traffic densityρ, and that in S3 changes steadily from 0.61 seconds to 3.45 seconds in this range. However, a sharp rise happens with respect to the mean of computational time per iteration in S2asρincreases from 0.54 to 0.66. Compared with S2, the mean of computational time per iteration in S3 is much less, especially under the circumstance that more traffic flow comes into the lane-changing system. According to the statistical analysis of the mean of computational time per iteration in Fig.5.11, the changing trend satisfies the exponential function model for S2 and S3 and a linear function model for S1, the residuals approximately fit the normal distribution. Indeed, the computational time can be shortened with a more powerful calculating device for all the strategies.
Another important issue is analyzing the performance between S2and S3. As men- tioned in the previous section, S3 generates optimal (i.e., case of CCE) or close-to- optimal (i.e., case of PCE or ICE) decisions for all the vehicles in a game, which results in an approximate lane-changing decision strategy to S2 in general. Thus, in terms of lane-changing efficacy, S3should be a little worse than S2based on the theoretical anal- ysis. That is to say, the close-to-optimal solutions occasionally exist in S3, and there is no such close-to-optimal case in S2. However, the experimental results after a long pe- riod in S3are not worse than S2as expected. On the contrary, S3perform slightly better in the efficacy of lane-changing maneuvers than S2. After the investigation, the result of each iteration is indeed in line with the theoretical analysis, i.e., S2 performs either equally or better. That is because the mechanism of generating optimal decisions in this lane-changing model is based on an immediate reward and S2 produces an optimal decision for each iteration instead of a close-to-optimal decision. However, there could be more available space for lane-changing maneuvers in the front of the lane-changing sector during the process so that S3 could achieve a higher long-term reward after a long period, despite a lower immediate reward in the current state. That results in a slightly better performance in S3 than S2 in the efficacy of lane-changing after a long period. As a matter of fact, the close-to-optimal solutions are almost the same as the optimal solutions, which has no significant influence on the efficacy of lane-changing maneuvers. Thus, during the experimental process, the performance of lane-changing maneuvers in S3is quite similar to that in S2in Table5.2, Fig.5.9and Fig.5.10.
Moreover, the results also suggest that S3has much less computational time than S2 in the same traffic situation, and the difference enlarges as the expected traffic density ρ increases. It is because that the space complexity in S3 is significantly reduced by decomposing a large game into several smaller games (refer to (5.11) and (5.16)). The number of players in a game is dynamic in this lane-changing system, and the difference in space complexity between the two strategies enlarges with an increasing number of players, according to (5.11) and (5.16). Asρgrows, the number of players of the games increases dramatically with more vehicles in the lane-changing sector. Hence, the gap of the space complexity between the strategies widens, which directly results in much more computational time in S2than S3. Due to the exponential property in (5.11) and (5.16), the changing trend of computation time in S2 and S3quite fit in an exponential function model as ρ grows in Fig. 5.11, which is reasonable since the computational complexity of finding a Nash equilibrium solution increases exponentially as the num- ber of players in a game increases. That is to say, with more incoming vehicles in the
lane-changing sector in a specific period, the possibility of a high number of players in a game increases so that the computational time could grow exponentially. In addition, S1 spends the least computation time on generating decisions among these strategies due to the sequential and fixed logic rules. Without a doubt, the computational time barely increases even whenρ grows to a large number since the rules are executed with a small computational complexity rather than finding a Nash equilibrium. According to Algorithm5.1, the complexity of S1 is also determined by the number of vehicles in the lane-changing sector. Asρ increases, the number of vehicles also increases, which means more rules must be evaluated in the system. Hence, the computational time increases slightly and linearly with a linear function model in Fig.5.11.
Based on the above analysis of lane-changing maneuvers and computational time, it is obvious to explain the regression models of the maximum incoming queues. As the expected traffic density increases, S1is the earliest to reach the control limit of regulat- ing traffic flow, so the maximum incoming queues increase exponentially and fit in an exponential model. For the other two strategies, the maximum incoming queues barely change when ρ is small since the level of traffic flow is far below the control limit of these two strategies. Hence, there is no significant change in the maximum incoming queues with the increasing number of incoming vehicles. In addition, there is an ex- plicit growing sign of the maximum incoming queues in Fig. 5.9 when ρ reaches the upper limit since there are few available gaps in the lane-changing sector and the bur- den of regulating traffic flow increases. Therefore, neither the linear nor the exponential function model is suitable for these two strategies, according to R-squared in Fig.5.11.
As for the results of other indicators such as incoming vehicles, passing vehicles and throughput, the detail and reason are discussed clearly in the previous section.
The proposed decomposition algorithm can also potentially reduce the computa- tional complexity in some applications where many players and the players have identi- fied connections with adjacent players. E.g., it can be used in perimeter-defence games where the defenders are constrained to move on the perimeter of the target area and can adjust the distance among each other to intercept the attack from the intruders.
Another example is in the industrial field, i.e., a smart grid where the household can ef- ficiently distribute extra electricity to adjacent households that require more energy. In detective robotics, an example is detecting the activity area of sharks, where the robotic agents move separately and cooperatively to detect the surrounding information. The decomposition algorithm can also be considered in other applications to improve com- putational efficiency.
However, there also exist some limitations to GTDA. Firstly, some assumptions are still ideal in this proposed model, such as the discrete space of traffic lanes, the discrete velocity of vehicles, and the smooth operating maneuvers. Additionally, if the players do not know the identified connections with adjacent players, it is limited to decomposing a large game into several smaller games. Finally, it is not so adequate to decompose a three-dimensional game since the interactions between players are quite complicated, e.g., controlling an unmanned aerial vehicle system.