## 3 Results
### The Influence of Dishonest Unlicensed Users  
In this paper, we assume that unlicensed user 15 is dishonest. When it acts as a dishonest user, Figure 8 shows that it can make extra profits in the process of bargaining. The more dishonest it is, the greater its profits are. However, in most cases (dishonesty degree is 0, 0.4, and 0.6), the utility of the unlicensed user and the corresponding licensed-user will, respectively, converge on the same value (U = 21.1, \(\pi\) = 203.3), and that the spectrum price and choices of all unlicensed users also remain the same. This is proof of the high probability of Case 2. As can be seen in Figure 8, dishonesty causes longer bargaining times (shown by rounds). Therefore, a dishonest unlicensed user selects a high risk of making no profit and meeting the deadline. Each rational unlicensed user will have no desire to cheat.  
The phenomenon, in which the curve in Figure 8a and corresponding curve in Figure 8b form a trend of complementing each other, is interesting. This is because, for one side of a trade, generating profits is always at the expense of the other's revenue. Another phenomenon, which is unusual, is seen  
Figure 6: The process of finding Nash bargaining solution (NBS): (**a**) spectrum price; (**b**) utility of licensed user.  
Figure 7: The relationship between NBS (spectrum price) and spectrum demand.  
when the dishonesty degree is 0.2. In order to determine why U decreases to 13.8, while \(\pi\) increases to 222.7, we performed further simulations. Regarding spectrum price, Figure 8c indicates that, in the process of bargaining, price converges on a higher value. In fact, the spectrum prices of all licensed users increase from [9.5, 9.53, 8.74, 8.93] to [10.3, 10.33, 9.78, 9.49], which directly lead to a higher \(\pi\) and a lower U. Although this process is difficult, it can also be explained by Case 1.2. In one round of bargaining, dishonesty causes a higher price for licensed user 3, which causes an increase in other licensed-users' prices. Owing to this general increase in price, unlicensed user 13 no longer chooses any licensed user. The proposed scheme finds another NBS of this market.  
Having analyzed the influence of a single unlicensed user being dishonest, Figure 9 simulates Case 3. Unexpectedly, although the dishonesty degree of all unlicensed-users increases, the total utility of licensed and unlicensed users both remains the same (\(\pi=599.6\), U = \(141.0\)). Further, we find that the proposed scheme can always find the same optimal price ([9.5, 9.53, 8.74, 8.93]), no matter how dishonest all unlicensed users are. However, the analysis in Table 1 shows that the total utility of licensed users will reduce. As mentioned in Figure 7, the supply-and-demand relationship is essential in deciding NBS. Although all unlicensed users are cheating, their spectrum demands have no differences. Since the analyses  
Figure 8: The influence of single unlicensed-user being dishonest: (**a**) utility of itself; (**b**) utility of corresponding licensed user; (**c**) spectrum price of corresponding licensed user.  
in Table 1 are based on a situation where dishonest unlicensed users occur after a NBS is found, the analyses in Table 1 are also correct. In other words, the supply-and-demand relationship deciding NBS, can be achieved in the process of many rounds of bargaining, before, rather than after, a NBS is found.  
Figure 9 also shows that a greater dishonesty degree can only let unlicensed users make more profits in the process of finding NBS.