# Optimal Pricing in a Rented 5G Infrastructure Scenario with Sticky Customers
## 6 Results
For each case, we define the loss as the percentage of profits that are lost when prices are set by another party:  
\[L_{h}^{((k))}=\frac{\Pi_{h}^{(h)}-\Pi_{h}^{(k)}}{\Pi_{h}^{(h)}}. \tag{18}\]  
where \(h\) may be either MNO or MVNO, and \(k\) may denote one of the three price-setting scenarios (\(o\), \(r\), or \(SW\), for where prices are set by the owner, the renters, and the regulator, respectively). In Figure 2, we see that the MNO's loss grows with the level of competition (as expected). However, somewhat unexpectedly, the worst case for the MNO is having its prices set by the regulator. Bandwidth has an alleviating effect because the owner is paid by renters proportionally to the bandwidth allotted to each customer, whereas the revenues of the MVNOs grow with the number of customers. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \(\Pi_{\text{o}}^{(o)}\) & \(\Pi_{\text{r}}^{(o)}\) & \(\Pi_{\text{o}}^{(r)}\) & \(\Pi_{\text{r}}^{(r)}\) & \(\Pi_{\text{o}}^{(\text{SW})}\) & \(\Pi_{\text{r}}^{(\text{SW})}\) \\ \hline monopoly & 116,439 & NA & NA & NA & NA & NA \\ \hline \(N_{R}=1\) & 63,083 & 51,899 & 59,276 & 54,194 & 57,088 & 42,208 \\ \hline \(N_{R}=3\) & 38,564 & 23,655 & 29,548 & 27,424 & 20,591 & 27,350 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Profits of operators under different scenarios. Similarly, in Figure 3, we show the loss suffered by each renter. Here, the bandwidth has an adverse effect for the same reason as seen for the owner: renters obtain revenues proportionally to the number of customers but pay the owner proportionally to the allotted bandwidth. Again, we see that the worst cases occur when prices are set by the regulator. Another factor impacting the strategies of operators is the stickiness of customers. Any deviation of the stickiness factor \(\xi\) from its central value \(\xi_{0}\) moves customers towards either operator. Precisely, if \(\xi>\xi_{0}\), customers move towards the MVNOs. In Figure 4, we see what happens to MNOs' profits when the stickiness factor \(\xi=\lambda\xi_{0}\) moves around its central value. As expected, profit losses grow when the stickiness factor grows and renters set prices. The behaviour is somewhat erratic when the regulator sets prices such that it tends to balance the wishes of both operators. We observe a similar behaviour for the renter(s) in Figure 5. The stickiness factor does not impress a significant direction to losses when the regulator sets prices. However, the presence of stickiness towards MVNOs does not help, contrasting the strategy by the MNO, which is able to overcome that stickiness and reduce the MVNOs' profits by suitably setting prices. We now examine the benefits of competition for end users by observing how operators are compelled to change their prices in response to increased competition, which is represented here by a growing number of virtual operators. Figure 3: MVNO’s loss. Figure 2: MNO’s loss. In Figure 6, we examine the effect of the number or renters on the price set by the owner. We see that the prices decline as expected, though in a less-than-linear fashion. The bandwidth considered here is \(B=60\). In comparison, the impact of the resources required by users is much smaller. In Figure 7, which is plotted for the case of a single renter, we see that the impact is negligible when prices are set by the regulator to maximize the social welfare. Instead, we observe contrasting effects for prices set by the owner and the renter. The price set by the owner tends to make bandwidth a bit cheaper when the request grows. The renter, instead, makes it significantly more expensive to buy more bandwidth. In the picture, we have omitted the superscript indicating the specific renter because we have assumed for simplicity that all renters set the same prices. When the competition grows (we consider again the case of three renters), we see in Figure 8 that prices set by the same operator (e.g., the price set by the owner for itself) exhibit the same behaviour as in the case of a lower competition.