## 4 Results
### Business Case: Maximize Revenue and Maintain Certain Profitability Level  
A second plausible business case is related to the willingness to improve performance. In particular, we envision a case where the advertiser aims to maintain certain profitability and increase revenue. The restriction of the profitability will define the minimum ROAS the client can afford. In addition, by exploiting the advertiser's historical data from previous ad campaigns and associated conversions, we could compute the cost per impression for that client. Similarly, based on the client historical data, we can compute the revenue per click for the client. With all this information, we will use the following formula introduced in Section 3.5:  
\[CTR=\frac{ROAS\cdot Cost_{Per\text{Impression}}}{Revenue_{Per\text{Click}}}\]  
Using this formula, we can obtain the lower bound for the CTR that will guarantee to maintain the desired profitability. In this setting, we will test two different strategies to define the optimal \(FC\) value:  
* Absolute capping: This strategy is the one currently implemented in the digital marketing industry. A particular user will keep receiving impressions until the \(FC\) is reached and will not receive any further impression from that moment. This strategy does not consider any interaction (i.e., click of the user) of the user with the ad. For instance, if we define a \(FC=5\) and a user clicks and converts in the first impression, he/she can receive 4 more ads. * Non-clicked based: We propose a novel strategy to define the \(FC\) based on the number of consecutive non-clicked impressions. This means we stop sending impressions to a user if he/she has received \(FC\) consecutive impressions and did not click any of them. Then, if the user clicks on an impression, the counter to reach \(FC\) is restarted. For instance, if we define a \(FC=5\) and a user does not click in the first 5 impressions, we stop sending impressions. However, if he/she clicks on the fourth impression, he/she can receive up to 5 new impressions even if he/she does not click again. Note that we did not compare these strategies in the previous business case because as the considered paths are formed only by zeros, both strategies become the same. Frequency Capping Strategies Evaluation Using Real Datasets  
For the business case under consideration, we work with the entire graph of paths Figure 1. We do not require to use simulations for this business case. Instead, we use the real data belonging to five different companies. We apply different \(FC\) values to the historical data and study how this impacts the CTR and ROAS. In this business case, we evaluate \(FC\) values ranging from 1 to 80. Figure 13 shows the CTR for the Absolute and Non-click-based strategies for each of the five advertisers analyzed and the different \(FC\) values considered. We observe that the Non-click-based strategy outperforms the Absolute approach. The difference is especially relevant for \(FC\leq 10\) in all the advertisers. Both strategies become similar for \(FC\geq 30\), and from that point, the CTR decrement is very low. Furthermore, while the absolute method does not describe the same shape for all the clients, in the non-clicked strategy, the results show a continuously decreasing shape in all five companies. Finally, we can appreciate that from a value of 30, no matter the method, the CTR tends to stabilize with a slight decrease but not significant. Once we have obtained the CTR for both strategies, we could easily convert the ROAS into CTR and use the previous Figure 13 to find the best CTR. However, it is even better to convert CTRs into ROAS and create new plots showing the ROAS value for the \(FC\) range under analysis. To perform this exercise, we use a CPM equal to \(1.53\epsilon\), which means a cost per impression of \(0.00153\epsilon\). In addition, we consider a revenue per click of \(3\epsilon\). These values are the real values from a company, and they were provided directly from Neo Media World.