MODELING THE MARGINAL CUSTOMER ACQUISITION COST (MARGINAL CAC) UNDER THE AUCTION DYNAMICS OF DIGITAL ADVERTISING PLATFORMS

АННОТАЦИЯ

В статье рассмотрено моделирование предельной стоимости привлечения клиента marginal CAC в условиях аукционной динамики цифровых рекламных платформ. Показано, что средний CAC сглаживает эффект насыщения аудитории и изменения конкурентной среды, в то время как marginal CAC отражает экономику масштабирования и чувствителен к убывающей предельной отдаче. Предложена формализованная модель кривой насыщения и получено выражение экспоненциального роста, что задает основания для оптимизационного перераспределения бюджета по принципу выравнивания предельных стоимостей. Обоснована необходимость предварительной очистки потока данных и фильтрации неинтенциональных запросов на уровне DSP для предотвращения смещения оценок, а также применения частотных ограничений по не кликнутым показам для стабилизации эффективности. Представлена иллюстрация на открытых данных кампаний Facebook (принадлежит Meta, признана экстремистской и запрещенной в России) и AdWords за 2019 г., подтверждающая различия в «краевых» значениях и практическую применимость предложенного алгоритма бюджетного аллоцирования, с учетом атрибуции пути пользователя и робастной проверки на временных рядах.

ABSTRACT

The paper examines the modeling of marginal Customer Acquisition Cost (marginal CAC) under the auction dynamics of digital advertising platforms. It is demonstrated that average CAC smooths the effects of audience saturation and changes in the competitive environment, whereas marginal CAC reflects the economics of scaling and is sensitive to diminishing marginal returns. A formalized saturation curve model is proposed, and an exponential growth expression is derived, providing the foundation for optimization-based budget reallocation according to the principle of marginal cost equalization. The necessity of preliminary data stream cleaning and filtering of non-intentional bid requests at the DSP level is substantiated to prevent estimation bias, along with the application of frequency caps on non-clicked impressions to stabilize performance. An empirical illustration based on open data from Facebook (принадлежит Meta, признана экстремистской и запрещенной в России) and AdWords campaigns (2019) confirms differences in tail marginal values and demonstrates the practical applicability of the proposed budget allocation algorithm, accounting for user-path attribution and robust time-series validation.

Ключевые слова: предельная стоимость привлечения клиента, marginal CAC; аукционная динамика, бюджетное аллоцирование, ограничение частоты показов рекламы.

Keywords: marginal customer acquisition cost, marginal CAC; auction dynamics; budget allocation; ad frequency capping.

Introduction. In auction-based advertising platforms (auction dynamics defined as real-time changes in prices and win probabilities), Customer Acquisition Cost (CAC) is commonly interpreted as the average cost of acquisition over a given period. In practice, however, this metric is distorted by traffic and conversion noise, including click fraud (fraudulent clicks that increase advertising spend without generating business outcomes). For instance, studies on the detection and prevention of click fraud show that attackers create an “illusion” of user interest through automated and human-generated clicks, while advertisers receive no commensurate value, leading to inflated and unstable CAC estimates [1]. In this context, there is a practical need to model the marginal customer acquisition cost (marginal CAC). The relevance of this issue is determined by the fact that under conditions of scaling digital campaigns and increasing budgets, marginal CAC reflects the true economics of reach expansion, whereas average CAC smooths the effects of audience saturation and changes in the competitive environment. As bids and impression frequency increase, auction algorithms reallocate inventory, leading to higher winning prices and a decline in incremental returns. Ignoring the dynamics of marginal CAC results in errors in budget allocation, overspending, and a decrease in marketing return on investment. Therefore, the formalization of a marginal CAC model becomes a necessary tool for the analysis and optimization of advertising investments, which defines the objective and scope of this study.

The scientific contribution of this study is as follows:

1. A conceptual approach is proposed to distinguish between average CAC and marginal CAC as two economically non-equivalent metrics for evaluating the performance of digital advertising campaigns under budget scaling conditions.
2. A formalized model of channel saturation is developed, demonstrating that as budget increases, marginal CAC grows at a faster rate than average CAC, making marginal metrics more suitable for decision-making.
3. A principle of budget reallocation across channels is proposed based on the equalization of marginal returns and its equivalent interpretation through marginal CAC, along with its applied validation using open data from advertising campaigns.

