## 2 Proposed Spectrum-Sharing Scheme
### Strategic Bargaining
#### 2.3.2. Finitely Many Rounds of Bargaining  
Through one round of bargaining, the Nash equilibrium always can't be found. Therefore, the bargaining will repeat once and once again until the Nash equilibrium is obtained. In practical applications, an infinite game cannot exist, depending on the exogenous deadline. With limited rounds of bargain, finitely many rounds of bargaining [27] can solve the spectrum sharing problem as follows:  
\[\begin{array}{c}\underset{P_{i}}{\text{argmax}}\ \Pi_{i}(P),\ for\ \forall i\in\{1,\ 2,\ \ldots,\ M\}\\ st.\sum\limits_{j=1}^{N}\left(D_{j}\times x_{ij}\right)\leq W_{i},\ i=1, \ldots,M\\ x_{ij}\in\left\{x_{ij}|\underset{x_{ij}}{\text{argmax}}\ U_{j}(P),\ for\ all\ i\in\{1,\ 2,\ \ldots,\ M\}\right\},\ j=1,\ldots,N\end{array}. \tag{5}\]  
In order to obtain its own optimal utility, each licensed user will dynamically adjust their spectrum price. Each time a user makes the strategic decision, it depends on the last action of the opponent. That is to say, each user in Player 1 updates their strategy, based on the previous spectrum demand from Player 2, while each user in Player 2 makes their choice according to Player 1's price. A process, where all users can dynamically find the optimal strategies, will continue until a consensus is reached. This paper uses finitely many rounds of bargaining (Figure 4) for modeling. Because of the existence of a deadline, the ultimatum game is designed as the penultimate part of a strategic game. When licensed users hold the former proposals (namely, choose the H), the responses of unlicensed users are already optimal. Therefore, compared with the ultimatum game mentioned above, the ultimatum game in this part does not have (H, R). Regarding the existence of subgame-perfect equilibrium in the ultimatum game, as shown in Figure 4, we prove the Proposition 4.  
**Proposition 4**.: _If the finitely many rounds of bargaining go to the ultimatum game, the (H, A) will be a subgame-perfect equilibrium._  
Figure 4: Finitely many rounds of bargaining.  
**Proof.**: _In the ultimatum game, if any licensed user selfishly changes their price in favor of more profit, without knowing other licensed users' strategies in that round, there is a high probability that \(x\) will change. Obviously, those changes will lead to (C, R), which will cause all users to gain nothing. Therefore, for sequentially-rational users, (H, A) will be a subgame-perfect equilibrium. It is worth mentioning that one or more subgame-perfect equilibria will exist, which is corroborated using our many simulations._  
Knowing the \"bargain\" or \"warn\" from all unlicensed users, each licensed user can draw a utility curve (as in Figure 2). From this curve, a licensed user can find an optimal price, which will allow unlicensed users to enter a new round of bargaining. This process will continue until strategies of licensed and unlicensed users no longer change, and a Nash bargaining solution is found. Because the subgame-perfect equilibrium in the ultimatum game is probably not optimal for licensed and unlicensed users, all users are willing to come to an agreement before the ultimatum game.