}$	${\mathbf{A}}_{\mathbf{p}}$	$\mathbf{K}$	$\mathit{\alpha }$	$\mathbf{r}$	${\mathbf{\gamma }}_{\mathbf{0}}$	${\mathbf{\gamma }}_{\mathbf{1}}$
${\overline{\mathbf{c}}\phantom{|}}^{\ast }$	↑	⟷	↓	↑	↓	⟷	↑
${\mathbf{Z}}^{*}$	↓	⟷	↑	↓	↑	↑	↑
${\mathbf{x}}_{\mathbf{q}}^{*}$	${\uparrow }^{\aleph }$	⟷	${\downarrow }^{\aleph }$	↓	↑	⟷	↓
${\mathbf{x}}_{\mathbf{p}}^{*}$	↑	⟷	↓	${\downarrow }^{\aleph }$	${\uparrow }^{\aleph }$	⟷	↑

^ℵ: numerically determined.

In Table 4 we provide the comparative statics results for Scenario I associated with an increase in the input parameters. An increase in the output variable is denoted by a ↑, a decrease is denoted by a ↓, and a ⟷ denotes no change in the outcome variable. The trends determined numerically are marked with ^ℵ.

We will first study the impact of changes to the advertiser‐determined parameters, starting with M. In order to meet contractual requirements the aggregator is already purchasing more than the number of unconstrained optimal impressions

$\widehat{M}$

. In this case, an increase in M would further widen the gap between the number of impressions that must be purchased and the optimal number that the aggregator should purchase (

$\widehat{M}$

). The aggregator can acquire more impressions by either increasing the target probability for the quality advertiser (taking into consideration the convex nature of the cost curve with respect to the win probability), or allocating more opportunities for the performance advertiser (recognizing that this will reduce the revenue from the performance advertiser). The optimal solution balances these trade‐offs by increasing the

${\overline{c}\phantom{|}}^{\ast }$

as well as the target probabilities (on average) for both advertiser types. Clearly, the expected total profit from the quality advertiser will decrease since the increase in M forces the aggregator to move further away from

$\widehat{M}$

. The expected total profit from the performance advertiser also decreases since fewer opportunities will now be allocated to this advertiser. The budget, A, has no impact on any of the outcome variables since the aggregator is already utilizing less than the current budget.

The number of opportunities (K) is the first of the market‐determined parameters that we next discuss. The extra opportunities from an increase in K are distributed between the two advertisers by increasing

${\overline{c}\phantom{|}}^{\ast }$

in such a way that both

$K_q=KF({\overline{c}\phantom{|}}^{\ast })$

and

$K_p=K(1-F({\overline{c}\phantom{|}}^{\ast }))$

increase. The second market parameter is α. Needless to say, since α is a cost parameter, any increase results in lowered profits. The question is, how best to minimize the loss? We find that the target probabilities for both advertisers (on average) decrease. However, the optimal threshold increases, thereby increasing the available opportunities to the quality advertiser. In this way, the impression constraint can be honored even with the lower target probabilities.

Finally, we discuss the impact of aggregator‐determined revenue parameters on our outcome variables. Obviously, the profit increases with an increase in any of these parameters (r, γ₀, and γ₁). When r increases, the target win probabilities increase; however,

${\overline{c}\phantom{|}}^{\ast }$

decreases. When γ₀ increases, the

${\overline{c}\phantom{|}}^{\ast }$

and target win probabilities remain the same. When γ₁ increases,

${\overline{c}\phantom{|}}^{\ast }$

always increases,

$x_q^{\ast }$

decreases, and

$x_p^{\ast }$

increases. The mechanisms and intuition are similar to those discussed for other parameters. Proposition 2

Scenario II: the following conditions hold:

SlackImpConstraint

TightBudgetConstraint

The solution is given by

$x_{q,i}^{\ast }=\frac{m(c_i)}{2\alpha }$

and

$x_{p,i}^{\ast }=\frac{c_{i}A}{K(1-F({\overline{c}\phantom{|}}^{\ast }))r({\sigma }_p+{\mu }_p^2)}$

where

${\overline{c}\phantom{|}}^{\ast }$

is the solution of the following equation:

