# Context-Aware Caching Distribution and UAV Deployment: A Game-Theoretic Approach
## 3 UAV-Assisted Caching Game
### Analysis of NE  
The NE is the stable solution in game theory. The existence of NE will be analyzed in the part. According to Reference [16], we first give the definitions as follows:  
**Definition 1 (Nash Equilibrium)**.: _An action profile \(a^{*}=(a^{*}_{1},...,a^{*}_{N})\) is a pure stragety NE if and only if no player can improve its utility by deviating unilaterally, i.e.,_  
\[U_{n}({a_{n}}^{*},{a_{I_{n}}}^{*})\geq U_{n}(a_{n},{a_{I_{n}}}^{*}),\forall n \in\mathcal{N},\forall a_{n}\in A_{n},a_{n}\
eq{a_{n}}^{*}. \tag{18}\]  
**Definition 2 (Exact Potential Game)**.: _A game is an exact potential game if and only if a potential function \(\Phi\): \(A_{1}\otimes...\otimes A_{N}\mapsto R\) exists so that for \(\forall n\in\mathcal{N}\):_  
\[U_{n}(\bar{a}_{n},a_{J_{n}})-U_{n}(a_{n},a_{J_{n}})=\Phi(\bar{a}_{n},a_{-n})- \Phi(a_{n},a_{-n}),\forall a_{n}\in A_{n},\bar{a}_{n}\in A_{n}. \tag{19}\]  
Therefore, a game is an exact potential game when the change in individual utility function is the same as the change in the potential function.  
According to the definitions [31], the following theorem is given to present the properties of the proposed game.  
**Theorem 1**.: \(\mathcal{G}\) _is an exact potential game that has at least one pure strategy NE. The optimal pure strategy NE of \(\mathcal{G}\) is the optimal solution of the proposed optimization problem._  
Proof.: Inspired by Reference [32], we firstly denote the potential function as follows:  
\[\Phi(a_{1},a_{2},...,a_{N})=R(a_{n},a_{-n}), \tag{20}\]where \(R(a_{n},a_{-n})=\sum\limits_{n\in\mathcal{N}}u_{n}\left(a_{1},a_{2},...,a_{N}\right)\) is the performance evaluation function of all UAVs, which is the same as Equation (14).  
The potential function indicates the performance of all UAVs. When UAV \(n\) changes its decision action from \(a_{n}\) to \(\bar{a}_{n}\), then the change in utility function of UAV \(n\) caused by this unilateral change is given by:  
\[U_{n}(\bar{a}_{n},a_{n})-U_{n}(a_{n},a_{n})=R(\bar{a}_{n},a_{-n})-R(\bar{a}_{n }=0,a_{-n})-\left[R(a_{n},a_{-n})-R(a_{n}=0,a_{-n})\right]. \tag{21}\]  
Note that \(a_{n}=0\) and \(\bar{a}_{n}=0\) all indicate the truth that UAV \(n\) takes no action in service, which means that this UAV \(n\) is broken and can not provide service. Therefore, the performance of UAV-assisted networks is actually the same in these two conditions, i.e.,  
\[R(a_{n}=0,a_{-n})=R(\bar{a}_{n}=0,a_{-n}). \tag{22}\]  
According to Equations (21) and (22), we can get the following equation:  
\[U_{n}(\bar{a}_{n},a_{I_{n}})-U_{n}(a_{n},a_{I_{n}})=R(\bar{a}_{n},a_{-n})-R(a_ {n},a_{-n})=\Phi(\bar{a}_{n},a_{-n})-\Phi(a_{n},a_{-n}). \tag{23}\]  
From Equation (23), we can draw the conclusion that the change in the utility function is exactly the same as the change in the potential function. Thus, according to the definition, the fact that our proposed game is an exact potential game is proved. Moreover, the potential game has the most important two properties.  
* The potential game has one pure strategy NE at least.
* Local or global maxima of potential function constitutes a pure strategy NE.  
Therefore, the UAV-assisted caching game has one NE at least. Moreover, the NE is a stable solution to the potential function, which is equal to the performance of all UAVs. It means that the optimal NE can make the performance of UAV-assisted networks largest. Thus, Theorem 1 is proved.