# 5G Price Competition with Social Equilibrium Optimality for Social Networks
## 2 The Model
### Supplier Competition Game  
There are two stages in the supplier competition game. Each supplier claims its price in the first stage. In the price vector \(b=[b_{1},\ldots,b_{M}]\), \(b_{m}\) represents the price for the unit commodity that supplier \(m\) charges. In addition, every user \(n\in\mathcal{N}\) chooses a demand from different suppliers, depicted by vector \(r_{n}=[r_{n1},\ldots,r_{nM}]\). Then we use a vector to depict the overall demand: \(r=[r_{1},\ldots,r_{N}]\). In the second stage, as prices \(b\) have already been set, user \(n\) selects its demand \(r_{n}\) to maximize its payoff based on the price. We define the payoff as a utility after subtracting expenses:  
\[v_{n}(r_{n},b)=u_{n}(\sum_{m=1}^{M}r_{nm}c_{mn})-\sum_{m=1}^{M}b_{m}r_{nm}. \tag{1}\]  
In this equation, \(c_{nm}\) is the offset of channel quality between provider \(m\) and user \(n\) (see Example 1 and Assumption 2). Here, \(u_{n}\) is the utility function, which is concave and increases with quantity. We can see that the utility function is based on the term \(\sum_{m=1}^{M}r_{nm}c_{nm}\), which is the amount of service that users acquire, and also the function of commodity uses. In the first stage, after taking into account the resource constraint \(\sum_{n=1}^{N}r_{nm}\leqslant R_{m}\), and in the second stage, after factoring users' demand, then supplier \(m\) sets the price to \(b_{m}\) to maximize its return \(b_{m}\sum_{n=1}^{N}r_{nm}\). We assume linear pricing, with each user facing the same price. In this model, any user can buy a commodity from more than one supplier simultaneously. In other words, for users, \(n\), more than one \(r_{nm}\) (\(m\in M\)) can be positive. It may be reasonable only if the user's device has several wireless interfaces. Interestingly, for most users (\(N-M\) at least), the optimal strategy is to select one or no supplier. In the following, a specific example shows how to apply our model in a practical 5G wireless system. **Example 1** (NOMA). : _Non-Orthogonal Multiple Access (NOMA) is a popular technology to improve the efficiency of the 5G spectrum, with low latency, low signaling cost, and attenuation resistance (see [30; 31]). \(W_{m}\), \(m\in\mathcal{M}\) are the non-orthogonal frequency bands on which wireless providers operate. \(r_{nm}\) is the portion of time user \(n\) can transmit exclusively on supplier \(m\)'s frequency band, in which \(\sum_{n\in\mathbb{N}_{m}}r_{nm}=1\), \(m\in\mathcal{M}\) is the constraint. We assume a peak power constraint \(P_{n}\) exists as well for each user. Then we define \(c_{nm}\) as \(W_{m}\log\left(1+\frac{P_{n}|h_{nm}|^{2}}{c_{nm}^{2}W_{m}}\right)\) by [PERSON]'s theorem, in which the channel's Gaussian noise variance is \(\sigma_{nm}^{2}\) between provider \(m\) and user \(n\), while \(h_{nm}\) is channel gain, channel gain describes the transmission capability characteristics of the channel itself. The payoff for the user, then, is the remaining utility after subtracting payment for the service, \(v_{n}=u_{n}(\sum_{m=1}^{M}r_{nm}c_{nm})-\sum_{m=1}^{M}b_{m}r_{nm}\)._  
Similarly, our model applies when suppliers sell bandwidth of ultra-reliable low latency communication (uRLLC) tones to users who face a maximum power constraint [32]. For example, \(c_{nm}\)--the offset factor in Example 1--not only represents channel capability but essentially any aspect of the channel capacity's increasing function. Compared with other previous technologies, 5G has a more significant channel gain. According to the Shannon formula, when the channel gain \(h_{nm}\) increases, the channel capacity \(c_{nm}\) will increase to achieve an extremely low delay. Although the payment for the 5G service is high, the significant improvement of quality of service(QoS): utility function \(u_{n}\), is enough to offset the payment, and the final income is significantly higher than that of time division multiple access(TDMA), and it can also meet the requirements of low delay and high spectrum efficiency in modern times.