${\overline{c}\phantom{|}}^{\ast }=\frac{{\gamma }_{0}K(1-F({\overline{c}\phantom{|}}^{\ast })}{\frac{2\alpha A}{r({\sigma }_p+{\mu }_p^2)}-{\gamma }_{1}K(1-F({\overline{c}\phantom{|}}^{\ast }))}$

. The profit is given by

${\mathbb{Z}}^{\ast }=\frac{KF({\overline{c}\phantom{|}}^{\ast })}{4\alpha }(2{\gamma }_0{\gamma }_1{\mu }_q+{\gamma }_0^2+{\gamma }_1^2({\mu }_q^2+{\sigma }_q))+A(1-\frac{\alpha A}{K(1-F({\overline{c}\phantom{|}}^{\ast }))r^2({\mu }_p^2+{\sigma }_p)})$

.

The comparative statics results for Scenario II are provided in Table 5. Since the impression constraint is slack, M does not affect any of the decision variables and outcomes. When A increases, the aggregator allocates a portion of the surplus impressions from the quality advertiser to the performance advertise by decreasing

${\overline{c}\phantom{|}}^{\ast }$

. Further, it bids more aggressively in order to use the increased budget by increasing

$x_p^{\ast }$

on average. Clearly, profit increases with an increase in A in this scenario.

TABLE 5

Summary of comparative statics results—Scenario II

	${\mathbf{M}}_{\mathbf{q}}$	${\mathbf{A}}_{\mathbf{p}}$	$\mathbf{K}$	$\mathit{\alpha }$	$\mathbf{r}$	${\mathbf{\gamma }}_{\mathbf{0}}$	${\mathbf{\gamma }}_{\mathbf{1}}$
${\overline{\mathbf{c}}\phantom{|}}^{\ast }$	⟷	↓	↑	↓	↑	↑	↑
${\mathbf{Z}}^{*}$	⟷	↑	↑	↓	↑	↑	↑
${\mathbf{x}}_{\mathbf{q}}^{*}$	⟷	↓	↑	↓	↑	↑	↑
${\mathbf{x}}_{\mathbf{p}}^{*}$	⟷	${\uparrow }^{\aleph }$	${\uparrow }^{\aleph }$	${\downarrow }^{\aleph }$	${\downarrow }^{\aleph }$	${\uparrow }^{\aleph }$	${\uparrow }^{\aleph }$

^ℵ: numerically determined.

When K increases,

${\overline{c}\phantom{|}}^{\ast }$

increases, thereby allocating more opportunities to the quality advertiser. Contrary to expectations, the target probability also increases on average for the quality advertiser. Thus, in this scenario an increase in K causes the aggregator to bid higher amounts and more often for the quality advertiser, resulting in a higher realized CTR and therefore a higher revenue per impression. While this means an overall higher profit, the profit from the performance advertiser decreases since the higher optimal threshold reduces the number of opportunities allocated to the performance advertiser. The aggregator must therefore now also bid higher for the performance advertiser in order to buy all the impressions required to use up the entire budget.

When α increases, the loss is minimized by decreasing the threshold

${\overline{c}\phantom{|}}^{\ast }$

, and more opportunities are allocated to the performance advertiser in order to allow utilizing the entire budget amount. The target win probabilities accordingly decrease in order to reduce the bidding cost. When any of these aggregator‐determined parameters increase, then

${\overline{c}\phantom{|}}^{\ast }$

and

$x_q^{*}$

increase. The average

$x_p^{*}$

decreases when r increases; it also increases when either of γ₀ and γ₁ increases. Proposition 3

Scenario III: the following conditions hold:

TightImpConstraint

TightBudgetConstraint

The solution is given by

$x_{q,i}^{\ast }=\frac{M}{KF({\overline{c}\phantom{|}}^{\ast })}+\frac{{\gamma }_1(c_i-{\mu }_q)}{2\alpha }$

and

$x_{p,i}^{\ast }=\frac{c_{i}A}{K(1-F({\overline{c}\phantom{|}}^{\ast }))r({\sigma }_p+{\mu }_p^2)}$

where

${\overline{c}\phantom{|}}^{\ast }$

is a solution of the following equation:

