## 5 Experiments
### Evaluation of the Synthetic Dataset  
For the first set of experiments, we used to verify the superiority and rationality of our mechanism in terms of economic utility. Concretely, we first demonstrated how FedNow selects more efficient edge servers, and we verified the rationality of the incentive design. Then, we performed experiments on the synthetic dataset to evaluate the performance of the different strategies.  
Figure 2 shows the MECs and AMECs of different types of edge servers. Apparently, the MECs and AMECs of the type-1, -3, and -10 edge servers were both greater than 0, which means that the cost contribution of the type-1, -3, and -10 edge servers is positive, considering that the smaller the MEC/AMEC (i.e., smaller cost increment), the higher the value/efficiency of the corresponding edge servers. In contrast, the MEC and AMEC of the type-2, -4-9 edge servers are both less than 0, which means that they have negative effects on the platform's costs. In other words, when MEC/AMEC is less than zero, the smaller the MEC/AMEC, the lower the priority of selecting the corresponding type of edge server. In addition, since the training cost of type-10 edge servers is the largest, then, when calculating the efficiency score of the other types of edge servers, the type-10 edge servers are always in the incentivized set \(\mathcal{J}^{\prime}\), and the platform's optimal payment for them will just cover their training costs, making the type-10 edge servers always a critical payment type. When calculating the efficiency score of the type-10 edge servers, the critical payment type changes from 10 to 9, which may lead to a significant reduction in server payment costs. The above two factors explain the intense changes in efficiency scores between the type-9 and type-10 edge servers. Therefore, when the platform has complete information about the local training rounds of the edge servers, the preference order is \(\{1,3,10,5,2,4,6,7,8,9\}\), whereas the preference order under incomplete information is \(\{3,1,10,2,5,4,6,7,8,9\}\).  
Table 2 characterizes the performance differences of different strategies under their respective optimal incentive solutions. According to the control group, it is clear that the lack of data size leads to a significant accuracy loss, thus making the total costs higher than UB, even if the payment costs are the lowest. The cost-driven incentive strategy (CD) could incentivize more types of edge servers to participate, but this also leads to higher payment costs, whereas when reducing the incentivized types, the decrease in data size causes an increase in accuracy loss and may make the total cost \(U_{S}\) higher than UB. Moreover, in Table 2, we denote MEC as the efficiency-aware incentive mechanism with complete information about edge servers' local training rounds. Obviously, with the complete information about \(r_{j}\), MEC achieves the best performance on both accuracy loss and payment costs, thereby minimizing the total costs of the platform. Due to the influence of incomplete information, although the preference order and the optimal incentivized set of FedNow are different from MEC, FedNow can achieve similar performance to MEC and outperform the cost-driven strategy.  
Figure 3 shows that the platform provides positive contract items for the incentivized type of edge servers, and the critical type of edge servers (i.e., type-10 edge servers) only obtain 0 utility since the optimal rewards provided to type-10 edge servers by the platform just cover their training costs.  
Figure 3: The platform’s optimal contract for different types of edge servers.  
Figure 2: Marginal efficiency score.