# 5G Price Competition with Social Equilibrium Optimality for Social Networks
## 4 Game Analysis
### Users' Equilibrium Strategy  
Taking every user's decision into account, we can solve the problem of user payoff maximization (UPM):  
\[UPM:\max_{r_{n}\geqslant 0}v_{n}=\max_{r_{n}\geqslant 0}u_{n}(\sum_{m=1}^{M }r_{nm}c_{nm})-\sum_{m=1}^{M}b_{m}r_{nm}. \tag{3}\]  
**Lemma 2**. : _Regarding the UPM problem, with each maximizer \(r_{n}\), \(\sum_{m=1}c_{nm}r_{nm}=y_{n}^{*}\), for a unique nonnegative value of \(y_{n}^{*}\). Furthermore, for any \(m\) such that \(r_{nm}>0\), \(\frac{b_{m}}{c_{nm}}=\min_{k\in M}\frac{b_{k}}{c_{k}}\)._  
Proof. : We can easily verify that Slater's conditions are satisfied via UPM [36]. The following are the Karush-Kuhn-Tucker (KKT) conditions required for an optimal solution \(r_{n}\geq 0\) of UPM of user \(n\):  
\[u_{n}^{\prime}(y_{n})c_{nm}\leq b_{m},m\in\mathcal{M} \tag{4}\]  
\[r_{nm}(u_{n}^{\prime}(y_{n})c_{nm}-b_{m})=0,m\in\mathcal{M}, \tag{5}\]  
\[\text{where}\,y_{n}=\sum_{m=1}^{M}r_{nm}c_{nm},r_{n}\geq 0. \tag{6}\]  
Here, (4) implies that \(u_{n}^{\prime}(y_{n})\leq\varphi\), where \(\varphi=\min_{k\in\mathcal{M}}\frac{b_{k}}{c_{nk}}\). Based on user \(n\)'s utility function, two scenarios are possible: \(u_{n}^{\prime}(0)<\varphi\) and \(u_{n}^{\prime}(0)\geq\varphi\). For the first scenario, \(u_{n}^{\prime}(0)c_{nm}-b_{m}<0\), so \(c_{nm}u_{n}^{\prime}(y_{n})-b_{m}<0\) for all \(m\in\mathcal{M}\) since, by Assumption 1, \(u_{n}^{\prime}(\cdot)\) is a marginal utility-with a strictly decreasing function. So, keeping (5) in mind, \(r_{nm}=0\) for all \(m\in\mathcal{M}\). Then \(r_{n}=0\), and with (6), we see \(y_{n}^{*}=0\). So, Equations (4)-(6) hold for the \(y_{n}^{*}=0\) unique value. With the second scenario, \(u_{n}^{\prime}(0)\geq\varphi\). However, keeping in mind that \(u_{n}^{\prime}(\cdot)\) dwindles to zero (Assumption 1), a unique \(\hat{y}_{n}\geq 0\) exists, such that \(u_{n}^{\prime}(\hat{y}_{n})=\varphi\). First, we make sure \(r_{n}\) exists, such that Equations (4)-(6) hold with \(y_{n}=\hat{y}_{n}\). We find that Equation (4) holds, because \(u_{n}^{\prime}(\hat{y}_{n})=\varphi\leq b_{m}/c_{nm}\) for all \(m\in\mathcal{M}\). Then, with (5), we remember that for any m such that \(b_{m}/c_{nm}>\varphi=u_{n}^{\prime}(\hat{y}_{n})\) there is \(r_{nm}=0\). For any other m, \(b_{m}/c_{nm}=\varphi=u_{n}^{\prime}(\hat{y}_{n})\), so, when it comes to (5), \(r_{nm}\) can take any non-negative value. In particular, so that (6) holds for the set \(\{m\in\mathcal{M}:b_{m}/c_{nm}=\varphi\}\), it is possible to choose \(r_{nm}\)'s. We provide the last part of the lemma by noting that \(r_{nm}\) is positive only when \(b_{m}/c_{nm}=\varphi\). It remains to be seen whether \(y_{n}\) is the only value of \(y_{n}\) for which \(r_{n}\) satisfies Equations (4)-(6). We can say that for any \(y_{n}<\hat{y}_{n}\), \(u_{n}^{\prime}(y_{n})>\varphi\), which violates (4) for \(m\in\arg\min b_{k}/c_{nk}\). Then, for each \(y_{n}>\hat{y}_{n}\), \(u_{n}(y_{n})<\varphi\), which means that \(u_{n}(y_{n})c_{nm}-b_{m}<0\) for every \(m\in\mathcal{M}\). Equation (5) implies, then, that \(r_{nm}=0\) for every \(m\in\mathcal{M}\), meaning that \(y_{n}=0\); this is contradictory to \(y_{n}>\hat{y}_{n}>0\). The unique searched value \(y_{n}^{*}\) is thus \(\hat{y}_{n}\). **Definition 4**. : _Each supplier \(m\in\mathcal{M}\) with \(\frac{b_{m}}{c_{nm}}=\min_{k\in\mathcal{M}}\frac{b_{k}}{c_{nk}}\) is included in user \(n\)'s preference set \(\mathcal{M}_{n}(b)\) for price vector \(b\)._  
According to Lemma 2 and Section 2, we can divide users into decided and undecided users based on the preference sets' cardinality. The support sets in Section 2 are quite similar to the preference sets. However, unlike support sets where users buy resources from suppliers, it is just possible for a user to request a resource from suppliers in the preference set. Evidently, the support set acts as a subset for the preference set: \(\hat{\mathcal{M}}_{n}(r(b))\subset\mathcal{M}_{n}(b)\). Knowing this, we set about using the preference sets to construct a BGR so that there are on-loops with probability 1.