${\overline{c}\phantom{|}}^{\ast }=\frac{(1-F({\overline{c}\phantom{|}}^{\ast }))}{F({\overline{c}\phantom{|}}^{\ast })}(\frac{(2M\alpha -{\gamma }_1{\mu }_{q}KF({\overline{c}\phantom{|}}^{\ast }))}{\frac{2\alpha A}{r({\sigma }_p+{\mu }_p^2)}-{\gamma }_{1}K(1-F({\overline{c}\phantom{|}}^{\ast }))})$

. The profit is given by

${\mathbb{Z}}^{\ast }=m({\mu }_q)M+KF({\overline{c}\phantom{|}}^{\ast })\frac{{\gamma }_1^2{\sigma }_q}{4\alpha }-\frac{\alpha M^2}{KF({\overline{c}\phantom{|}}^{\ast })}+A(1-\frac{\alpha A}{K(1-F({\overline{c}\phantom{|}}^{\ast }))r^2({\mu }_p^2+{\sigma }_p))})$

.

TABLE 6

Summary of comparative statics results—Scenario III

	${\mathbf{M}}_{\mathbf{q}}$	${\mathbf{A}}_{\mathbf{p}}$	$\mathbf{K}$	$\mathit{\alpha }$	$\mathbf{r}$	${\mathbf{\gamma }}_{\mathbf{0}}$	${\mathbf{\gamma }}_{\mathbf{1}}$
${\overline{\mathbf{c}}\phantom{|}}^{\ast }$	↑	↓	↑	${\downarrow }^{\aleph }$	↑	⟷	↑
${\mathbf{Z}}^{*}$	↓	↑	↑	↓	↑	↑	↑
${\mathbf{x}}_{\mathbf{q}}^{*}$	${\uparrow }^{\aleph }$	↑	↓	${\uparrow }^{\aleph }$	↓	⟷	↓
${\mathbf{x}}_{\mathbf{p}}^{*}$	${\uparrow }^{\aleph }$	${\uparrow }^{\aleph }$	${\downarrow }^{\aleph }$	${\downarrow }^{\aleph }$	${\downarrow }^{\aleph }$	⟷	${\downarrow }^{\aleph }$

^ℵ: numerically determined.

The comparative statics results are provided in Table 6. In this scenario, both the impression and budget constraints are tight. When M (A, respectively) increases trends are similar to those explained in Proposition 1 (2, respectively). When K increases, although

${\overline{c}\phantom{|}}^{\ast }$

increases, the number of opportunities for both the advertisers increase. Thus, the win probabilities decrease. The situation of an increased α is tricky. We can either increase

${\overline{c}\phantom{|}}^{\ast }$

resulting in a higher

$x_p^{\ast }$

and lower

$x_q^{\ast }$

, or decrease

${\overline{c}\phantom{|}}^{\ast }$

causing a decrease in

$x_p^{\ast }$

and increase in

$x_q^{\ast }$

. This conflict is reflected in the result we find: The

${\overline{c}\phantom{|}}^{\ast }$

increases when

${\gamma }_0+{\gamma }_1{\mu }_q>r{\overline{c}\phantom{|}}^{\ast }$

. For the parameter values we use in our numerics, the optimal solution is to decrease the threshold, resulting in a higher target probability for the quality advertiser and a lower target probability for the performance advertiser. The rationale for such a shift in the threshold can be explained as follows. Recall that our expected bid cost for an impression is convex with respect to the target win probability, which in turn is directly proportional to the click probability of each opportunity. Every opportunity earmarked for the quality advertiser has a click probability lower than that of any opportunity intended for the performance advertiser. Therefore, increasing the target win probability for the quality advertiser will have a lower adverse impact on the overall expected profit than would be the case if the

${\overline{c}\phantom{|}}^{\ast }$

and

$x_q^{\ast }$

were increased and

$x_p^{\ast }$

was decreased. Proposition 4

Scenario IV: the following conditions hold:

$$\begin{array}{c}\hfill M<\frac{m({\mu }_q)KF({\overline{c}\phantom{|}}^{\ast })}{2\alpha },\end{array}$$

