## 5 Numerical Studies  
Simulation studies are conducted to illustrate the performance of the privacy-preserving incentive mechanism. First, we show the correctness of the proposed incentive mechanism. Then, the computational efficiency of the collaborative routing scheme and the route validation scheme is analyzed under different privacy protection settings. The MPC protocols are implemented in SCALE-MAMBA ([[https://github.com/KULeuven-COSIC/SCALE-MAMBA](https://github.com/KULeuven-COSIC/SCALE-MAMBA)]([https://github.com/KULeuven-COSIC/SCALE-MAMBA](https://github.com/KULeuven-COSIC/SCALE-MAMBA)) (accessed on 9 January 2024)) and MP-SPDZ ([[https://github.com/data6/MP-SPDZ](https://github.com/data6/MP-SPDZ)]([https://github.com/data6/MP-SPDZ](https://github.com/data6/MP-SPDZ)) (accessed on 9 January 2024)), and the route validation scheme is implemented using Python. To validate the correctness of the MPC protocol, the example network (see Figure 6) in [21] is used to illustrate that the MPC protocol can calculate the same incentives with proper settings. Twenty vehicles depart from node 13 to node 16 in the network. There are three local destinations, nodes 21, 22, and 23, and four alternative routes connecting nodes 13 and 16, as listed in Table 1 and illustrated in Figure 6. The desired flows and the corresponding travel costs of the four routes are provided by the route flow assignment model in [21]. The values of time () for the individual vehicles are shown in Table 2. With these inputs, the implemented MPC scheme can generate the same vehicle route assignment results \(\eta_{i},\ i=1,\ \ldots,\ 20\) (the id of the route that vehicle \(i\) should take) as in [21]. In terms of the incentives, when the integer representation precision of the MPC protocol is set as 15-bit fixed point numbers with a 5-bit decimal part, the output incentives \(p_{i}^{(5,\ 15)}\) are different from the ones calculated in [21] (although \(\sum_{i=1}^{20}p_{i}^{(5,\ 15)}=0\) holds as \(\sum_{i=1}^{20}p_{i}=0\)). If the precision is increased to 31-bit fixed point numbers with a 16-bit decimal part, the output incentives \(p_{i}^{(16,\ 31)}=p_{i},\ i=1,\ \ldots,\ 20\) (i.e., the implemented MPC scheme can calculate the same incentives as in [21] under this setting). The total data exchanged among the vehicles when executing the MPC scheme increase from 135.478 MB to 180.649 MB in this case, which is acceptable in the real world. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Route ID** & **Links** & **Flow** & **Travel Cost** \\ \hline
1 & 1-2-10 & 8 & 289.961 \\
2 & 1-2-4-9 & 2 & 290.086 \\
3 & 1-3-7-9 & 1 & 331.512 \\
4 & 1-3-8 & 9 & 331.500 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Alternative routes with desired route flows and travel costs. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Vehicle ID** & \(\lambda_{i}\) & \(\eta_{i}\) & \(a_{i}\) & \(p_{i}\) & \(p_{i}^{(0,15)}\) & \(p_{i}^{(16,31)}\) \\ \hline
1 & 0.80  
Next, we validate the computational efficiency of the collaborative routing scheme. Due to our modifications (fake vehicles and routes) to the incentive mechanism, the compiled MPC protocol can be executed by fewer vehicles than the predefined number of parties, \(N\). Figure 7 shows how the computational time of different stages changes with the number of participating vehicles (\(n_{s}\)) given a compiled 8-party MPC protocol (\(N=8\)). The computational times of the input and output stages increase with \(n_{s}\) because fake vehicles' inputs are pre-determined and do not require outputs. The computation time for the computation stage slightly increases as the number of fake vehicles decreases. Figure 8 shows how the computational time of the output stage changes with \(N\) and \(n_{s}\) (\(n_{s}\leq N\)). It increases significantly with an increase in the number of MPC parties. To enable real-time implementation, the vehicle group size should be limited to 10. When more than 10 vehicles want to participate, they can be assigned into multiple vehicle groups with the same OD.