## 4 The Coalition Formulation Game Model
### Coalition Formation Basic Concepts
Thus, UAV-3 and UAV-16 can transmit packets in the two time slots \(\{3,16\}\) simultaneously, the payoff of UAV-3 or UAV-16 is \(u_{3}(\{3,16\},\pi)=u_{16}\{\{3,16\},\pi\}=2R_{0}\), and the utility of coalition \(\{3,16\}\) is \(v(\{3,16\},\pi)=u_{3}(\{3,16\},\pi)+u_{16}\{\{3,16\},\pi\}=4R_{0}\). While in coalition \(\{5,6,7\}\), the arbitrary element is in the communication range of another element, the time slots \(\{5,6,7\}\) cannot be shared to transmit packets simultaneously, and the payoff of an arbitrary UAV in the coalition is \(u_{i}(\{5,6,7\},\pi)=R_{0}\). Therefore, the utility of coalition \(\{5,6,7\}\) is \(v(\{5,6,7\},\pi)=3R_{0}\). The utility of the remaining coalitions can be calculated in the same manner. Thus, the utility of the coalitional structure \(\pi\) can be calculated as \(v(\pi)=\sum_{j=1}^{8}v(S_{j},\pi)=25\). The solution to the time slot coalition game \(\mathcal{G}_{\mathcal{D}}=(\mathcal{S},\ \mathcal{V})\) is to find a stable coalitional structure. Moreover, the newly stable coalitional structure should have the properties of external stability and internal stability. In the coalition formation game, the action of each UAV is to form a coalition \(S_{i}\), where \(i\) is the index of the coalition. Given the current coalition structure, i.e., the state of the game, the UAV faces three options: (i) stay in the current coalition, (ii) join with any of the other coalitions, or (iii) leave the current coalition. Here, we define the join-and-leave rule. **Definition 4** (Join-and-leave rule). : _In the time slot coalition game \(\mathcal{G}_{\mathcal{D}}=(\mathcal{S},\ \mathcal{V})\) given the coalitional structure \(\pi_{x}\), two coalitions \(S_{j},S_{k}\in\pi_{x}\), \(j\
eq k\) and node \(i\in S_{k}\). If \(\pi_{x}\succ_{i}\pi_{y}\), node \(i\) can decide to leave coalition \(S_{k}\) and join coalition \(S_{j}\), and \(\pi_{x}\) can evolve to \(\pi_{y}\): \(\pi_{y}=\big{\{}\pi_{x}\setminus\big{(}S_{k},S_{j}\big{)}\big{\}}\cup\big{\{}S _{k}\setminus i\big{\}}\cup\big{\{}S_{j}\cup\{i\}\big{\}}\). That is, \(\pi_{x}\succ_{i}\pi_{y}\) is represented as \(\pi_{x}\succ_{i}\pi_{y}\Leftrightarrow u_{i}(\big{\{}S_{j}\cup\{i\}\big{\}}, \pi_{y})>u_{i}(S_{k},\pi_{x})\),\(v\big{(}\big{\{}S_{j}\cup\{i\}\big{\}},\pi_{y})>v(S_{j},\pi_{y})\) and \(v(\pi_{y})>v(\pi_{x})\). During time slot coalition formation, each UAV-\(i\) prefers to join another new coalition and leave its current coalition if its individual payoff, the new coalitional payoff, and the system payoff increase._  
**Definition 5** (The stable partition). : _For a given partition \(\pi^{*}=(S_{1},S_{2},\ldots,S_{M})\), if \(\forall i\in S_{k}\), \(\forall S_{j}\in\pi^{*},j\
eq k\), \(u_{i}(S_{k},\pi^{*})\geq u_{i}(\big{\{}S_{j}\cup\{i\}\big{\}},\pi^{\prime})\), \(v(S_{k},\pi^{*})\geq v(\{S_{k}\setminus i\},\pi^{\prime})\) and \(v(\pi^{*})\geq v(\pi^{\prime})\), where \(\pi^{\prime}=\big{\{}\pi^{*}\setminus\big{(}S_{k},S_{j}\big{)}\big{\}}\cup\{ S_{k}\setminus i\big{\}}\cup\big{\{}S_{j}\cup\{i\}\big{\}}\), then \(\pi^{*}\) is a stable partition. In a stable partition, there does not exist a UAV that wants to leave the current coalition to join another one._