SlackImpConstraint

$$\begin{array}{c}\hfill A>\frac{K(1-F({\overline{c}\phantom{|}}^{\ast }))r^2({\sigma }_p+{\mu }_p^2)}{2\alpha }.\end{array}$$

SlackBudgetConstraint

The solution is given by

$x_{q,i}^{\ast }=\frac{m(c_i)}{2\alpha }$

and

$x_{p,i}^{\ast }=\frac{r}{2\alpha }{c}_i$

, where

${\overline{c}\phantom{|}}^{\ast }=\frac{{\gamma }_0}{r-{\gamma }_1}$

. The profit is given by

${\mathbb{Z}}^{\ast }=\frac{KF({\overline{c}\phantom{|}}^{\ast })}{4\alpha }(2{\gamma }_0{\gamma }_1{\mu }_q+{\gamma }_0^2+{\gamma }_1^2({\mu }_q^2+{\sigma }_q))+\frac{K(1-F({\overline{c}\phantom{|}}^{\ast }))({\mu }_p^2+{\sigma }_p)r^2}{4\alpha }$

.

TABLE 7

Summary of comparative statics results—Scenario IV

	${\mathbf{M}}_{\mathbf{q}}$	${\mathbf{A}}_{\mathbf{p}}$	$\mathbf{K}$	$\mathit{\alpha }$	$\mathbf{r}$	${\mathbf{\gamma }}_{\mathbf{0}}$	${\mathbf{\gamma }}_{\mathbf{1}}$
${\overline{\mathbf{c}}\phantom{|}}^{\ast }$	⟷	⟷	⟷	⟷	↓	↑	↑
${\mathbf{Z}}^{*}$	⟷	⟷	↑	↓	↑	↑	↑
${\mathbf{x}}_{\mathbf{q}}^{*}$	⟷	⟷	⟷	↓	↓	↑	↑
${\mathbf{x}}_{\mathbf{p}}^{*}$	⟷	⟷	⟷	↓	${\downarrow }^{\aleph }$	↑	↑

^ℵ: numerically determined.

The comparative statics results are provided in Table 7. In this scenario the parameter M and A clearly do not affect any of the outcome variables of interest. An increase in K increases profit but does not affect either the win probabilities or the threshold. The effects of α, γ₀, γ₁, and r are obvious for the reasons explained earlier.

Multiple heterogeneous advertisers

We are finally ready to provide a solution for the aggregator's profit maximization problem (Problem MA‐QC‐PC) when faced with

$|{\mathrm{\Theta }}_Q|$

quality advertisers and

$|{\mathrm{\Theta }}_P|$

performance advertisers. Note that this problem is a generalization of Problem QC‐PC discussed in the previous section. Analogous to the cases of a single advertiser type, as stated in Corollary 1, Problem MA‐QC‐PC can be written as Problem 18 with a single (virtual) quality advertiser whose impression requirement equals the sum of the requirements for all quality advertisers, as well as a single (virtual) performance advertiser whose budget equals the sum of the budgets for all performance advertisers. Corollary 1

The solution of a campaign with N advertisers of whom

$|{\mathrm{\Theta }}_Q|$

have quality contracts and the remaining

$|{\mathrm{\Theta }}_P|$

have performance contracts can be obtained by solving the problem (

$VA-QC-PC$

), allocating the bids won among the quality advertisers with probability

$y_{q,i}^{\pi }(·)=y_q^{\pi }=\frac{M_q}{{\sum }_{q\in {\mathrm{\Theta }}_Q}{M}_q}{\mathcal{Y}}_i^{\pi }(·)$

and the performance advertisers with probability

$y_{p,i}^{\pi }(·)=y_p^{\pi }=\frac{A_p}{{\sum }_{p\in {\mathrm{\Theta }}_P}{A}_p}(1-{\mathcal{Y}}_i^{\pi }(·))$

with

${\mathcal{Y}}_i^{\pi }(·)$

denoting the probability of assigning impression i to the advertisers with quality contracts.

