## 4 Optimal Design of Efficiency-Aware Incentive Mechanism
### Optimal Rewards  
By integrating the constraints of IR and IC, the cost optimization problem, \((\mathbb{P}_{1})\), of the incentive mechanism can be transformed as a non-collaborative cost optimization problem,\((\mathbb{P}_{2})\), which aims to motivate edge servers to make more data contributions (i.e., more training rounds and greater data size) at the lowest possible costs, i.e.,  
\[ \begin{split}(\mathbb{P}_{2})&\min_{\Phi_{rj}}\ U_{S}= \frac{1}{\sqrt{\sum_{j\in\mathcal{J}}r_{j}!j_{j}}}+\xi\sum_{j\in\mathcal{J}}r_{ j}I_{j}p_{j},\\ &\text{s.t.}\ \left\{\begin{array}{l}1\leq r_{j}\leq\overline{ \theta},\\ 0<s_{j}\leq\text{s}_{\max},\\ u_{j}\big{(}c_{j},\phi_{j}\big{)}\geq 0,\\ u_{j}(c_{j},\phi_{j}\big{)}\geq u_{m}(c_{m},\phi_{m},),\forall j\
eq m\in \mathcal{J}.\end{array}\right.\end{split}  \tag{9}\]  
Solving the above problem involves two challenges. First, due to the enormous number of \(\mu\)-IR and \(\mu\)-IC constraints (i.e., \(\mathcal{J}^{2}\)[18]), directly solving the problem is significantly complex. Second, the multi-dimensional decisions of edge servers (i.e., \(\phi_{j}\) and \(r_{j}\)) will affect their total payoffs and, consequently, influence the platform's optimal incentive strategy \((\phi_{j}^{*},\overline{\theta}^{*})\), leading to a nonconvex mixed-integer nonlinear programming problem. In order to overcome this complex problem, we first equivalently transform \(\mu\)-IR&\(\mu\)-IC constraints into a simplified set with fewer constraints (i.e, \(\mathcal{J}\), Lemma 1) and then derive the optimal rewards for different types of edge servers, thereby simplifying the platform's strategies, \((\phi_{j}^{*},\overline{\theta}^{*})\), into two-dimensional variable decisions that only involve \((s_{j}^{*},\overline{\theta}^{*})\). In a scenario of information asymmetry between the cloud server and edge servers, a contract is feasible if and only if the following conditions hold:  
1. \(p_{I}-c_{I}s_{I}\geq 0\);
2. \(p_{1}\geq p_{2}\geq\cdots\geq p_{I}\geq 0\) and \(s_{1}\geq\cdots\geq s_{I}\geq 0\);
3. \(p_{j+1}+c_{j}\big{(}s_{j}-s_{j+1}\big{)}\leq p_{j}\leq p_{j+1}+c_{j+1}\big{(}s _{j}-s_{j+1}\big{)}.\)  
Proof. : See Appendix A.  
Constraint _(i)_ is the equivalent conversion of the \(\mu\)-IR constraints. Constraints _(ii)_ and _(iii)_ jointly correspond to the \(\mu\)-IC constraints. Constraint _(i)_ ensures that if the edge server with the largest training cost can obtain non-negative utility by choosing the contract item intended for its type, then each type-_j_ edge server can also receive a non-negative payoff by choosing the contract item of type-_J_. This is because \(p_{I}-c_{I}s_{I}\geq p_{I}-c_{I}s_{I}\geq 0,\forall j\in\{1,\ldots,J\}\). Constraint _(ii)_ indicates that the platform should provide more rewards for the edge servers with a lower training cost and require more data in return. Constraint _(iii)_ specifies the relationship between any two adjacent contract items. Based on the equivalent relaxation of \(\mu\)-IR&\(\mu\)-IC, we can derive the optimal rewards for each type of edge server, shown as follows:  
For any data size, \(s=\big{\{}s_{j}\big{\}}_{j\in\mathcal{J}}\), and any edge server type, \(j\in\mathcal{J}\), the optimal choice for the platform is to select the rewards that satisfy the following criteria:  
\[\big{(}p_{j}\big{)}^{*}=\left\{ \begin{array}{l}c_{j}s_{j},\quad\text{if }j=J,\\ c_{j}s_{j}+\sum_{i=j+1}^{J}(c_{i}-c_{i-1})s_{i},\\ \quad\text{if }j\
eq J\text{ and }j=\{1,\cdots,(J-1)\}.\end{array} \right.\]  
Proof. : See Appendix B. When deriving the optimal rewards for each type of edge server, most of the existing works tend to prioritize incentivizing low-cost edge servers [20; 30; 41]. However, they often overlook the importance of resource efficiency, i.e., when the low-cost edge servers are providing more data for local model training, some expensive but computationally powerful edge servers may have already completed the local training, resulting in significant waiting times and resource waste for the entire participating group.