## 3 Results
### Nash Bargaining Solution  
Figure 6 simulates the finitely many rounds of bargaining in Figure 4. In the 51 st round, the proposed strategic game finds a Nash bargaining solution, namely, a subgame-perfect equilibrium, where the spectrum price is [9.50, 9.53, 8.74, 8.93], the utility of a licensed user is [73.67, 182.84, 203.26, 139.79], and the sold bandwidth of a licensed user is [5, 23, 28, 17]. Licensed user 1 sells spectra to unlicensed user 3; licensed user 2 supplies service to unlicensed users 2, 4, 8, 11, and 14; licensed user 3 sells spectra to unlicensed users 5, 10, 12, 13 and 15; and licensed user 4 satisfies the spectra of unlicensed users 1, 6, and 9. Noted that, unlicensed user 7's demand has not been met. This is because its \(k^{(u)}\) is so small that its utility is negative. The principle of the proposed scheme is that unlicensed users will not buy spectra when the utility is negative. On the other hand, this result also shows that the proposed scheme will not always meet the spectrum demands of unlicensed users.  
By analyzing \(k^{(u)}\), we find that if we only consider the \(k^{(u)}\) of each unlicensed user, the spectrum demands of licensed users 1 and 2 will appear to be greater than they are. Therefore, to gain maximal utility, licensed users 1 and 2 need to reduce buyers by using a higher price, while licensed users 3 and 4 need to use a relatively lower price to attract customers. The influence of the bandwidth of the licensed user can also not be ignored. Owing to relatively lower bandwidth availability, licensed user 1 has the highest price; on the other hand, licensed user 3 has the lowest price. Figure 7 proves licensed user 3's principle of small profits, yet quick turnovers. With a higher price than licensed user 1, licensed user 2 sells less bandwidth but gains almost the same profits.  
As mentioned before, spectrum price will converge on the NBS. Figure 7 shows that, except for \(k^{(u)}\) and licensed users' bandwidth, the supply-and-demand relationship is also a vital factor. When the  
\begin{table}
\begin{tabular}{c c} \hline \hline \multirow{2}{*}{M} & the number of licensed users \\ \cline{2-3}  & 4 \\ \hline \multirow{2}{*}{N} & the number of unlicensed users \\ \cline{2-3}  & 15 \\ \hline \multirow{2}{*}{W/MHz} & the bandwidth of each licensed users \\ \cline{2-3}  & [12, 28, 36, 24] (total: 100) \\ \hline \multirow{2}{*}{\(M^{\prime}\)} & the number of ongoing licensed connections \\ \cline{2-3}  & [6, 14, 18, 12] \\ \hline \multirow{2}{*}{\(B^{mq}/\)MHz} & the spectrum demand for an ongoing licensed connection \\ \cline{2-3}  & [2, 2, 2, 2] \\ \hline \multirow{2}{*}{P\({}^{0}\)} & the initial spectrum price \\ \cline{2-3}  & [6, 6, 6, 6] \\ \hline \multirow{2}{*}{P\({}^{1}/\)P\({}^{2}/\)P\({}^{3}/\)P\({}^{4}\)} & the spectrum price of licensed user \(1/2/3/4\) \\ \hline \multirow{2}{*}{\(\delta\)} & the dishonesty degree of unlicensed user \\ \hline \multirow{2}{*}{\(BER^{tar}\)} & the target bit-error rate \\ \cline{2-3}  & 0.0001 \\ \hline \multirow{2}{*}{\(c_{1}/c_{2}\)} & the weights for revenue/cost function \\ \cline{2-3}  & 2/2 \\ \hline \multirow{2}{*}{\(\omega\)} & the income from the per-transmission rate of the unlicensed user \\ \cline{2-3}  & [5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5] \\ \hline \multirow{2}{*}{D/MHz} & the spectrum demand of unlicensed user \\ \cline{2-3}  & [2, 4, 5, 8, 3, 9, 5, 2, 6, 6, 3, 7, 5, 6, 7] \\ \hline \multirow{2}{*}{Because \(\gamma_{ij}\in[9,\,22]\), \(k^{(u)}_{ij}\in[1.4731,\,2.4173]\) and is randomly generated.} \\ \end{tabular}
\end{table}
Table 2: Parameter settings.  
spectrum demand of HWSNs range from 50 to 120 (the maximal supply of licensed users is 100), the spectrum price of all licensed users will increase along with it. Due to the character of each licensed user and unlicensed users' SNR, the relative relationship of licensed users' prices remain almost the same. Note that, no matter what the relationship between supply and demand may be, Figure 7 proves the proposed scheme to be applicable.