# Optimal Pricing in a Rented 5G Infrastructure Scenario with Sticky Customers
## 5 The Owner-Renter Game  
In Section 4, we have set the stage to analyse the strategies of the operators. Now we know the expected number of customers they obtain and the profits they achieve under a demand function that takes into account the preferences of customers that are dictated not just by prices but also by their stickiness to either the infrastructure owner or the renters. In this section, we formulate a game between the operators and look for its Nash equilibrium.  
We assume that the aim of all the operators is to maximize their own profit. In order to achieve that goal, they use pricing as their strategic leverage. Precisely, they adjust the prices offered to end customers for which they are competing. So, the infrastructure owner adjusts \(p_{\text{MNO}}\), whereas the generic renter will adjust \(p_{\text{MVNO}^{(i)}}\), \(i=1,2,\dots,N_{R}\). We assume that end customers may be offered different prices such that \(p_{\text{MNO}}\
eq p_{\text{MNO}^{(i)}}\) in general. The price difference will not result in the lowest-price operator taking all of the market due to the sticky demand function.  
The maximization of the profits for the operators can be achieved by zeroing the derivative of the profit. We pursue the mathematical details in Appendix B. We show that the equations holding the solution for the maximum profit cannot be solved simultaneously, i.e., there is no Nash equilibrium.  
We then consider three different scenarios for price setting that, although they do not envisage an equilibrium, may be employed in a practical setting. The three scenarios differ for the stakeholder in charge of the pricing decision:  
* Owner;
* Renters;
* Regulator.  
In the first scenario, the owner sets the price for itself and the renters as well. It is the natural extension of a monopolistic setting where the owner is the only operator and is fully in charge of setting prices to maximize its profit. Though it may overestimate the power of the owner and appear impracticable, we can consider it as a reference value (because it is possibly the highest price that can be set). In addition, though it confers great power on the owner, that power would not lead the owner to squeeze the renters out of the market by using the price for the renters as leverage. In fact, setting a high price for the renters would increase their unit profit and would reduce the overall market size due to demand reduction entailed by the demand function described in Section 4.2. On the other hand, setting a low price for the renters would leave them a significant portion of the market, though with reduced unit profit margins.  
In the second scenario, the roles are reversed. It is now the renters that set the price both for themselves and the owner to maximize the renters' profits. The same arguments considered for the first scenario apply here as well, with the addition that the owner cannot be ousted from the market, because that would leave the renters themselves without infrastructure.  
Finally, in the third scenario, we envisage a regulating authority that sets the prices for both the owner and the renters to maximize the social welfare SW shown in Equation (17), i.e., the sum of the profits of all the operators present in the market. The prices will be different for the owner and the renters, reflecting their different revenue and cost structures.  
\[\begin{split}\text{SW}&=\Pi_{\text{MNO}}+\sum_{i=1 }^{N_{R}}\Pi_{\text{MVNO}}^{(i)}\\ &=p_{\text{MNO}}\pi_{\text{MNO}}+\sum_{i=1}^{N_{R}}p_{\text{MVNO }}^{(i)}n_{\text{MVNO}}^{(i)}\\ &-(C_{0}+mB)\left[n_{\text{MNO}}(1+\alpha)+\sum_{i=1}^{N_{R}}n_ {\text{MVNO}}^{(i)}(1+\alpha+\eta_{i}\alpha+\beta_{i})\right]\end{split} \tag{17}\]  
Contrary to what happens for the Nash equilibrium, we find that all those scenarios allow for a solution to the game.  
In the following section, we report the results obtained for all the scenarios and a selected choice of values for the parameters.