# A Learning Game-Based Approach to Task-Dependent Edge Resource Allocation
## 4 Incentives under Information Sharing Conditions
### Stackelberg Equilibrium Analysis under Information Sharing Conditions  
In this paper, we first analyze the existence and uniqueness of the SE under information sharing conditions, in which each participant in the game can obtain all the information from other participants, such as the preference factor \(\delta\) reflecting the user's preference for communication and computing resources, and the inconvenience factors \(\alpha_{m}\), \(\beta_{m}\) reflecting the ES \(m\)'s reception and processing tasks. The objective of the Stackelberg game model proposed by the user and ESs is to find a unique SE that maximizes user utility. In this equilibrium, neither the user nor the ESs have the motivation to unilaterally change their strategy. The SE is defined in this paper as follows:  
**Definition 1**. : _If the leader's strategy is denoted by \(\mathbf{R}=\left\{R_{1},R_{2}\right\}\) and the followers' strategy is denoted by \(\mathbf{Z_{m}}=\left\{f_{m},r_{m}\right\}\), then there exist optimal strategies \(\mathbf{R}^{*}=\left\{R^{*}_{1},R^{*}_{2}\right\}\) and \(\mathbf{Z}^{*}=\left\{\mathbf{Z}^{*}_{1},\mathbf{Z}^{*}_{2},...,\mathbf{Z}^{* }_{m}\right\}\) that are the respective Stacklberg equilibria for the leader and the follower if the following conditions are satisfied:_  
\[ \begin{split}\forall\mathbf{R},U(\mathbf{R}^{*},\mathbf{Z}^{*}) \geq U(\mathbf{R},\mathbf{Z}^{*}),\\ \forall\mathbf{Z},u_{m}(\mathbf{R}^{*},\mathbf{Z}^{*})\geq u_{m}( \mathbf{R}^{*},\mathbf{Z}).\end{split} \]  
Next, the Nash equilibrium (NE) of the non-cooperative game between the follower parties is analyzed, first to analyze the follower's best response for the computational resources, as well as to differentiate Equation (9) with respect to \(f_{m}\) to obtain the following:  
\[\frac{\partial u_{1,m}}{\partial f_{m}}=\frac{\sum_{n=1,n\
eq m}^{M}f_{n}}{( \sum_{m=1}^{M}f_{m})^{2}}R_{1}-2\alpha_{m}D\eta_{m}\kappa_{m}f_{m}, \tag{15}\]\[\frac{\partial^{2}u_{1,m}}{\partial f_{m}^{2}}=-2\frac{(\sum_{n=1,n\
eq m}^{M}f_{n} )(\sum_{m=1}^{M}f_{m})}{(\sum_{m=1}^{M}f_{m})^{4}}R_{1}-2\alpha_{m}D\eta_{m} \kappa_{m}<0. \tag{16}\]  
Clearly, utility function \(u_{1,m}\) with respect to \(f_{m}\) is concave; as such, there exists a maximum value for \(u_{1,m}\). This implies that a non-cooperative game concerning the allocation of computational resources has an NE. Therefore, by making \(\frac{\partial u_{1,m}}{\partial f_{m}}=0\), we can obtain the following:  
\[\frac{\sum_{n=1,n\
eq m}^{M}f_{n}}{(\sum_{m=1}^{M}f_{m})^{2}}R_{1}-2\alpha_{m} D\eta_{m}\kappa_{m}f_{m}=0. \tag{17}\]  
By summing both sides of Equation (17), we obtain  
\[\sum_{m=1}^{M}(\frac{\sum_{n=1,n\
eq m}^{M}f_{n}}{(\sum_{m=1}^{M}f_{m})^{2}}R_ {1})=2\sum_{m=1}^{M}\gamma_{m}f_{m}, \tag{18}\]  
where \(\gamma_{m}=\alpha_{m}D\eta_{m}\kappa_{m}\), and, since the difference between any two \(\gamma_{m}\) is extremely small, the right side of Equation (18) can be approximated as \(2\hat{\gamma}\sum_{m=1}^{M}f_{m}\). As such, Equation (18) is further computed as follows:  
\[\frac{|M|-1}{\sum_{m=1}^{M}f_{m}}R_{1}=2\hat{\gamma}\sum_{m=1}^{M}f_{m}. \tag{19}\]  
Thus, it can be derived that  
\[\sqrt{\frac{|M|-1}{2\hat{\gamma}}R_{1}}=\sum_{m=1}^{M}f_{m}. \tag{20}\]  
By substituting Equation (20) back into Equation (17), we obtain the closed-form solution for the optimal response of \(f_{m}\):  
\[f_{m}= \begin{cases}0&\text{if }\mid\text{M}\mid\text{<2}\\ \sqrt{\frac{(|M|-1)R_{1}\hat{\gamma}}{2((|M|-1)\gamma_{m}+\hat{\gamma})^{2}}} &\text{otherwise.}\end{cases}  \tag{21}\]  
Given user's strategy \(R_{1}\), each ES always has its own optimal response \(f_{m}\). Due to the concavity of the utility function of the ES \(u_{1,m}\), the optimal response is unique. Therefore, the NE of the non-cooperative game between the ESs regarding computational resources is also unique.