## 2 Proposed Spectrum-Sharing Scheme
### Strategic Bargaining
#### 2.3.1 One Round of Bargaining (the Ultimatum Game)  
Unlike classical one-round bargaining, in the ultimatum game [27], Player 1 or Player 2 (shown in Figure 3) represent two groups of users, rather than two players. In other words, each licensed user needs to complete the ultimatum game with all unlicensed users at the same time. If there are M licensed users, M ultimatum games will be carried out simultaneously. Formally, this paper uses one ultimatum game to represent this process for simplicity. Meanwhile, only if all users in player 1 choose to hold the former strategies will H (hold the former proposals) be obtained; otherwise, C (change the proposals) be the choice. Similarly, A (adhere to former \(x\)) will be achieved only if all unlicensed users adhere to their former choices (that is, \(x\) remains unchanged). Secondly, as a game of perfect information (unlicensed users know the proposals of licensed users before making a response), this paper not only let licensed users know unlicensed users' choice (A or R (react, change the \(x\))), but also know the \"bargain\" and \"warm\" from unlicensed users (Figure 2). For any \(x_{ij}\) which equals 0, \"bargain\" means how much price licensed user \(i\) needs to reduce, where unlicensed user \(j\) will change to choose licensed user \(i\). \"Warn\", aiming at \(x_{ij}=1\), tell the licensed user \(i\) the spectrum price, at which it will lose the buyer \(j\). Knowing these bargains and warns, and having all the \"bargaining power\" (due to the first-mover advantage), each licensed user can find the optimal price. Figure 2: Utility function of a licensed user. Although the one round of bargaining in this paper is different from the classical one, all propositions in Chapter 11.1 of Reference [27] are also applicable. Therefore, it can also be divided into an ultimatum game. With respect to our game model, we performed some adaptations, proposed some new propositions, and make the corresponding proofs. In one round of the bargaining game, any proposal that makes the utility of Players 1 and 2 be \"not less than zero\", can be supported as a Nash equilibrium. We suppose the utility of Players 1 and 2 to be \(\pi\) and U, respectively. No matter whether Player 1 chooses H or C, Player 2's strategy will be \"1 accept any proposal making U \(\geq\) 0 and reject any proposal making U < 0.\" For Player 1, any strategy that makes \(\pi\geq 0\) will be adopted. Therefore, \(\pi\geq 0\) and U0 are the mutual best responses, and the proposition has been proven. Although Player 1 will accept any strategy that makes \(\pi\geq 0\), whether there is a \(\pi\) that let Player 1 obtain the maximum utility under the premise of U \(\geq\) 0, without cooperation, licensed users in Player 1 have no need, nor ability, to achieve a global optimum. On the contrary, the individual optimum of one licensed user is more meaningful. This expectation leads to the following proposition:  
One round of the bargaining game will have at least M Nash equilibria, if Player 1 has M licensed users. We assume that there is a situation where licensed user i obtains optimal utility under the premise of x remaining unchanged (it can be obtained if licensed user i adjusts the price according to \"warn\", and other licensed users hold the former strategies). In this situation, if any other licensed user changes the strategy, the x will change and (C, R) will be obtained (as shown in Figure 3). If any unlicensed user changes the choice, x will also change and (H, R) or (C, R) will be obtained. Meanwhile, (H, R) or (C, R) will cause the utility of Players 1 and 2 to reduce to zero. In other words, other licensed and unlicensed users cannot obtain more benefits; the proposed situation is a Nash equilibrium. Because I \(\in\) {1, 2,..., M}, there are at least M Nash equilibria, which proves the proposition. We have proved the existence of some Nash equilibria, but what we indeed need is at least one subgame-perfect equilibrium in strategic bargaining.