18

The intuition of the above result lies in the structure of the problem. Specifically, all quality advertisers can be grouped into one virtual quality advertiser, and similarly all performance advertisers can be aggregated into one virtual performance advertiser. From the aggregator's perspective, all quality advertisers differ from each other only in their impression requirements. In other words, the demand for impressions is additive across all quality advertisers. They can therefore be combined into a single (virtual) advertiser for whom the aggregator develops an optimal bidding strategy. Once an impression is won, it can then be randomly assigned to one of the

$|{\mathrm{\Theta }}_Q|$

advertisers in a proportionate manner based on their impression requirements. Similar logic holds for the performance advertisers as well. This concludes the solution for the aggregator's profit maximization problem MA‐QC‐PC.

In Figures 4, 5, and 6, we show the trends of the optimal decision variables

$x_p^{*}$

,

$x_q^{*}$

, and the optimal profit

$Z^{*}$

when A, M, or K change, respectively. Note that when a parameter increases (changes)—for example, when K increases, the applicable scenario may also change since the boundary conditions of the current scenario may not remain valid while those of another scenario may become applicable. Such a shift is shown in Figure 6. It is therefore worthwhile to analyze such cases in order to generate important managerial insights. We perform this analysis in the next section.

FIGURE 4

Plots of optimal target win probabilities and profits against budget (A). (a)

$x_p^{*}$

vs. A, (b)

$x_q^{*}$

vs. A, (c) Profit (

$Z^{*}$

) vs. A

FIGURE 5

Plots of optimal target win probabilities and profits against impression (M). (a)

$x_p^{*}$

vs. M, (b)

$x_q^{*}$

vs. M, (c) Profit (

$Z^{*}$

) vs. M

FIGURE 6

Plots of optimal target win probabilities and profits against opportunities (K). (a)

$x_p^{*}$

vs. K, (b)

$x_q^{*}$

vs. K, (c) Profit (

$Z^{*}$

) vs. K

Trajectory across scenarios

We now move to the situations where changes in parameters may change the scenario itself. Note that while each solution is optimal given the boundary conditions, the change in scenario happens when an exogenous parameter value changes, which drives a shift from one optimal decision (based on the previously applicable boundary conditions) to another (corresponding to the new applicable boundary conditions). In Figure 7 we show the different applicable scenarios based on the various values of M and A. These scenarios are bounded by the blue lines and the regions are labeled accordingly. The scenario changes are represented by the numbered paths shown with red lines. Figures 8 and 9 individually show the changes in scenarios with increases in M and A, respectively. In these figures the red node denotes the starting scenario for a path, the green node denotes the ending scenario, and the violet node(s) denote the intermediate scenarios. An important characteristic of all paths is that the movement from one scenario to the next is independent of any scenarios that were previously applicable. In other words, the shift from one scenario to the next is memory‐less.

FIGURE 7

Possible scenarios for different values of A and M when

$K=117,768$

,

${\gamma }_0=0.01$

,

$\gamma =0.2$

,

$\alpha =0.073939$

, and

$r=3.0$

FIGURE 8

Trajectory for changes in M

FIGURE 9

Trajectory for changes in A

Changes in M: As shown in Figure 8, with an increase in M the solution follows one of two trajectories. The trajectory {II, III, I} is shown in Path # 1 of Figure 7. Initially in Scenario II the aggregator buys a surplus (

$\widehat{M}-M$

) of impressions. As M increases this surplus reduces to the point where Scenario III becomes relevant. Further increase in M forces the aggregator to meet impressions requirements by leaving some of the performance budget unused, making Scenario I applicable.

Changes in A: As shown in Figure 9, with an increase in A the solution follows one of two trajectories, which both begin at Scenario

$II$

. The trajectories {II, IV} and {II, III, I} are shown in Paths # 3 and # 4 of Figure 7, respectively. The trade‐off between the profitabilities of the impression revenue parameters (γ₀ and γ₁) and the click revenue parameter (r) determines which of these two trajectories will prevail. The trajectory {II, IV} holds when r dominates, and the other trajectory holds when γ₀ and γ₁ dominate. The second trajectory, {IV, I}, shown as Path # 2 of Figure 7 begins at Scenario IV, that is, the aggregator buys

$\widehat{M}>M$

and utilizes

$\widehat{A}<A$

. The initial surplus of impressions purchased for the quality advertiser are used to meet the increased contractual requirements. After a point exactly the required number of impressions are purchased resulting in Scenario I.

