## 5 Results and Discussion
### Performance Comparison of the Proposed and Heuristic Rule-Based Algorithm  
In this subsection, we verified the adaptability of our proposed algorithm by comparing it with the heuristic rule-based algorithm using the same scenario in Section 5.2. The traditional ruled-based or model-based decision control approaches were capable of generating stable decision results, accurate control curves, and were easy to implement. For example, the vehicles coordination of negotiating the access in an intersection was reformulated as a virtual platoon control problem in [9]. According to the passage sequence, the HDV acted as the leader and the two CAVs kept the desired following distance. Figure 10 shows the results using a virtual platoon. The HDV traveled at the desired speed of 10 m/s. CAV\({}_{1}\) adjusted its speed to keep the desired distance from the HDV and CAV\({}_{2}\) followed CAV\({}_{1}\).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Metrics**} & \multicolumn{3}{c}{**Horizon Time**} \\ \cline{2-4}  & \(T_{h}=1\) s & \(T_{h}=3\) s & \(T_{h}=5\) s \\ \hline Collision rate & 0.03 & 0 & 0 \\ \hline Average Speed (m/s) & 9.10 & 9.91 & 8.67 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Performance comparison for the different forward predictive horizons.  
Figure 9: Comparison of the episode reward in the training for the different forward predictive horizons.  
Although the vehicles could cooperatively and efficiently cross the intersections under the ideal traffic conditions using the heuristic rule-based model, some of the idealized condition assumptions were difficult to satisfy in many cases. Generally speaking, the traffic scenarios involving the HDVs were more complex and uncertain for the uncontrolled human driving. For example, some drivers were conservative and traveled at a low speed for fear of getting scratched in intersections. The speed and position curves of all the vehicles when the HDV traveled at a low speed using a heuristic rule-based algorithm are shown as Figure 11. In order to form a virtual platoon, the speed of both CAVs converged to that of the leading HDV. Practically, the CAV\({}_{2}\) was not in the same lane as the HDV, so it did not need to keep up with the speed of the HDV, which could greatly enhance the road efficiency. The results demonstrated that the heuristic rule-based approaches, while easy to implement, were not adaptive to some special cases.  
A learning-based approach enabled more efficient decision-making. The results are presented in Figure 12 using the proposed algorithm when the HDV traveled at a low speed. Based on the trajectory prediction of the other vehicles, CAV\({}_{2}\) predicted that the slow speed of the HDV would not cause a collision. [PERSON]\({}_{2}\) made the decision to breaks traffic rules and pass first, since the reward for following the traffic rules was smaller than the penalty for crossing the intersection at a low speed. Thus, if no collision was predicted, the HDV only affected the decision of CAV\({}_{1}\) behind it in the same lane but not CAV\({}_{2}\)'s  
Figure 11: The speed and position curves of all the vehicles when the HDV traveled at a low speed using the heuristic rule-based algorithm; (**a**) the speed, (**b**) the position.  
Figure 10: The speed and position curves of all the vehicles in the ideal traffic using the heuristic rule-based algorithm; (**a**) the speed, (**b**) the position.  
in other lane. As shown in Figure 12b, after the HDV crossed the stop line, that is, when its position was less than 0, the CAV began to accelerate. This accounted for the fact that the proposed approach could deal with some special cases, which were more adaptive and generalized.