## 4 Optimal Design of Efficiency-Aware Incentive Mechanism
### Efficiency Score Function Design
For any local training round \(r_{j}\in\mathcal{R}=\{1,2,\cdots,\overline{\theta}\}\), the platform can calculate the mapping probability \(P(r_{j}^{\prime}|r_{j})\) as  
\[P(r_{j}^{\prime}|r_{j})=\frac{exp(e\frac{-|r_{j}-r_{j}^{\prime}|^{\frac{1}{2} }}{2\Delta q^{\frac{1}{2}}})}{\sum_{r^{\prime}\in\mathcal{R}}exp(e\frac{-|r_{ j}-r^{\prime}|^{\frac{1}{2}}}{2\Delta q^{\frac{1}{2}}})}. \tag{14}\]  
For the mapping probability \(P(r_{j}^{\prime}|r_{j})\), the platform can use the expected rounds, \(\mathbb{E}(r_{j})\), to approximate the edge servers' true local training rounds \(r_{j}\), i.e., given \(P(r_{j}^{\prime}|r_{j})\), \(r^{\prime}\in\mathcal{R}\), the expected local training rounds can be calculated by  
\[\mathbb{E}(r_{j})=\sum_{r^{\prime}\in\mathcal{R}}r_{j}^{\prime}P(r_{j}^{\prime }|r_{j})=\sum_{r^{\prime}\in\mathcal{R}}r_{j}^{\prime}\frac{exp(e\frac{-|r_{ j}-r_{j}^{\prime}|^{\frac{1}{2}}}{2\Delta q^{\frac{1}{2}}})}{\sum_{r^{\prime} \in\mathcal{R}}exp(e\frac{-|r_{j}-r_{j}^{\prime}|^{\frac{1}{2}}}{2\Delta q^{ \frac{1}{2}}})}. \tag{15}\]  
Therefore, the approximate marginal efficiency cost (AMEC) can be expressed as  
\[ \begin{split}\tilde{e}_{j}&=\frac{1}{\sqrt{\sum_{j \in\mathcal{J}}\mathbb{E}(r_{j})I_{j}s_{j}}}+\xi\sum_{j\in\mathcal{J}}\mathbb{ E}(r_{j})I_{j}p_{j}-\\ &\frac{1}{\sqrt{\sum_{j\in\mathcal{J}/\{j\}}\mathbb{E}(r_{j})I_{ j}s_{j}}}-\xi\sum_{j\in\mathcal{J}/\{j\}}\mathbb{E}(r_{j})I_{j}p_{j}^{\prime}. \end{split}  \tag{16}\]  
The platform sorts each type of edge server according to their AMEC in ascending order. A smaller AMEC indicates a higher margin utility and higher quality. The platform will select edge servers according to their AMEC from low to high until finding a threshold type, \(x^{\prime}\), such that \(U_{S}(\mathcal{J}^{\prime})\leq\overline{U_{S}}\) and \(U_{S}(\mathcal{J}^{\prime}\cup\{\overline{e}_{x^{\prime}+1}\})>\overline{U_{S}}\), where \(\mathcal{J}^{\prime}\) represents the set of selected edge servers (Assuming there are only five types of edge servers: {1, 2, 3, 4, and 5} and \(x^{\prime}=4\). Since the order sorted by the edge servers' AMEC may not follow the same order as their contract types, e.g., AMEC: {2, 1, 4, 3, and 5}, then \(\overline{e}_{x^{\prime}}\triangleq 3\), \(\overline{e}_{x^{\prime}+1}\triangleq 5\), and \((\overline{e}_{x^{\prime}}+1)\triangleq 4\)). For the selected set \(\mathcal{J}^{\prime}\), we denote \(J^{\prime}=|\mathcal{J}^{\prime}|\) and re-index the edge servers' type in \(\mathcal{J}^{\prime}\) by \(\{j\}_{j\in\{1,\cdots,J^{\prime}\}}\) in ascending order of training cost, \(c_{j}\). However, since the platform's cost, \(U_{S}\), is affected by both \(r_{j}\) and \(s_{j}\), even if the incentivized set \(\mathcal{J}^{\prime}\) is determined, the platform may be able to obtain a better feasible solution by adjusting its data requirements \(s_{j},\forall j\in\mathcal{J}^{\prime}\). Based on the above analysis, we obtain  
**Proposition 1**. : _Given a set of contract items \(\phi_{j}=(p_{j},s_{j})\) and \(\overline{  
_if the threshold type_ \(x^{\prime}\) _exists, there exists an optimal type_ \(x^{*}\) _and required data size_ \(s_{j}^{*}\) _such that_ \(U_{S}^{*}(s_{j}^{*},\mathcal{J}^{*})\leq U_{S}^{\prime}(s_{j},\mathcal{J}^{\prime})\)_, where_ \(\mathcal{J}^{*}\) _is the platform's optimal incentitized type set._  
See Appendix D.  
Proposition 1(_i_) shows that, given \(s_{j}\), the optimal choice for the platform is to incentivize the edge servers that can provide more total data at lower payment costs. In addition, the platform only provides positive contract items for the incentivized type of edge servers, whereas the non-incentivized edge servers will receive zero contract items. Proposition 1(_ii_) suggests that if a feasible solution exists, the platform can adjust its incentive strategy (i.e., required data size \(s_{j}\)), which may lead to a better solution that minimizes the platform's costs.