# Traffic Offloading in Unlicensed Spectrum for 5G Cellular Network: A Two-Layer Game Approach
## 4 Coalition Game
### Coalitional Game Definition  
Define the distributed traffic offloading process as a coalitional transferable utility game (\(\mathcal{S}_{i}\
u\)), where \(\mathcal{S}\) denotes the set of players and \(\
u(S)\) is the utility function of the coalition \(S\). Note that, in our assumption, only the players with heavy traffic ask for a favor and the players which would like to help onload data are permitted to form a coalition together. **Definition 1**: _(Utility of the player) The utility of player \(i\) in a coalition \(\
u(i)\) is defined as the difference after the player \(i\) joining the coalition \(S:\
u(i)=\
u(S\cup i)-\
u(S)\)._  
At the beginning of each stage, players with their own stations compete for the channel based on their own transmitting needs respectively, according to its own station numbers, channel conditions and other competing AP numbers in the network. After that, players consult with the coalitionists to reach a consensus. To motivate players to onload data from APs with heavy load which ask for a favor, an encouragement factor \(\alpha\) is put forward. We define \(\alpha_{s}\) as the prize for giving a hand and a prize \(\alpha_{a}\) for asking for a favor as follows:  
\[(\alpha_{s},\alpha_{a})=\left\{ \begin{aligned} &\alpha_{s}=-\beta R(n^{*}-n_{i}),& \alpha_{a}=0,& n_{i}<n^{*}\\ &\alpha_{a}=\beta R(n_{i}-n^{*}),&\alpha_{s}=0,& n_{i}>n^{*}\end{aligned} \right., \tag{8}\]  
where \(\beta\) is a discount factor related to the sensitivity of giving a reward and \(\beta\in[0,1]\). \(n_{i}\) is the AP station number in stage \(i\). The rule is that if the AP's encourage factor \(\alpha\) is negative and there are other APs which asks for a favor at the current stage, the AP has to help onload data. On the contrary, the coalition player whose encouragement factor is positive can offload data successfully. Players know rewards and rules in advance and the encourage factor for the operator is  
\[\alpha=\sum_{i}(\alpha_{s,i}+\alpha_{a,i}), \tag{9}\]  
where \(i\) is the index of the stage and the centralized controller updates information periodically. The expected utility of one coalition is\[\tilde{v}_{coalit}=\sum_{i}\left[-\hat{P}_{i}(\alpha_{s,i}-\varepsilon_{i})+\hat{Q }_{i}(\alpha_{a,i}-\varepsilon_{i})\right], \tag{10}\]  
where \(\hat{P}_{i}\) and \(\hat{Q}_{i}\) are the probabilities of offering help and asking help, respectively. \(\varepsilon_{i}\) is the cost of investment and we assume that all players have the same \(\varepsilon_{i}\). Thus, each player has the incentive to obtain the optimal stations in way of helping other APs carrying stations or asking for a favor to transfer some stations in order to increase its utility. **Definition 2**. : _(Transfer rule of the coalition structure) Collecting all coalition structure possibilities to one set, namely \(\mathrm{Y}=\{S_{1},S_{2},\ldots,S_{L}\}\), \(L\) is the total number of possibilities that a coalition structure could form. For a given coalition structure \(S_{L}\) and \(S_{L^{\prime}}\), if \(\
u(S_{L})\leq\
u(S_{L^{\prime}})\), then coalition structure \(S_{L^{\prime}}\) is better than \(S_{L}\). The formula can be expressed as:_  
\[S_{L}\to S_{L^{\prime}}\Leftrightarrow\
u(S_{L})\leq\
u(S_{L^{\prime}}),1\leq l ^{\prime},l\leq L. \tag{11}\]  
**Theorem 1**. : _The criterion for coalition formation to find proper players to minimize \(\sigma_{i}\) is_  
\[-\sum_{i}n_{i,s}\hat{P}_{i}\alpha_{i,s}+\sigma_{i}=\sum_{i}n_{i,a}\hat{Q}_{i} \alpha_{i,a}, \tag{12}\]  
_where \(n_{i,a}\) is the number of stations that needs to be transferred at the stage \(i\), and \(n_{i,s}\) is the number of stations which is promised to be onloaded by the APs which join in the coalition and have less traffic. \(\sigma_{i}(\sigma_{i}\geq 0)\) is the correction factor which is related to the AP station number and the traffic in the whole channel._  
Proof. : It will be proven through the following two cases:  
Case 1: \(\sigma_{i}=0\).