Research methodology. The study is based on the formalization of marginal customer acquisition cost under the auction dynamics of digital advertising platforms and includes the mathematical specification of average and marginal CAC accounting for the saturation effect, input data quality control prior to model estimation, the formulation of budget reallocation as an optimization problem, consideration of auction dynamics in the RTB environment, and statistical validation of model parameters.

The following notations are used in this study:

 – index of the channel or advertising campaign, i = 1, ..., k.

 – budget of the iii-th channel over the considered period.

 – number of acquired customers as a function of the budget.

 – average customer acquisition cost.

 – marginal customer acquisition cost.

In the empirical section of the study, the advertising campaign budget is measured in U.S. dollars (USD), the number of customers is interpreted as the number of conversions, and the average CAC and marginal CAC metrics are expressed in USD per conversion. For consistency, this interpretation is used throughout the tables and calculations. However, depending on the national currency, each company may adopt its own units of measurement for these indicators.

Let us denote for channel/campaign  the budget  and the number of acquired customers . The average acquisition metric is defined as:

This metric aggregates costs over the period and smooths the structure of marginal effects. When scaling the budget, it does not reflect changes in marginal returns and therefore does not allow for an accurate assessment of the economics of channel expansion.

The marginal acquisition cost is defined as the derivative of cost with respect to the result:

The primary mechanism is saturation: as the budget increases, the growth in the number of customers slows down, indicating diminishing marginal returns. As a result,  increases faster than , leading to the formation of an exponentially widening gap between the average and marginal economics of the channel.

Marginal cost is understood as the cost of acquiring the next customer under a small increase in budget, i.e., an incremental rather than an average cost.

For approximating the saturation curve, a saturating exponential form is used:

where:

– channel capacity (the maximum attainable number of customers),

– saturation rate parameter.

The marginal return with respect to the budget is given by:

Accordingly,

Thus, the marginal customer acquisition cost increases exponentially as the budget scales, which provides the foundation for optimization-based budget reallocation. From the above expression, it follows that as the budget  increases, marginal CAC grows according to an exponential law. Consequently, as the channel approaches saturation, each additional conversion becomes significantly more expensive than the previous ones, whereas average CAC responds to this effect in a more smoothed manner.

Prior to calibrating the function  , it is necessary to stabilize the input auction data stream. In the RTB environment, a significant share of impressions and clicks may be associated with non-intentional or anomalous traffic. If such observations are included in the model, the parameter  is overestimated, and the marginal cost becomes distorted [6].

Therefore, a constraint is introduced in the estimation procedure:  is evaluated only on the cleaned data stream that has undergone filtering by audience, sources, blocklists, and behavioral features. Otherwise, marginal CAC reflects noise rather than the true cost of customer acquisition.

Let the total budget be given by:

The objective is to maximize the total number of customers:

subject to the constraints:

For smooth functions, the interior optimum is achieved when the marginal return on budget is equalized across all active channels:

where  denotes the common marginal return per unit of budget at the optimum.

An equivalent formulation in terms of marginal customer acquisition cost is given by:

since

In applied systems, additional constraints related to platforms, budget shares, and minimum presence levels are typically introduced. For example, a study on advertising strategy optimization for a small enterprise formalizes the optimization problem under a budget constraint as well as constraints on platform shares and allocation coefficients across platforms and formats [2]. Therefore, it is appropriate to impose the following constraints:

as well as constraints on the budget spending rate (pacing). The optimization problem is solved numerically using gradient-based methods or via the Karush–Kuhn–Tucker conditions.

In the RTB environment, the probability of winning depends on the bid, competitive activity, and the distribution of transaction prices. Consequently, the function  is endogenous with respect to both the budget and the bid.

The calibration of parameters  and  is performed with consideration of auction regimes, including price quantiles, time of day, and inventory types, which helps to avoid bias associated with price fluctuations. Additionally, segmentation is carried out across periods of high and low competition. In a study on bid strategy design under cost constraints, limitations arising from price dynamics and sparsity in the distribution of transaction prices are highlighted; therefore, an approach is proposed that jointly improves landscape prediction and bidding strategy [4].

