## 4 Results and Discussion
### IoT-SP\({}_{1}\) and IoT-SP\({}_{2}\) Pricing and Profits  
In order to study the Game II results we show the equilibrium price \(f_{1}^{*}\) and the IoT-SP\({}_{1}\) profit \(\Pi_{IoT-SP_{1}}\), varying the sensitivity of the users to the providers' price \(\alpha\) and the parameters \(c\), \(N\) and \(M\).  
Figures 11 and 12 show the IoT-SP\({}_{1}\) equilibrium price \(f_{1}^{*}\) as a function of \(\alpha\) for different values of \(c\) and \(N\) respectively. The equilibrium price does not depend on the value of \(c\) and increases with N, due to the higher utility perceived by users. Note that it only happens if the maximum system load \(L<1/2\). We also observe that when users' sensitivity to \(R_{1}/f_{1}\) increases, the IoT-SP\({}_{1}\) optimal price decreases very fast.  
Figure 13 shows the IoT-SP\({}_{1}\) profit as a function of \(\alpha\) for different values of \(c\). Similarly to the price the equilibrium profit does not depend on the value of \(c\) and decreases with the sensitivity of the users to the price. On the other hand Figure 14 shows that the IoT-SP\({}_{1}\) profit increases with the value of N. This means that final users are willing to pay a higher price if the amount of data collected by the IoT-SP is higher. This will drive to a competition between the IoT-SPs to increase the number of sensors that is not studied in this paper. As shown in Figure 15 the profit also increases with M due to the higher pool of users subscribed to the IoT-SP\({}_{1}\). Note that here there is not a congestion effect when M increases. With low values of the sensitivity parameter users choices have a very weak dependence on the prices \(f_{i}\) and the IoT-SPs increase hugely its prices. When it occurs the rate of users subscribed to the IoT-SPs is very low, but the higher prices offset it.  
Figure 16 shows the IoT-SP\({}_{1}\) profit as a function of \(N\). The figure shows how the profit increases with N until the OP network is congested. After that, the profit remains constant.  
The conclusions for the IoT-SP\({}_{2}\) obtained from Figures 17-22 are the same that those obtained for the IoT-SP\({}_{1}\) taking into account that the values of \(N1\) and \(N2\) are different.  
From previous results we observe that users' sensitivity parameter \(\alpha\) is critical in the second game. For values of \(\alpha<1\) and \(\alpha>2\) it is not possible to reach an equilibrium with positive profits, as deduced from the analysis. In addition, if the value is in the range \(1<\alpha<2\) it still has a huge relevance in the IoT-SPs equilibrium decisions and profits, not only in the value of them, but also in the feasibility of the whole model as deduced in (38) and (39).  
Figure 10: OP optimal profit as a function of \(L\) for different values of \(c\).  
Figure 11: IoT-SP\({}_{1}\) equilibrium price as a function of \(\alpha\) for different values of \(c\).  
Figure 12: IoT-SP\({}_{1}\) equilibrium price as a function of \(\alpha\) for different values of \(N\).  
Figure 13: IoT-SP\({}_{1}\) equilibrium profit as a function of \(\alpha\) for different values of \(c\).  
Figure 16: IoT-SP\({}_{1}\) equilibrium profit as a function of \(N\).  
Figure 14: IoT-SP\({}_{1}\) equilibrium profit as a function of \(\alpha\) for different values of \(N\).  
Figure 15: IoT-SP\({}_{1}\) equilibrium profit as a function of \(\alpha\) for different values of \(M\).  
Figure 19: IoT-SP\({}_{2}\) equilibrium profit as a function of \(\alpha\) for different values of \(c\).  
Figure 17: IoT-SP\({}_{2}\) equilibrium price as a function of \(\alpha\) for different values of \(c\).  
Figure 21: IoT-SP\({}_{2}\) equilibrium profit as a function of \(\alpha\) for different values of \(M\).  
Figure 22: IoT-SP\({}_{2}\) equilibrium profit as a function of \(N\).