# A QoE-Oriented Uplink Allocation for Multi-UAV Video Streaming
## 4 Potential Game Based Uplink Allocation  
Compared with traditional optimization algorithms, game theory is another efficient way to solve resource allocation. The ordinary optimization algorithm is to find the optimal solution by solving the optimization expression, while the method of game theory is to find Nash equilibrium by iterations. Sometimes, there's no optimal solution in the former one, or the solving will take much time. Thus, this paper adopts the later one, the game theory, to solve the uplink allocation problem in multi-UAV video transmission. Each UAV is regarded as a game player to compete with others. Generally speaking, game methods can be divided into cooperative game and non-cooperative game. In practice, there is no absolute boundary between the two. From the perspective of a single player, each UAV intends to occupy the bandwidth as much as possible, but it is difficult to achieve due to the the limited total bandwidth the operator can provide. In addition, finally, the UCDC needs to improve the overall QoE of all videos. In this process, UAVs have a rational cooperation trend as well as a non-cooperative and selfish competition. Therefore, the Potential Game method can be used to build the model. On this basis, this paper believes that the behavior of game players is not completely independent, and the choice of game strategies depends on a kind of probability information outside the game process. Thus, the players can partially cooperate to change their strategies. Compared with the completely independent and non-cooperative Nash equilibrium, we hope to achieve the correlation equilibrium in this way and further improve the performance of the algorithm. The game can be expressed as \(G=[\Omega,\{\mathcal{K}_{i}\}_{i\in\Omega},\{\mathcal{U}_{i}\}_{i\in\Omega}]\), where \(\Omega\)= \(\{1,2,...,N\}\) is the set of players, \(\mathcal{K}_{i}\in\textit{Cband}\) represents the set of the strategies of the \(i\)-th player, and \(\mathcal{U}_{i}\) is the corresponding utility set. Thus, we have the pure strategy of player \(i\) as \(r_{i}\in\mathcal{K}_{i}\), and have \(r_{-i}\in\mathcal{K}_{1}\times\cdots\times\mathcal{K}_{i-1}\times\mathcal{K}_ {i+1}\times\cdots\times\mathcal{K}_{N}\) as the strategy of all the players except \(i\), where \(\times\) denotes Cartesian product. We hope that, by less information interaction, each player can choose a reasonable strategy without breaking the equilibrium. In this section, we construct a potential function as \(I=\sum\limits_{i=1}^{N}u_{i}\), and the problem can be presented as:\[ \begin{split}\text{max}&\quad I=\Omega\cdot\sum_{i=1}^{N} \{a+b\cdot\sqrt{\frac{r_{i}}{c}}(1-\frac{c}{r_{i}})\}-\eta\cdot\sum_{i=1}^{N}r _{i}+N\cdot\Lambda,\\ s.t.&\quad R_{i}^{\text{min}}\leq r_{i}\leq R_{i}^{ \text{max}},\\ &\quad 0\leq\sum_{i=1}^{N}r_{i}\leq Cband.\end{split}  \tag{6}\]  
**Theorem 1**. : _Although, as a participant in the game, each UAV player can change its video transmission rate strategy without cooperation, the whole game process takes the total utility of UCDC as the goal, which makes each player conduct rational cooperation. Therefore, this game process is a complete potential game._  
Proof. : For the \(i\)-th UAV, we assume that it has another rate strategy to choose, expressed as \(r_{i}^{\prime}\). Because the potential function \(I\) is the sum of the single utilities, we can get  
\[\Delta I=I(r_{i})-I(r_{i}^{\prime})=u(r_{i})-u(r_{i}^{\prime})=\Delta u. \tag{7}\]  
The change of potential function is equal to the change of utility function of each player. According to [52], the game process \(G\) is a complete potential game, and the potential function \(I\) is a complete potential function. According to Theorem 1, in the process of rate allocation, as the participant of the game, a single UAV's rate updating will change its own utility, while the changing value is just the same with the total utility's variation. That means the application scenarios presented in this paper satisfies the definition of potential game.