For the marginal CAC model, it is assumed that  depends on the auction win probability and the quality of inventory, both of which vary with changes in the bid and budget. Therefore, the calibration of the saturation function should account for auction regimes (e.g., by price quantiles, time of day, and inventory types) and be validated for robustness.

The estimation of saturation parameters is performed on time series of marketing effects. Since the data exhibit seasonality and autocorrelation, a bootstrap approach that preserves the temporal structure (maximum entropy bootstrap) is employed [5].

The robustness validation algorithm includes:

1. Estimation of  and  on the original time series.
2. Generation of meboot replications.
3. Re-estimation of parameters on each replication.
4. Comparison of the distributions of and the shadow price .

If the confidence intervals for λ and the recommended budget shares are stable, the solution is considered robust to noise and the temporal dependence structure. The optimization of budget allocation across channels is formulated as the problem of maximizing the total number of customers under a fixed overall budget. At the optimum, the marginal return on budget must be equalized across channels. In this context, λ is interpreted as the marginal effectiveness of a unit of budget, i.e., a common shadow value of the incremental increase in customers at the optimum. Therefore, the condition of equal marginal returns is equivalent to the condition of equalizing marginal CAC across channels.

Results and discussion. To illustrate the differences between average CAC (average CAC defined as the ratio of budget to the number of acquired customers) and marginal CAC (marginal CAC defined as the cost of acquiring the “next” customer under a small increase in budget), we present an example of a synthetic dataset for a single channel (Table 1), which demonstrates a situation where, as the budget increases, average CAC changes smoothly, while marginal CAC grows more rapidly in the saturation tail.

Table 1.

Example of a saturation curve and CAC calculations (Channel A)

Thus, as can be observed, the increase in budget from 150 to 200 is associated with a deterioration in , while  remains relatively stable. At the 250–300 level, the marginal cost increases sharply, which serves as a signal for budget reallocation.

Based on a dataset [11] containing daily metrics of spend and conversions for Facebook (принадлежит Meta, признана экстремистской и запрещенной в России)and AdWords campaigns in 2019, illustrative calculations are performed. In the analysis, a “customer” is interpreted as a conversion; the selected channels include FB_May19, FB_Nov19, and AW_May19 (Fig. 1; Fig. 2). The data source for the illustrative empirical analysis is an open GitHub repository (Digital Marketing Performance Analysis: Facebook (принадлежит Meta, признана экстремистской и запрещенной в России) vs. AdWords (A/B Testing)). The dataset includes fields characterizing the observation date, advertising campaign, advertising spend, and conversion outcomes (in the dataset, these fields are labeled as: Date; Facebook (принадлежит Meta, признана экстремистской и запрещенной в России) Ad Campaign; Facebook (принадлежит Meta, признана экстремистской и запрещенной в России) Ad Conversions; Cost per Facebook (принадлежит Meta, признана экстремистской и запрещенной в России) Ad; AdWords Ad Campaign; AdWords Ad Conversions; Cost per AdWords Ad). Within the scope of this study, the spend metric is interpreted as the daily advertising expenditure of a campaign in USD, while conversions represent the number of daily conversions. Accordingly, in the empirical section, a “customer” is defined as a conversion; thus, average CAC is calculated as the ratio of spend to conversions, while marginal CAC is computed as the incremental cost of an additional conversion across adjacent levels of cumulative spend (in line with the methodology presented earlier). The use of this open dataset is illustrative and is intended to demonstrate the differences between average CAC and marginal CAC.

Figure 1. Distribution of data across channels (open data, 2019)

Table 2.

Summary metrics for three campaigns (open data, 2019)

In the empirical section, marginal CAC is calculated not as the ratio of total budget to the total number of customers, but using a difference-based approach between adjacent budget levels. Therefore, in this context, it represents an incremental rather than an average acquisition cost.

