## 4 Results
### Business Case: Get New Users
#### 4.1.1 Computing Number of New Users Achieved  
We simulate the behaviour of multiple users following the rules used to construct the graph in Figure 10. Each user will have a probability of receiving a new impression and a probability of clicking on it. If the user clicks on the impression, he/she will not receive more impressions. Contrarily, if the user does not click on the received impression, he/she could receive more impressions.  
In the following we define the different parameters required for our simulation:  
* Probability of receiving a new impression: \(P(x)=\dfrac{x}{x+1}\), where \(x\) is the number of previous impressions (As computed in Section 3.4.1).
* Probability of clicking on the received impression: \(P(x)=\dfrac{x+2}{x+3}\), where \(x\) is the number of preceding non-clicked impressions. We remind that if a user clicks on an ad impression she will not receive any further impression (As computed in Section 3.4.2 for company 4).
* We use an audience of 100k users. This means we run a separate process for each of the users.
* We run our simulation for 50 different \(FC\) values ranging from 1 to 50 ad impressions.  
Figure 10: Relevant states and transitions when trying to get new users.  
Using the above parameters, we will execute ten different realizations of our simulation to ensure consistent results. Figure 11 shows the results of the simulation for the following metrics: number of impressions, number of clicks, and CTR. Note that the results of the 10 realizations are represented in the figure, but in most cases, the circles used to represent the values are overlapped.  
The simulation results demonstrate that as the \(FC\) increases, the CTR decreases but the number of clicks increases. The figure is an important tool that will allow the client to select the optimal \(FC\) for its objective. For instance, let us use \(FC=5\) to illustrate how the figure has to be interpreted. On the one hand (negative side), there will be a loss of 5% with respect to the maximum number of clicks (i.e., new users) that could be retrieved. On the other hand (positive side), the client will be saving 20% of the budget (i.e., impressions), which would imply acquiring the referred 5% of extra budget. If we set up \(FC=15\), the client may get all the potential users but will incur in a 10% extra cost. In other words, the last 5% of the clients implies spending 10% more budget. Whether this is worthwhile is a decision that each client can take using the results from our methodology.  
In terms of business implications, this result is very relevant. The marketing manager will have a group of users (audience) that they want to target; this could be, for instance, people interested in travel or user who have entered the page in the past but not in the last two months. Using this method, the marketing manager will be able to take several actions. First, he/she can determine the maximum number of users to bring to the page, making sure they can fulfill this business objective. Once he/she had checked that the target number of users to bring to the page is a reasonable objective, he/she can tell the activation teams the maximum number of impressions for each user that will ensure that the objective is fulfilled, but not money wasted in unnecessary impressions.