## Appendix A
\] (A4)  
According to Formula (A4), for a feasible contract, \(s_{j}\geq s_{m}\) and \(p_{j}\geq p_{m}\) if and only if \(c_{j}\leq c_{m}\), \(\forall j\
eq m\in\{1,\cdots,J\}\). In other words, both \(p_{j}\) and \(s_{j}\) are negatively correlated with \(c_{j}\). So, Condition (ii) holds. 3. Condition (iii) \(p_{j+1}+c_{j}(s_{j}-s_{j+1})\leq p_{j}\leq p_{j+1}+c_{j+1}(s_{j}-s_{j+1})\): according to IC constraint, \(p_{j}-c_{j}s_{j}\geq p_{m}-c_{j}s_{m},\forall j,m\in\{1,\cdots,J\}\), for any two neighbor contract items, we have \[\left\{ \begin{array}{c}p_{j+1}-c_{j}s_{j+1}\leq p_{j}-c_{j}s_{j},\\ p_{j}-c_{j+1}s_{j}\leq p_{j+1}-c_{j+1}s_{j+1},\end{array} \right.\] (A5) which can be equivalent, as \[p_{j+1}+c_{j}(s_{j}-s_{j+1})\leq p_{j}\leq p_{j+1}+c_{j+1}(s_{j}-s_{j+1}).\] (A6) So, Condition (iii) holds. (ii) Sufficiency:  
We denote the subset of contract items that contain the last \(r\) data-reward contract items as \(C(r)\), i.e., \(C(r)=\big{\{}\big{(}s_{j},p_{j}\big{)}\mid j=J-r+1\big{\}}\), and use induction to complete the proof:  
We first prove that \(C(1)\) is feasible: according to condition (i), i.e., \(p_{j}-c_{j}s_{j}\geq 0\), the IR constraint is satisfied. Since there is only one contract item, the IC constraint does not need to be considered. Therefore, \(C(1)\) is feasible. Then, we show that if \(C(r)\) is feasible, then \(C(r+1)\) is also feasible. This corresponds to two aspects:  
1. For the new type \((J-r)\), the IR and IC constraints are satisfied, i.e., \[\left\{ \begin{array}{c}p_{(J-r)}-c_{(J-r)}s_{(J-r)}\geq 0,\\ p_{(J-r)}-c_{(J-r)}s_{(J-r)}\geq p_{r}-c_{r}s_{r},\\ \forall r=J-r+1,\cdots,J.\end{array} \right.\] (A7)
2. For the existing type \((J-r+1,\cdots,J)\), the IR and IC constraints are still satisfied when the type \(J-r\) exists, i.e., \[\left\{ \begin{array}{c}p_{r}-c_{r}s_{r}\geq 0,\\ p_{r}-c_{r}s_{r}\geq p_{r^{\prime}}-c_{r^{\prime}}s_{r^{\prime}},\\ \forall r,r^{\prime}\in(J-r,J-r+1,\cdots,J)\text{ and }r\
eq r^{\prime}. \end{array} \right.\] (A8)  
We first prove Formula (A7) as follows: since \(C(r)\) is feasible, the IC constraint is satisfied for the type \((J-r+1)\) client, i.e., \[p_{(J-r+1)}-c_{(J-r+1)}s_{(J-r+1)}\geq p_{r}-c_{(J-r+1)}s_{r},\forall r=J-r+1,\cdots,J.\] (A9)Moreover, according to Condition (iii), we have  
\[p_{(J-r)}\geq p_{(J-r+1)}+c_{(J-r)}\Big{(}s_{(J-r)}-s_{(J-r+1)}\Big{)}. \tag{100}\]  
By summing up Formulas (A9) and (A10), we have  
\[p_{(J^{\prime}-r+1)}+p_{(J-r)}-c_{(J-r+1)}s_{(J-r+1)}\geq p_{r}-c_{(J-r+1)}s_{r} +p_{(J-r+1)}+c_{(J-r)}\Big{(}s_{(J-r)}-s_{(J-r+1)}\Big{)} \tag{101}\]  
which can be rewritten as  
\[ \begin{split} p_{(J-r)}-c_{(J-r)}s_{(J-r)}&\geq p _{r}-c_{(J-r)}s_{(J-r+1)}+c_{(J-r+1)}\Big{(}s_{(J-r+1)}-s_{r}\Big{)}\\ &\geq p_{r}-c_{(J-r)}s_{(J-r+1)}\\ &\geq p_{r}-c_{(J-r)}s_{r},\forall r\in\{J-r+1,\cdots,J\}.\end{split}  \tag{102}\]  
where Formula (102) is based on condition (ii), i.e., \(s_{(J-r+1)}\geq s_{r},\forall r=J-r+1,\cdots,J\). Therefore, the IC constraints are satisfied for type \((J-r)\) edge servers. Additionally, we can prove that \(p_{r}-c_{r}s_{r}\geq p_{r}-c_{(J-r)}s_{r},\forall r\in\{J-r+1,\cdots,J\}\). To sum up, the IR constraint for type \((J-r)\) edge servers is satisfied. We then prove Formula (A8): obviously, proving Formula (A8) is equivalent to demonstrating the following:  
\[p_{r}-c_{r}s_{r}\geq p_{(J-r)}-c_{r}s_{(J-r)},\forall r\in\{J-r+1,\cdots,J\}. \tag{103}\]  
According to condition (iii), we have  
\[p_{(J-r)}\leq p_{(J-r+1)}+c_{(J-r+1)}\Big{(}s_{(J-r)}-s_{(J-r+1)}\Big{)}. \tag{104}\]  
Moreover, since the IC constraints are satisfied for type-\(r\) edge servers (\(\forall r\in\{J-r+1,\cdots,J\}\)), we have  
\[ \begin{split} p_{(J-r)}&+p_{(J-r+1)}-c_{r}s_{(J-r+1 )}\leq p_{(J-r+1)}\\ &+c_{(J-r+1)}\Big{(}s_{(J-r)}-s_{(J-r+1)}\Big{)}+p_{r}-c_{r}s_{r},\end{split}  \tag{105}\]  
which can be rewritten as  
\[