The values of are calculated using a difference-based approach:

where  and  are defined between adjacent budget levels ; ), formed based on quantiles of cumulative spend (12 levels for each campaign). The index jjj denotes adjacent budget levels formed based on quantiles of cumulative spend within each campaign. Thus, average CAC and marginal CAC are not equivalent and are calculated according to different rules. The median and p90 are calculated based on the distribution of  values across these levels.

Figure 2. Comparative analysis of CACmargCAC^{marg}CACmarg across channels (open data, 2019)

It should be emphasized that the presented empirical calculations are illustrative and demonstrative in nature and are performed using an open dataset. Therefore, the obtained results should not be interpreted as a full validation of the algorithm on real client data; rather, they serve to demonstrate the principles and features of the calculations, the comparison of channels, and the potential directions for budget reallocation.

Thus, at comparable levels of spend, Facebook (принадлежит Meta, признана экстремистской и запрещенной в России) campaigns demonstrate a significantly lower level of  at the tail of the distribution (p90), while for AW_May19 the marginal cost is notably higher. Therefore, under a fixed total budget, reallocating a small share  from a campaign with a higher  toward campaigns with lower  is consistent with the principle of marginal cost equalization used in iterative reallocation algorithms. At the same time, filtering non-intentional traffic at the DSP level (DSP — demand-side platform), as a pre-bid data quality control stage, reduces the risk of overestimating marginal cost due to anomalous requests [6], while optimizing frequency capping constraints on non-clicked impressions supports the stabilization of marginal efficiency in the saturation tail [8]. The validity of cross-campaign comparison in terms of conversions additionally depends on the selected user-path attribution model [7], and the reliability of conclusions regarding saturation parameters and derived quantities should be verified using bootstrap procedures for time series [5]. Accounting for auction dynamics and the partial observability of the winning price further reinforces the need for separate calibration across price landscape regimes (i.e., the distribution of winning prices) [4].

Based on the above, an algorithm for budget reallocation across channels and campaigns can be proposed; an iterative algorithm for equalizing marginal CAC subject to constraints is employed.

Step 0. Data preparation.

a) filtering of anomalous traffic prior to the auction (pre-bid classification and removal of non-intentional bid requests) [6];

b) attribution of conversions across channels/campaigns;

c) construction of  and estimation of  over a window comparable in terms of seasonality.

Step 1. Estimation of the current marginal CAC.

For each channel:

The increment  is set as a “small” reallocated share (e.g., 3–5% of the channel budget).

Step 2. Determining the direction of budget reallocation.

The following channels are selected:

donor (the highest marginal cost),

receiver (the lowest marginal cost),

subject to the satisfaction of share/minimum constraints.

Step 3. Reallocation of share .

Step 4. Stopping criterion.

The process terminates when:

or when the constraints are reached.

For practical implementation, it is useful to incorporate frequency capping constraints, as the accumulation of repeated contacts increases marginal cost in the saturation tail. One possible approach is to set a limit on the number of non-clicked impressions, which reduces wasteful delivery and stabilizes performance [8]; in the algorithm, this is implemented as a controllable parameter affecting the saturation rate  (with stricter filtering of ineffective contacts, saturation occurs later, and increases more slowly).

Below is an example of a tabular representation of a budget reallocation decision based on the comparison of marginal costs.

Table 3.

Example of budget reallocation based on marginal CAC (3 channels)

Thus, even with similar values of  , channels may differ significantly in terms of . Reallocation is performed from the “expensive tail” to the “cheaper tail” until marginal costs converge or constraints are reached.

Subsequently, validation becomes necessary, based on the assessment of changes “before” and “after,” as well as robustness checks.

“Before–after” validation involves:

Period : baseline budget allocation and existing bidding and frequency rules.

Period : application of the marginal CAC equalization algorithm and updating of the frequency cap on non-clicked impressions [8].

The primary effect metric is the increase in the number of customers  at a fixed budget  or the reduction in total  at a fixed .

Additionally, metrics such as ROAS (Return on Ad Spend), conversion rate, the share of filtered traffic, and the share of the saturation tail may be used.

