## 2 System Model
### Radar Signal Model  
In the \(n\)th time slot, the \(K^{\prime}\) downlink ISAC streams transmitted by vehicle \(k\) can be expressed as \(\mathbf{s}_{k,n}=\left[s_{1,n},\cdots,s_{k^{\prime},n},\cdots,s_{K^{\prime},n }\right]\in\mathbb{C}^{K^{\prime}\times 1}\), where \(\mathbb{E}\left[\left|s_{k^{\prime},n}\right|^{2}\right]=1\)[34]. After beamforming, the transmitted signal can be expressed as  
\[\mathbf{\tilde{s}}_{k,n}=\mathbf{F}_{k,n}\mathbf{s}_{k,n}\in\mathbb{C}^{N_{t} \times 1}, \tag{1}\]  
where \(\mathbf{F}_{k,n}\in\mathbb{C}^{N_{t}\times K^{\prime}}\) represents the transmit beamforming matrix, and \(N_{t}\) denotes the number of transmit antennas. The echo signal reflected by the target vehicle is input into a  
Figure 2: The network model of the ISAC beam allocation scheme (IMSBA) based on MAPPO and a Stackelberg game.  
radar matched filter to eliminate the effects of a Doppler shift and delay. The processed echo signal of the target vehicle \(k^{\prime}\) can be expressed as [35]  
\[\begin{split} r_{k^{\prime},n}=&\zeta\sqrt{p_{k^{ \prime},n}}\sqrt{G}\phi_{k^{\prime},n}\mathbf{g}^{H}(\tilde{\theta}_{k^{\prime},n })\mathbf{b}\left(\theta_{k^{\prime},n}\right)\mathbf{a}^{H}(\theta_{k^{\prime },n})\\ &\cdot\mathbf{f}(\tilde{\theta}_{k^{\prime},n})s_{k^{\prime},n}+ \mathbf{g}^{H}(\tilde{\theta}_{k^{\prime},n})\left[\sum_{m=1}^{M}\psi_{m,k} \zeta\sqrt{p_{m,n}}\right.\\ &\cdot\mathbf{H}_{k,k}\mathbf{f}_{m}q_{m,n}+\sum_{i=1,i\
eq k}^{ K}\sum_{j=1}^{K^{\prime}}\psi_{i,k}\xi\sqrt{p_{j,n}}\mathbf{H}_{i,k}\mathbf{f}_{j,n}\\ &+\left.\sum_{v=1,v\
eq k^{\prime}}^{K^{\prime}}\zeta\sqrt{p_{v, n}}\mathbf{H}_{v}\mathbf{f}_{v}s_{v,n}\right]+z_{r},\end{split} \tag{2}\]  
where \(\zeta=\sqrt{N_{t}N_{r}}\) represents the antenna array gain; \(p_{k^{\prime},n}\) denotes the transmit power of the beam directed at target vehicle \(k^{\prime}\); \(G\) represents the matched filter gain; \(\mathbf{g}^{H}(\tilde{\theta}_{k^{\prime},n})\) represents the receive beamforming vector of the echo signal; \(\mathbf{a}\) and \(\mathbf{b}\) are the transmit and receive steering vectors, respectively; \(\theta_{k^{\prime},n}\) represents the true angle of target vehicle \(k^{\prime}\) relative to vehicle \(k\); \(\mathbf{f}(\tilde{\theta}_{k^{\prime},n})=\mathbf{a}(\tilde{\theta}_{k^{\prime },n})\) denotes the \(k^{\prime}\)th column of \(\mathbf{F}_{k,n}\); \(\tilde{\theta}_{k^{\prime},n}\) represents the beam direction selected through multi-agent reinforcement learning; \(\psi_{m,k}=1\) indicates the same frequency band used by vehicle \(m\) and vehicle \(k\), and \(\psi_{m,k}=0\) indicates different frequency bands, with a similar meaning for \(\psi_{i,k}\); \(p_{m,n}\) denotes the transmit power of the beam directed at vehicle \(m\) by the RSU; \(\mathbf{H}_{k,k}\) represents the channel matrix from the RSU to vehicle \(k\), \(\mathbf{H}_{i,k}\) represents the channel matrix from vehicle \(i\) to vehicle \(k\), \(H_{v}\) represents the reflection channel matrix of target vehicle \(v\); \(z_{r}\) represents complex additive white Gaussian noise with zero mean and variance \(\sigma_{r}^{2}\); \(q_{m,n}\) represents the downlink symbol transmitted by the RSU, \(\mathbb{E}\left[\left|q_{m,n}\right|^{2}\right]=1\); the reflection coefficient \(\phi_{k^{\prime},n}=\varepsilon_{k^{\prime},n}\left(2d_{k^{\prime},n}\right)^ {-1}\).  
The variance of sensing noise is directly proportional to the reciprocal of the sensing signal-to-interference-plus-noise ratio (SINR), which can be expressed as  
\[\omega_{k^{\prime},n}^{r}=\frac{\zeta^{2}p_{k^{\prime},n}G\big{|}\phi_{k^{ \prime},n}\mathbf{g}^{H}(\tilde{\theta}_{k^{\prime},n})\mathbf{b}\left(\theta _{k^{\prime},n}\right)\mathbf{a}^{H}(\theta_{k^{\prime},n})\mathbf{f}\big{|}^ {2}}{\sigma_{r}^{2}+I_{2V}+I_{V2V}+I_{Echo}}, \tag{3}\]  
where \(I_{2V}\), \(I_{V2V}\), and \(I_{Echo}\) represent the interference power of I2V, V2V, and echo signals, respectively. The sensing SINR is affected by transmit power, beam direction, and frequency band allocation.