## 3 System Model
### Incentive Model  
**Cost model for FEL**: The training costs of edge servers include computation, communication, and data collection. The computation and data-collection cost is proportional to the data-usage size \(s_{i}\)[30], while the communication cost is related to the size of the compressed uploaded model, which we assume to be a fixed model size, \(S_{d}\)[33]. For simplicity, we denote \(c_{i}^{p}\), \(c_{i}^{l}\). and \(c_{i}^{m}\) as the unit computation cost, unit data-collection cost, and unit communication cost of edge server \(i\), respectively. The total training costs can be expressed as  
\[c_{i}^{t}=(c_{i}^{p}+c_{i}^{l})s_{i}+c_{i}^{m}S_{d}. \tag{3}\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Variable** & **Physical Meanings** \\ \hline \(\mathcal{I}\), \(\mathcal{J}\), \(\mathcal{J}^{t}\) & The number set, type set, and incentivized type set of edge servers. \\ \(\Phi,\hat{\phi}_{j}\) & The set of contact items; contract items for type-\(j\) edge servers. \\ \(c_{j},r_{j}\) & The unit training costs and local training rounds of type-\(j\) edge servers. \\ \(\hat{s}_{j},p_{j}\) & The required data size and corresponding payments for type-\(j\) edge servers. \\ \(\beta,t_{i}\) & The upper limit of local training rounds; the longest allowable training time. \\ \(e_{j},\tilde{e}_{j}\) & The margin efficiency costs of type-\(j\) edge servers; the approx marginal efficiency costs. \\ \(P(\rho^{t}|r)\) & The conditional probability of edge servers with training strategy \(r\) choosing \(r^{t}\) training strategy. \\ \(\tilde{e}_{j}\) & The serial number corresponding to the type-\(j\) edge servers in score queue \(\tilde{e}_{j}\). \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Key notations used in this paper. Figure 1: Framework of the efficiency-aware incentive mechanism for FEL. When considering that edge servers may perform several local training rounds but only one communication process in each global epoch, we assumed that training time is the predominant factor in each global epoch and set the edge server's unit training costs as \(c_{i}\). **Contract formulation**: We distinguish edge servers based on their unit training costs, \(c_{i}\). Specifically, we define an edge server with unit training costs \(c_{i}\) as a type-\(i\) edge server and categorize all edge servers in set \(\mathcal{I}=\{1,\cdots,I\}\) into \(\mathcal{J}=\{1,\cdots,I\}\) types, where each type \(j\) contains \(I_{j}\) edge servers, and \(\sum_{j\in\mathcal{J}}I_{j}=|\mathcal{I}|\). We use a smaller index number to indicate smaller unit training costs, such that \(c_{1}<c_{2}<\cdots<c_{J}\). It should be noted that the total number of participating edge servers, \(|\mathcal{I}|\), is public information, and the platform can know the number, \(I_{j}\), of edge servers for each type based on market research or past training records [34], but for privacy considerations, each edge server will not report its privacy attributes, such as unit training costs, \(c_{j}\). In light of this, the platform requires the design of a set of contracts, \(\Phi\triangleq\{\phi_{j}\}_{j\in\mathcal{J}}\), to motivate edge server participation. Each contract item, \(\phi_{j}\triangleq(s_{j},p_{j})\), corresponds to edge server type, \(j\), expressing the relationship between the data size (required for local training) of each type-\(j\) edge server and the corresponding rewards. **Edge server utility**: We represent the CPU cycle frequency of the type-\(j\) edge server as \(f_{j}\). We assume the number of CPU cycles for edge servers to compute one sample of data in local training is \(\tau\). Therefore, the computation time for a type-\(j\) edge server to perform one round of local training is \(T_{j}^{p}=\frac{\tau s_{j}}{f_{j}}\).