A significant component of validation is associated with the correct attribution of the user path (i.e., the sequence of contacts prior to conversion). The use of linear regression is possible, as it maintains predictive competitiveness and supports decision-making at the campaign level [7]. Within robustness checks, attribution is specified in two configurations: (1) a baseline configuration and (2) an alternative configuration (e.g., with a shortened contact window), after which the stability of and the recommendations for  are compared.

Robustness checks are carried out as follows:

1) bootstrap of saturation parameters using meboot (maximum entropy bootstrapping — a bootstrap method that preserves the structure of temporal dependence) for weekly and/or daily time series while maintaining seasonality and autocorrelation [5];

2) placebo windows, i.e., running the algorithm over periods without changes in creatives and targeting, with the assessment of the “false effect.”

3) sensitivity to filtering, i.e., comparison of  with and without pre-bid filtering of non-intentional auction bid requests [6];

4) accounting for auction dynamics, with separate calibration across regimes of the distribution of winning prices in the auction and robustness checks of recommendations under partial observability and price censoring [4].

Based on the above, marginal CAC proves to be more robust for scaling for several reasons:

1. Alignment with the auction mechanism. In real-time bidding, the bid affects both the probability of winning and the price, thereby altering the effectiveness of incremental budget. Approaches that combine price landscape prediction with bidding strategy improve cost efficiency and the stability of decisions under partial price observability [4]. Marginal CAC is conceptually consistent with this approach, as it evaluates the “price” of the next customer at the current auction frontier.

2. Management of external effects and delivery constraints. In automated media buying, external effects arise related to infrastructure load and delivery dynamics. The analysis of the delivery paradox in predictive marketing under environmental constraints demonstrates the applicability of specialized regimes [3]. Marginal CAC can be interpreted in conjunction with delivery constraints (frequency limits, data, and formats), as these factors alter the shape of saturation and the position of the optimum.

3. Integration with personalization. The effectiveness of additional budget is determined by the system’s ability to deliver relevant impressions under constraints of privacy and data quality. The systematization of computational techniques in personalized advertising establishes the dependence of RTB optimization on dataset quality, targeting mechanisms, and related factors, which directly affect the saturation parameters  [9]. In practice  corresponds to the channel capacity under a given targeting quality, while  corresponds to the rate of audience saturation.

4. Compatibility with dynamic causal models of effect. Under repeated exposures, carryover effects (influence on future responses) and fatigue (i.e., a decline in response) emerge. Dynamic incremental treatment effects assume the joint use of causal forests and reinforcement learning for optimizing sequential decisions and estimating effects in counterfactual simulation [10], which, in the long term, extends the application of marginal CAC to the level of the next intervention in the customer lifecycle, where the cost of a decision accounts for long-term value and response dynamics.

Conclusion. Thus, the conducted study confirms that marginal CAC is a more informative metric for scaling advertising campaigns in an auction environment compared to average CAC, as it captures diminishing marginal returns and the increasing cost of acquiring the “next” customer.

Calculations based on an open dataset show that, at comparable levels of spend, Facebook (принадлежит Meta, признана экстремистской и запрещенной в России) campaigns exhibit lower values of marginal CAC at the upper levels of the distribution compared to AW_May19. Thus, the illustrative example confirms that average CAC and marginal CAC may provide different views of channel performance, while the use of the marginal metric allows for the identification of a higher-cost scaling frontier. Within the scope of this dataset, this provides grounds for the hypothetical reallocation of a small budget share from a channel with a higher marginal CAC to one with a lower marginal CAC.

At the same time, filtering of non-intentional traffic at the DSP level, the configuration of frequency capping per user, the choice of attribution model, the consideration of price dynamics, and robustness checks on time series are treated in this study as methodological recommendations and practical conditions for the correct applied implementation of the proposed approach in real advertising systems. Additionally, elements related to traffic filtering, attribution, frequency capping, and the consideration of auction regimes should be regarded as directions for further applied validation of the model on client data.

Thus, the proposed formalization based on the saturation curve provides a transparent criterion for budget reallocation through the equalization of marginal costs across channels under given constraints. The practical applicability of the approach is enhanced by input data quality control and filtering of non-intentional traffic at the DSP level, optimization of contact frequency, proper attribution, and robust validation of parameters on time series.

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