# 5G Price Competition with Social Equilibrium Optimality for Social Networks
## 3 Social Welfare Optimization
### Socially Optimal Demand Vector \(r^{*}\)'s Uniqueness  
However, it is interesting that \(u_{n}(\cdot)\) fails to be strictly concave to the demand vector \(r_{n}\), as is the case of SWO to \(r\). As we all know, a maximization problem that is not strictly concave may have more than one global optimal solution (see [33; 34]). To get more than one solution of vector \(r^{*}\) in SWO, we simply modify \(c_{nm}\)s, \(R_{m}\)s, and \(u_{n}(\cdot)\)s to some value. For example, if \(c_{nm}\) is constant and the same for different \(n\), \(m\) as it is for \(R_{m}\), we can get a non-unique maximizer of SWO. However, as we showed previously, \(c_{nm}\)s are independent random variables from continuous distributions, and the probability of that case occurring is zero (see Assumption 2). As we also learn in Lemma 1, no two maximizing demand vectors can exist in SWO that possess the same nonzero components. If two maximizing demand vectors combine, the result is still a maximizing demand vector. Finally, with the previous intermediate result, we see that maximizing demand vectors yields no convex combinations possessing different nonzero components, and this contradicts Lemma 1. So, we can use this to prove the primary finding of this section (Theorem 1). Next, we define a demand vector \(r_{n}\)'s support set. User \(n\)'s support set is composed of suppliers from which user \(n\)'s demand is strictly positive:  
\[\hat{\mathcal{M}}_{n}(r_{n})=\{m\in\mathcal{M}:r_{nm}>0.\}\]  
Given demand vector \(r\), we define the support sets' ordered collection \(\hat{\mathcal{M}}_{1},\hat{\mathcal{M}}_{1},\ldots,\hat{\mathcal{M}}_{N}\) as \(\{\hat{\mathcal{M}}_{n}\}_{n=1}^{N}\). If \(r^{*}\) is SWO's maximizing demand vector containing the corresponding collection of support sets of \(\{\hat{\mathcal{M}}_{n}\}_{n=1}^{N}\), then it is almost true that \(r^{*}\) can be a unique maximizing demand vector compared to \(\{\hat{\mathcal{M}}_{n}\}_{n=1}^{N}\). Equation (2) holds for the maximizing demand vector \(r^{*}\), and \(r^{*}\) is uniquely constructed from \(\{\hat{\mathcal{M}}_{n}\}_{n=1}^{N}\). Then two categories exist for the users: decided and undecided. We define that the decided users are those who buy from one supplier \((|\hat{M}_{n}|=1)\), while the undecided users are those who buy from more than one supplier \((|\hat{M}_{n}|>1)\). In fact, some users will buy nothing. Without loss of generality, these users are defined as decided users. \(y_{n}^{*}=\sum_{m=n}^{M}r_{nm}^{*}c_{nm}\) holds for all users. If user \(n\) is a decided user--which means that he only buys from supplier \(m\) and buys nothing from other suppliers--we can reduce the equation to \(y_{n}^{*}=r_{nm}^{*}c_{nm}\), because other terms are zero. Then the unique demand vector that corresponds is \(r_{n}^{*}=[0\ldots 0,\frac{y_{n}^{*}}{c_{nm}},0\ldots 0]\). Theorem 4.1: _There is a unique maximizing solution \(b^{*}\) with probability 1 in SWO. There are no multiple maximizing demand vectors, and the convex combination of SWO for maximizing demand vectors retains the same support._  
Proof: Suppose that more than one SWO optimal demand vector exists. Two of them are \(r^{\prime}\) and \(r^{*}\). We learn from Lemma 1 that \(r^{*}\) and \(r^{\prime}\) almost certainly have distinct support sets \(\left\{\hat{M}_{n}^{\lambda}\right\}_{n=1}^{N}\) and \(\left\{\hat{M}_{n}^{\prime}\right\}_{n=1}^{N}\). Then, let \(r^{\lambda}=\lambda r^{*}+\bar{\lambda}r^{\prime},\lambda\in(0,1),\bar{ \lambda}=1-\lambda\). If \(y_{n}^{*}=\sum_{m=1}^{M}r_{nm}^{*}c_{nm}=\sum_{m=1}^{M}r_{nm}^{*}c_{nm}\) and \(\sum_{m=1}^{M}r_{nm}^{\lambda}c_{nm}=\lambda\sum_{m=1}^{M}r_{nm}^{*}c_{nm}+ \bar{\lambda}\sum_{m=1}^{M}r_{nm}^{\prime}c_{nm}=y_{n}^{*}\), then we can say that \(r^{\lambda}\) is an SWO maximizing solution for each \(\lambda\in(0,1)\).