Changes in K: We now discuss the scenario paths that apply when K changes (Figure 10). There are two possibilities: {I, IV, II} and {I, III, II}. In each case for low values of K the relevant scenario is always I since the foremost responsibility for the aggregator is to meet the requirements of the quality advertiser. From here the solution can shift to either Scenario IV or Scenario III based on the relative profitability of these two advertisers. In both cases with a further increase in K the solution will shift to II and stay there. This scenario is an interesting one and is related to the results from Liu and Mookerjee (2018). With an increase in opportunities (K), the aggregator is capable of sending more traffic to the performance advertisers. Now the budget constraint avoids the situation of too many allocations to the performance advertisers who care about the ability to serve arriving customers satisfactorily (controlling traffic through the budget).

FIGURE 10

Trajectory for changes in K

MANAGERIAL IMPLICATIONS

We are now in a position to discuss the managerial implications of the solution to the aggregator's profit maximization problem and provide recommendations.

Since the aggregator has the flexibility of allocating zero impressions to the performance advertiser, the aggregator should always offer both contracts together instead of offering the quality contract alone. This holds even when the quality contract appears to be the more attractive option. On the other hand, if the performance contract is found to be better one, then the aggregator should offer the performance contact alone. Our recommendation derives from the trade‐off between marginal profits from these two contract types.

The aggregator should bid a nonzero amount for every arriving opportunity allocated to the quality advertiser. This is true even if the click probability is zero. Two reasons drive this result. First, the aggregator must meet the impression constraint. Second, the aggregator will be paid for displaying an impression even if its click probability is zero. However, for any arriving opportunity allocated to the performance advertiser, the optimal bid amount will be nonzero only when the click probability is strictly positive.

Our final recommendation is related to the “best” scenario that the aggregator would like to operate within. While our solution is optimal for a given set of parameters and the applicable scenario, there could be other (albeit subjective) business factors driving the aggregator to prefer a particular scenario. To the extent possible the aggregator would like to stay in Scenario III where all of the performance advertisers' budget is utilized and the quality advertisers are given exactly the number of impressions they contracted for. Although the quality advertisers' contract allows for the aggregator providing more impressions than required by the advertisers, the advertisers may not want large deviations from the required quantity. Likewise, the performance advertisers may feel short‐changed should their budget be underutilized. Therefore, if the active contracts result in one of the other scenarios being applicable, then the aggregator may desire to take actions that will ensure the pertinence of Scenario III. In such cases, the aggregator should find a way to either utilize the unused budget or allocate the impressions purchased in excess to new advertisers. The aggregator can employ one or more of three actions depending on the current scenario:

• (i)

if the aggregator is in Scenario I: the optimal action is to increase the number of available opportunities (K) in order to utilize the excess budget;

(ii)

if the aggregator is in Scenario II: the aggregator should either sign up more quality advertisers or more performance advertisers depending on which of the two advertiser types is more profitable;

(iii)

if the aggregator is in Scenario IV: the aggregator will need to both acquire more opportunities and also sign up with either more performance or more quality advertisers based on their respective profitabilities.

Changing the revenue or cost parameters is not a viable option in a competitive environment, such as the one in our context.

Acquiring more performance or quality advertisers will serve to eliminate (or reduce to the extent possible) the gap between the optimal unconstrained impressions or budget, and the corresponding constraint boundary. The decision to increase the number of available opportunities requires more thought and care. As described in the introduction in Section 1, the aggregator can simultaneously contract with multiple RTB exchanges in order to gain access to opportunities. In the event the aggregator desires to increase the K available to her, she will need to sign up with additional RTBs. Contracts with each RTB comes at an additional cost to the aggregator. Furthermore, each new contract means a step increase in the opportunities that will now become available to the aggregator. Therefore, before signing up with a new RTB, the aggregator should first weigh the additional cost of such a contract versus the potential benefits. The benefits are twofold. First, it will move the aggregator toward the direction of Scenario III as desired, and second, increasing K will increase profit.

