## 2 Proposed Spectrum-Sharing Scheme
### Integrity Monitoring Mechanism  
As shown in Figure 4, once Players 1 and 2 come to an agreement, or the deadline is reached, Player 1 will add an additional round, which is an integrity monitoring mechanism. The integrity monitoring mechanism is prepared for dishonest unlicensed users, who will falsely report the \"bargain\" and \"warn\" as follows:  
\[\beta_{ij}=(1+\delta_{j})\times\beta_{ij},\;i=1,\ldots,M;\;j=1,\ldots,N, \tag{6}\]  
\[\varpi_{ij}=(1+\delta_{j})\times\varpi_{ij},\;i=1,\ldots,M;\;j=1,\ldots,N, \tag{7}\]  
where \(\beta_{ij}\) and \(\varpi_{ij}\) are the \"bargain\" and \"warn\" from unlicensed user \(j\) to licensed user \(i\), respectively, and \(\delta_{j}\) is the dishonesty degree of unlicensed user \(j\), \(\delta_{j}\in[0,1]\). When \(\delta_{j}=0\), unlicensed user \(j\) is honest. The closer to 1 the value is, the more dishonest the user is. The reason why unlicensed users perform in this way is because all unlicensed users want licensed users to reduce their prices.  
Before designing the integrity monitoring mechanism, we needed to analyze all the cases:  
**Case 1:**: Single unlicensed user is dishonest: Located on the straight line with maximal utility; if they become dishonest, as shown in Figure 5, case 1.1, case 1.2, or case 1.3 will occur.
**Case 2:**: Single unlicensed user is dishonest: Not located on the straight line with maximal utility; therefore, their dishonesty cannot affect the licensed user's strategy.
**Case 3:**: All unlicensed users are dishonest: The peak of each straight line will reduce, leading to a reduction in a licensed user's utility.
**Case 4:**: Some unlicensed users are dishonest; this case can be analyzed using Cases 1, 2, and 3.  
The probability of each case happening, and the utility of Players 1 and 2, are intuitively stated in Table 1. Case 1.1 causes a slight reduction in price, which leads to a slight reduction in \(\pi\) and an increase in U. Since the suboptimal price is usually lower, Case 1.2 will occur with a lower probability than Case 1.3. A dishonest unlicensed user choosing another licensed user will cause a new Nash bargaining solution (NBS). Normally, a new NBS will be a higher price, caused by the chain reaction of the corresponding licensed user increasing their price. The slight reduction in \(\pi\) in Case 1.3 is dependent on a suboptimal price, and a lower price results in an increase in U, while a higher price leads to a reduction. The reason why Case 2 happens with a high probability is that the majority of unlicensed users do not lie on the straight line of maximal utility. Non-cooperative unlicensed users cannot easily achieve Case 3 without collusion. Their different \(\delta\) values make the U of each unlicensed user uncertain.  
On the one hand, through analyzing these cases, we found that dishonest unlicensed users can make negligible profit with a very low probability. A rational unlicensed user has no desire to be dishonest. On the other hand, an integrity monitoring mechanism is also needed for irrational unlicensed users. Licensed users with the first-mover advantage can add an additional integrity monitoring round, whenever an optimal proposal is found. It is worth noting that the mechanism is aimed at unlicensed users in Case 1. In the integrity-monitoring round, each licensed user slightly increases their price. If the choice of the unlicensed user changes, the conclusion that this unlicensed user is honest will be drawn; otherwise, the unlicensed user is dishonest. After the monitoring round, the price is restored to its original value and the dishonest, unlicensed user cannot buy any spectra from the licensed user. Without knowing of the approach of the monitoring round, dishonest unlicensed users cannot pretend to be honest.