## 6 Simulation Results
The PB algorithm is also a SNR balancing constrained power control iterative method which iteratively searches for decentralized transmit power level updated from the \(l\)-th iteration to the \((l+1)\)-th iteration. Let \(\gamma_{m}^{tar}\) denote the target SNR for the \(m\)-th SSN to maintain a certain QoS requirement, for \(m\in\mathcal{N}\). In this simulation, the target SNR can be set to \(\gamma_{m}^{tar}=7\) dB. Therefore, the PB algorithm iteratively updates the transmit power for the \(m\)-th SSN according to [35]:  
\[p_{m}^{(l+1)}=\min\left\{p_{m}^{\max},\left(\frac{\gamma_{m}^{tar}}{\gamma_{m} ^{(l)}}\right)\cdot p_{m}^{(l)}\right\} \tag{49}\]  
First the optimal transmit power is compared between PB algorithm with 250 iterations and our proposed distributed power control framework via the noncooperation and cooperation scenarios under the slot \(\tau_{rp}=12\) ms with the evolution of discount factor \(r\), as exhibited in Figure 8. It is shown that an increased discount factor from 0.1 to 0.9 will enhance the optimal transmit power of selected SSNs (i.e., IT-1, MT-2, MT-5, and MT-7) under the proposed OTPA algorithm for both the NoCoPC problem and the CoPC problem within the slot \(\tau_{rp}=12\) ms. This is due to the fact that the discount factor \(r\) has an affect on the optimal transmit power of selected SSNs via Equations (28) and (34). Specifically, the optimal transmit power of selected SSNs grows in proportion to discount factor \(r\). It is also revealed that the optimal transmit power of selected SSNs under a given discount factor satisfy the average power constraint of \(P_{av}^{IT}=18\) dBm and \(P_{av}^{MT}=8\) dBm. Thus, the optimal transmit power of selected SSNs will not be adjusted through OTPA algorithm. Comparing the performance of the NoCoPC problem and CoPC problem with the slot \(\tau_{rp}=12\) ms, we can also observe that the optimal transmit power of selected SSNs by using PB algorithm presents a fixed constant value. The reason for this is that the optimal transmit power of the selected SSN by using PB algorithm converges to an expected equilibrium point after 250 iterations. It is also interesting that Proposition 2 has turned out  
Figure 7: The detection probability and false alarm probability distributions of the selected spectrum sensor nodes involving IT-1, MT-2, MT-5, and MT-7. (**a**) Detection probability distribution; (**b**) False alarm probability distribution. IT: interior tier; MT: middle tier; SSN: spectrum sensor node. that the optimal transmit power of selected SSNs derived by NoCoPC problem converges to a fixed Nash equilibrium point. Moreover, Proposition 4 has also guaranteed that the optimal transmit power of selected SSNs under the CoPC problem will reach to a fixed value. From Figure 8, it is implicitly revealed that the optimal transmit power of selected SSNs by the OTPA algorithm is obviously lower than that of PB algorithm. This observation is reasonable since PB algorithm generates more power consumption to maintain the target SNR. However, the optimal transmit power of selected SSNs via the proposed OTPA algorithm fully depends on the maximum transmit power of the selected SSNs and the pricing factors in differential game model. Figure 9 displays the optimal transmit power comparison between PB algorithm with 250 iterations and our proposed distributed power control framework under discount factor \(r=0.5\) with the evolution of the slot \(\tau_{rp}\) from 10 ms to 15 ms. As can be seen, the proposed OTPA algorithm for both the NoCoPC problem and CoPC problem under the condition of discount factor \(r=0.5\) outperforms PB algorithm with 250 iterations significantly in terms of the optimal transmit power of selected SSNs with the growth of the slot \(\tau_{rp}\). This result further validates PB algorithm will result in much more power consumption aiming to maintain the target SNR.