# Traffic Offloading in Unlicensed Spectrum for 5G Cellular Network: A Two-Layer Game Approach
## 4 Coalition Game
### Merge-and-Split Based Coalition Formation Algorithm
However, the bidders could adjust their prices and make better choices in the next auction round after the end of a round according to the result and their own needs. Interactive information could exchange among APs through a central controller. In the round \(i\leq I\), each agent submits a bid based on the value of the current unit, and then LAA BS offloads the traffic of the winner with the largest bid and charges the second bid. In addition, the AP coalition which has won the game can also join the later auction. Every AP coalition has the right to choose whether to participate in the auction at the beginning of each round of the auction. In this paper, we assume that, if the AP can gain benefits from the auction, it would take part in the auction. ### Problem Formulation

We first investigate the benefit that the WiFi network can achieve after the auction. As mentioned before, the WiFi system achieves its benefit through transferring parts of users to the LAA system and reduce collisions. Thus, the average throughput of the AP coalition \(i\) after the auction can be expressed as

\[R_{i}(N-\hat{n})=\frac{1}{L}\sum_{i=1}^{N-\hat{n}}\frac{P_{tr}P_{S}E_{i}}{(1-P _{tr}T_{\sigma})+P_{tr}P_{S}T_{S}+P_{tr}(1-P_{S})T_{c}}\frac{T_{off}-d_{i}\bar {T}_{slot}}{T_{on}+T_{off}}, \tag{17}\]

where \(E_{i}\) is the mean number of bits sent by station \(i\) in a successful transmission [34], \(\bar{T}_{slot}\) is the average time slot, \(N\) is the total number of stations competing for the channel in a AP coalition and there are \(\hat{n}\) stations transferred to the LAA network with \(d_{i}\) relinquished slots. The utility of the WiFi is the promoted throughput after the auction, that is

\[ \begin{split} u_{WiFi}&=\frac{1}{L}\sum_{i=1}^{N- \hat{n}}\frac{P_{tr}P_{S}E_{i}}{(1-P_{tr}T_{\sigma})+P_{tr}P_{S}T_{S}+P_{tr}(1 -P_{S})T_{c}}\frac{T_{off}-d_{i}\bar{T}_{slot}}{T_{on}+T_{off}}\\ &\quad-\frac{1}{L}\sum_{i=1}^{N}\frac{P_{tr}P_{S}E_{i}}{(1-P_{tr }T_{\sigma})+P_{tr}P_{S}T_{S}+P_{tr}(1-P_{S})T_{c}}\frac{T_{off}}{T_{on}+T_{ off}}\\ &=\frac{T_{off}-d_{i}\bar{T}_{slot}}{T_{off}}R_{i}(N-\hat{n})-R_{ i}(N)\\ &=\frac{T_{off}-\theta}{T_{off}}R_{i}(N-\tilde{n})-R_{i}(N),\end{split}  \tag{18}\]

where \(\theta=d_{i}\bar{T}_{slot}\) means the time which is compensated for LAA. The revenue of the WiFi comes from parts of data offloaded to LTE in unlicensed bands, which reduces the probability of sending packet and collisions in WiFi network. That is to say, the burden of AP can be reduced effectively. The utility of the auctioneer is the throughput earned by LAA after the auction and it can be formulated as

\[ \begin{split} u_{LAA}&=r\frac{T_{off}}{T_{on}+T_{ off}}+r\delta_{i}(d_{i}-\omega_{i})\bar{T}_{slot}-r\frac{T_{off}}{T_{on}+T_{ off}}\\ &=r\delta_{i}(d_{i}-\omega_{i})\bar{T}_{slot}=r\delta_{i}\theta _{i}-\delta_{i}R(\tilde{n}),\end{split}  \tag{19}\]

in the premise of \(d_{i}\geq\omega_{i}\). Here, \(\delta_{i}\) is the discount factor of LAA participating in game and \(\delta_{i}\in[0,1]\). \(\omega_{i}=\frac{R(\tilde{n})}{\bar{T}_{slot}^{*}}\) means the number of time slots which is actually used in LAA by transmitting the WiFi data of \(\tilde{n}\) stations. ### A Second Price Auction Design

Next, a win-win strategy for both systems is discussed. The general win-win problem to balance the WiFi and LAA benefits can be expressed as a multi-objective function problem:

\[max\quad\left\{u_{WiFi},u_{LAA}\right\}, \tag{20}\]subject to

\[d_{i}\geq w_{i},\] \[0\leq\tilde{n}\leq N.\]

Equation (20) is a multi-objective problem. Next, we will use the Nash equilibrium solution method [16; 35] to solve the win-win strategy fairly. **Definition 4**.