## 4 Results
### Business Case: Get New Users
#### 4.1.2 Computing Audience Size  
In the previous simulation, we used our methodology to assume a predefined audience size and obtained the actual number of users that would perform a click on the ad impressions for a given \(FC\) value. In this subsection, we analyze a different business case  
Figure 11: The figure shows the evolution of impressions, clicks and CTR for the simulated cases as we change the maximum impressions capping. All simulations are based on an audience of 100k users.  
in which the predefined value is the number of users that click in an ad impression (i.e., new users), and we need to obtain what is the required audience size that that should be targeted and the \(FC\) value to achieve the predefined number of clicks.  
In our simulation, we will use the same equations as in the previous case for the probability of receiving a new impression and the probability of a user clicking on an ad. In contrast to the previous scenario, this time, we will keep adding users to our simulation until we reach the pre-established number of clicks.  
We run simulations that implement the described scenario. Our simulations are configured with a predefined number of clicks to be reached equal to 10k and \(FC\) values ranging between 1 and 50 ad impressions. In addition, as we did it before, for each \(FC\) value, we run ten executions of our simulations. Figure 12 shows the result of our simulations per \(FC\) value for the following metrics: number of impressions, CTR, and required audience size (i.e., number of users to achieve the expected number of clicks).  
Once more, Figure 12 appears to be a valid tool for advertisers to decide the best \(FC\) to achieve their objectives. Our results show that the required audience size becomes stable around \(FC=10\). This means using any \(FC\geq 10\) imply to reach the same amount of users to obtain 10k clicks. However, the cost (i.e., number of impressions) is not simply derived from \(FC=10\). Actually, a \(FC=10\) can save 15% of the impressions compared to higher \(FC\) values. This roughly means the advertiser can save 15% of the budget.  
To conclude this section, it is important to note that our methodology is highly flexible and can be very easily adapted to the actual metrics of the users. This means our methodology in both business cases simulation requires an impression and click probability that may differ from advertiser to advertiser. Therefore, the obtained results are valid for a impression probability and click probability equal to \(p_{I}(x)=\frac{x}{x+1}\) and \(p_{C}(x)=1-\frac{x+2}{x+3}\), respectively (obtained from the data that we have available for the company 4 in Sections 3.4.1 and 3.4.2). However, we could obtain ad hoc curves for any advertiser providing probability metrics from their historical ad campaigns.  
Figure 12: The figure shows the evolutions of impressions, clicks, and CTR for the simulated cases as we change the maximum impressions capping. All simulations are based on an target of reaching 10k clicks.  
From a business perspective, this method presents different options compared to the previous case. For this case, the marketing manager will need to obtain a certain number of new users on the web. This could be to showcase the products catalog, give visibility for a new product, or just a corporate campaign to help build the brand. Knowing the number of new visitors that have to reach, he/she uses this method to figure out the minimum number of people in the target audience. Therefore she can point look for the audience that fits the desired behaviour, and that has enough users to ensure a good CTR and not waste impressions.