# A Learning Game-Based Approach to Task-Dependent Edge Resource Allocation
## 4 Incentives under Information Sharing Conditions
### Stackelberg Equilibrium Analysis under Information Sharing Conditions
Thus, Condition 3 is proven. Next, we prove Condition 4. From Condition 1 and 3, we have \(\bar{\mathbf{S}}=1,2,...,q\), where \(q\in[2,n]\). According to Inequality (26), we can conclude that \(q\leq h\). This implies that \(\beta_{q+1}<\frac{\sum_{j}^{q+1}\beta_{j}}{q}\), which means that when \(r=\bar{r}\), the derivative of utility function \(u_{q+1,m}\) with respect to \(r_{q+1}\) is \(\frac{\sum_{j}^{q+1}\beta_{j}}{q}-\beta_{q+1}>0\). This suggests that \(q=h\), and thus Condition 4 is proven. Therefore, the NE of the non-cooperative communication resources between the follower parties is unique. **Theorem 2**: _In the proposed multivariate Stackelberg game, there is a unique Stackelberg equilibrium between the user and the edge servers._  
Proof of Theorem 2.: By substituting Equations (20) and (24) into Equation (12), we obtain the following:  
\[U=\delta ln(\sigma_{1}\sqrt{R_{1}}+1)+(1-\delta)ln(\sigma_{2}R_{2}+1)-R_{1}-R_{ 2}, \tag{29}\]  
where \(\sigma_{1}=\sqrt{\frac{|M|-1}{2\gamma}}\ \sigma_{2}=\frac{(|M|-1)}{\sum_{m=1}^{M} \beta_{m}}\). The Hessian matrix of Equation (29) is obtained as follows:  
\[ \begin{bmatrix}\frac{\partial^{2}U}{\partial R_{1}^{2}}&\frac{\partial^{2}U}{ \partial R_{1}R_{2}}\\ \frac{\partial^{2}U}{\partial R_{2}R_{1}}&\frac{\partial^{2}U}{\partial R_{2} ^{2}}\end{bmatrix} = \begin{bmatrix}-\frac{\delta\sigma_{1}(\sigma_{1}+\frac{1} {2\sqrt{R_{1}}})}{2(\sigma_{1}R_{1}+\sqrt{R_{1}})^{2}}&0\\ 0&-\frac{(1-\delta)\sigma_{2}^{2}}{(\sigma_{2}R_{2}+1)^{2}}\end{bmatrix} .\]  
Since the eigenvalues of this matrix are all less than zero, the matrix is negative definite, implying that the original function is concave and hence possesses a maximum value. As the best response strategy of the square is unique, the value that maximizes \(U\) is also unique. Therefore, the equilibrium of this Stackelberg game is unique. According to the analysis above, this article develops a centralized algorithm that is capable of calculating the SE under information sharing conditions, the details of which are presented in Algorithm 1. This algorithm achieves an approximate equilibrium for the multivariable Stackelberg game. The approximation precision depends on \(\epsilon\). When \(\epsilon\) is smaller, the result is highly accurate but the algorithm converges slowly. When \(\epsilon\) is larger, the accuracy is low but the algorithm converges quickly. ```
0: initialization \(D,\mathbf{\alpha},\mathbf{\beta},\mathbf{\kappa},I,\epsilon,\mathbf{R},\mathbf{x_{i}}\)
0: optimal strategy \(\mathbf{R}^{[k]},\mathbf{x}_{i}^{[k]}\)
1:while\(||\mathbf{R}^{[k]}-\mathbf{R}^{[k-1]}||>\epsilon\)do
2:while\(||\mathbf{x}_{i}^{[k]}-\mathbf{x}_{i}^{[k-1]}||>\epsilon\)do
3:for\(i=1,2,...,I\)do
4: calculation of the followers' utility \(u_{1,m}\), \(u_{2,m}\) by equations (9) and (10)
5: save the strategy that maximizes \(u_{1,m}\) and \(u_{2,m}\) as \(\mathbf{x}_{i}^{[k]}\)
6:endfor
7: calculation of the leader's utility \(U\) by equation (12)
8: save the strategy that maximizes \(U\) as \(\mathbf{R}^{[k]}\)
9:endwhile
10:endwhile
```  
**Algorithm 1** Coordinate Alternation Method