# Optimal Pricing in a Rented 5G Infrastructure Scenario with Sticky Customers
## 4 Operators' Costs, Revenues, and Profits
### The Demand Function
In the case of lock-in, the customer is currently served by an operator and experiences strong barriers to leaving that operator, whereas we are considering a customer who has to make the initial choice between two (or more) operators and is therefore relatively free to choose. Lock-in has been extensively studied as a general phenomenon of markets and has been reported for telecom services as well. It has been shown to be quite relevant both in broadband services, as shown by [56], and in mobile networks (see the paper by [57]). In some cases, the lock-in effect is incorporated in the contract between the telecom operator and the customer, as considered by [58]. A related phenomenon is churn, where operators suffer from many customer abandonments. Again, in churn, customers leave their present operator rather than choosing one from a set of possible choices, as in our case. Most studies on churn have concentrated on using machine learning techniques, where a large number of features are included at the same time. Typically, those features are mostly related to consumption characteristics as they are extracted from the telecom operator's logs and other information available to the operator (such as the pricing plan). For example, [PERSON] et al. used logistic regression and boosting to predict churn, starting with 700+ features and reducing them to 21 [59]. However, a few studies have highlighted that price is just one of the variables affecting churn. [PERSON] et al. classified those variables into two categories: service quality and brand image, with pricing falling into the service quality class [60]. However, brand image includes factors such as the operator's perceived fairness, friendliness, and innovativeness. Furthermore, [PERSON] et al. have highlighted the role of social ties in churn: people whose friends have left their operator tend to do likewise [61]. Going back to stickiness, we can model stickiness through an aptly defined repartition function. In the case of two operators, that function tells us how the aggregate set of customers is divided among operators as a function of their price ratio. In the absence of stickiness, we expect customers to choose either operator with the same probability (50%) when their prices are equal. Of course, the stickiness function becomes progressively unbalanced when the price of either operator tends to zero. The competitor asking for zero price would take all of the market. However, for any case where the price is different from zero, though extremely small, there will always be a non-zero probability that the customer opts for the higher-priced operator. Here, we take the infrastructure owner (the MNO) as the reference, i.e., the operator preferred by customers, that to which customers tend to stick to. For simplicity, we aggregate MVNOs by considering their average price \(\overline{p}_{\text{MVNO}}\):  
\[\overline{p}_{\text{MVNO}}=\frac{\sum_{i=1}^{N_{R}}p_{\text{MVNO}}^{(i)}}{N_{ R}}. \tag{7}\]The repartition function \(f_{\text{MNO}}\) of the MNO provides the average fraction of customers choosing the MNO as a function of the \(p_{\text{MNO}}/\overline{p}_{\text{MVNO}}\) ratio. It must exhibit the following properties:  
* \(f_{\text{MNO}}=1\) when \(p_{\text{MNO}}=0\);
* \(f_{\text{MNO}}=0\) when all the renters give their service for free, i.e., \(p_{\text{MVNO}}^{(i)}=0\), \(i=1,2,\ldots,N_{\text{R}}\), so that \(\overline{p}_{\text{MVNO}}=0\). Due to these properties, we adopt the following repartition function for the owner \[f_{\text{MNO}}=e^{-\xi\frac{p_{\text{MNO}}}{p_{\text{MNO}}}},\] (8) where \(\xi\) is the stickiness factor. When we have a single renter and there is no stickiness effect, if the MVNO offers the same price as the MNO, the repartition function divides customers equally between the MNO and the MVNO. In that case, solving Equation (8) for the stickiness factor, we obtain \(\xi=\ln 2\). In fact, when \(p_{\text{MNO}}=\overline{p}_{\text{MVNO}}\), we should obtain \(f_{\text{MNO}}=1/2\).