Finally, we highlight important operational aspects of our approach that are of managerial relevance. The RTB environment requires that aggregators respond with bids within 50 to 100 ms of an opportunity becoming available. In order to compute the bid amount an aggregator needs to: (1) predict the click probability, (2) evaluate the allocation of the bid, and (3) compute both the target bid probability and corresponding bid amount. Since the predictive model is predetermined and we have closed‐form expressions for each of allocation decisions, the win probability and the bid amount, the aggregator can respond with the optimal bid amount within the time limit imposed by the RTB environment.

SUMMARY AND CONCLUSIONS

In this paper, we develop and solve a generalized profit maximization problem that jointly optimizes a bidding policy and an allocation policy. This problem is solved from the perspective of a demand aggregator who simultaneously contracts with multiple advertisers where each advertiser opts for one of two contract types. By using Lagrangian relaxation techniques we provide a solution to this complex problem.

Despite dynamic features in the problem setting (e.g., the click probability of an arriving opportunity, as well as the uncertainty associated with the outcome of a bid), we show that the optimal bidding policy is independent of the memory carried in the system (the sequence of previous system states and decisions). This property of the optimal bidding policy provides significant practical benefits. In practice, demand aggregators use a distributed algorithmic bidding architecture; that is, bidding is performed by many slave bidders that report to a master bidder. Since the optimal bidding policy does not depend on the sequence of past events (states and decisions), the slave bidders do not need to coordinate with each other. This makes the communication between the master bidder and a slave bidder relatively simple. The memory‐free property of the optimal bidding policy also holds for the optimal allocation policy. That is, the allocation decision (assigning an advertiser to an arriving opportunity) depends only on the current state (the click probability of the arriving opportunity). Similar to the bidding policy, the memory‐free property of the allocation policy allows the slave bidders to operate independent of one another.

We conclude with highlighting the limitations of this research and thoughts regarding future research possibilities. The pricing model we have employed assumes that all impressions displayed or clicks achieved are equally valuable to the advertisers. However, if there are significant differences in the valuation by advertisers of different impressions or clicks, then these results may not be optimal. For instance, location plays an important role in decision making within the context of mobile advertising. An advertiser finds more value from a click by a user closer to its establishment than a click obtained at a farther distance. One extension would be to introduce a pricing scheme where the revenue per click is a function of the distance from a given focal point. That is, clicks achieved within a smaller radius would be worth more than clicks outside this radius. Another limitation of our work is the exclusion of other types of contract being offered, for example, a CPA contract. An extension could be to include other contracts in addition to those already in the model. Finally, the aggregator is usually not alone in bidding for ad spaces. Not considering a game between bidding aggregators is also a limitation of this study. An interesting direction to pursue would be studying such a bidding game between competing aggregators.

Footnotes

ACKNOWLEDGMENT

The authors are grateful to Venkat Kolluri (CEO) and Chris Caswell (optimization specialist) of Cidewalk for providing us with the problem framework and for their availability for multiple discussions and guidance.

1

https://ozcart.com/seo/pay‐per‐click‐or‐cost‐per‐impression‐which‐is‐best

2

Cidewalk is a demand aggregator we conferred with when defining and developing this analysis.

3

A list of the main data variables used in the model estimation is provided in the Supporting Information in Section B.

4

We also test other bin sizes and find the results to be consistent.

5

The delivery constraint is expressed in terms of an expectation. The realized number of impressions could therefore violate the minimum required number of impressions. Despite this we express our problem in terms of an expectation constraint for two reasons. First, it is easy to show that for a large number of opportunities (K) the probability of violating this constraint is negligible. Second, the probability of violating the constraint can be easily controlled by padding or increasing the value of the right side by a certain amount.

6

Analysis of cases when only one type of advertiser is present during a campaign is instructive and facilitates the solution of the more complex two‐advertiser type problem, which is the focus of our paper. Such analyses for each advertiser type are provided in Section E of the Supporting Information.

7

Note that since the means (

${\mu }_q$

and

${\mu }_p$

) and variances (

${\sigma }_q$

and

${\sigma }_p$

) are derived using the optimal click probability threshold (

${\overline{c}\phantom{|}}^{\ast }$

), these are also optimal values. However, for ease of readability, we omit using the asterisk (*) for these variables.

Supplementary Material

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