## 3 Beam Allocation Algorithm Based on MAPPO and Stackelberg Game
### Beam Allocation Algorithm Based on MAPPO
To further assess the sensing performance, Figure 13 plots the CDF of position sensing error versus the number of antennas, with all algorithms using a reward weight of 1. This error refers to the root mean square error (RMSE) between the true position and the sensed estimated position. In Figure 13, the horizontal axis range is smaller, while in Figure 14, the horizontal axis range is larger. It can be seen that for smaller position sensing errors, more antennas result in a higher CDF. For example, when the number of antennas is 128, the probability of the position sensing error being less than \(2.5\times 10^{-3}\) m is 28%, which is higher than other antenna numbers. However, for larger position sensing errors, fewer antennas result in a higher CDF. This is because the IMSBA algorithm requires target vehicle switching, which can lead to beam misalignment after switching. With more antennas, the beam width is narrower, and beam gain drops more significantly when misaligned, leading to larger position sensing errors, as shown in Figure 4:5 In the stable condition without target vehicle switching, more antennas result in greater antenna array gain and smaller position sensing errors, as shown in Figure 4:6 Figure 15 compares the CDF of the position sensing error for the proposed IMSBA algorithm with other benchmark algorithms, with all algorithms using a reward weight of 1 and 64 antennas. It can be seen that, regardless of the error magnitude, the IMSBA algorithm achieves the highest CDF, with a probability of 91.6% for the position sensing error being less than 0.2 m. The IMSBA w/o switch algorithm, constrained by a higher path loss, results in larger position sensing errors. In the

Figure 4:7 I2V achievable rate versus weight. Figure 4:8 Curve of cumulative reward versus number of antennas. Figure 16 illustrates the performance comparison of the IMSBA algorithm under different numbers of I2V vehicles, using experimental conditions with a reward weight of 1 and 64 transmit-receive antennas. The values in the figure represent results after algorithm convergence. As established earlier, I2V employs orthogonal frequency bands for communication, with the number of I2V vehicles equating to the number of orthogonal frequency bands available. Consequently, as the number of I2V vehicles increases, V2V can reuse more orthogonal frequency bands, further reducing mutual interference between V2V pairs and between I2V and V2V. As shown in Figure 16, the overall cumulative reward increases with the number of I2V vehicles, with V2V's energy efficiency



range of the vehicle. The IMSBA architecture has good modularization characteristics, which makes it easy to be deployed on edge computing platforms. Although it still needs to be validated under real-road conditions, this scheme has shown the potential to become the foundational framework for intelligent beam management in future ISAC systems. ## 5 Conclusions

In a multi-beam communication scenario with coexisting I2V and V2V communications and limited spectrum resources, this paper proposes an ISAC beam allocation scheme (IMSBA) based on multi-agent reinforcement learning to address interference suppression and maximize overall ISAC performance. To mitigate environmental instability, the scheme introduces the multi-agent reinforcement learning algorithm MAPPO, employing a centralized training and distributed execution architecture. It optimizes V2V beam direction and frequency allocation using sensing information from target vehicles. Additionally, to reduce the dimensionality of the action space, the paper separately optimizes I2V and V2V joint power allocation using the Stackelberg game method. Finally, vehicles achieve sensing domain sharing through V2V communication, utilizing sensing information from other vehicles to switch target vehicles and reduce the path loss caused by communication distance or obstruction.