# Traffic Offloading in Unlicensed Spectrum for 5G Cellular Network: A Two-Layer Game Approach
## 4 Coalition Game
### Merge-and-Split Based Coalition Formation Algorithm
For the Markov chain defined by the state space and the transition probability, there could exist an ergodic set \(\Xi\in S\), if \(\rho_{Y,Y^{\prime}}=0\) for any \(Y\in\Xi\) and \(Y^{\prime}\
otin\Xi\), and no nonempty subset of \(\Xi\) has this property. A stable coalition structure will evolve if the formation process reaches a singleton ergodic set, namely, absorbing state \(\rho_{Y,Y^{\prime}}=1\) for \(Y\in\Xi\). At an absorbing state, there is no player that has an incentive to merge or split. The transition probability matrix \(\mathbf{Q}\) of the absorbing DTMC [31] can be partitioned as follows:

\[\mathbf{Q}= \begin{bmatrix}\mathbf{T}&\mathbf{F}\\ \mathbf{0}&\mathbf{I}\end{bmatrix} \]

where \(\mathbf{T}\) is the transition probability matrix corresponding to the transitions among the transient states, \(\mathbf{I}\) is an identity matrix, \(\mathbf{0}\) is a zero matrix and \(\mathbf{F}\) is the transition probability matrix corresponding to the transitions from the transient states to the absorbing state. For an absorbing DTMC with transition probability matrix \(\mathbf{Q}\), the matrix \(\mathbf{M}=(\mathbf{I}-\mathbf{T})^{-1}\) is its fundamental matrix. The entry \(m_{Y,Y^{\prime}}\) of \(\mathbf{M}\) gives the expected number of times that the process is in transient state \(Y^{\prime}\) if it starts in transient state Y before the Markov chain reaches any absorbing state. From the Theorem 1 of [33], if \(\sum_{i\in S}\
u(\{i\})<\
u(S)\), there is at least one absorbing state which is a stable solution of the coalition game. Since the payoff of a player will be zero if the player acts alone, the players would rather to be a member of coalitions to obtain a higher payoff if they have the chance. The \(\
u(S)\) will be equal to or higher than zero, and then a stable solution may exist depending on the value of the payoff. Because the light-loaded AP can only form coalitions with the heavy-loaded AP, the stable coalition may exist when there are more than one players in a coalition. That is to say, a stable coalition may exist when there are both the light-loaded AP and the heavy-loaded AP in a coalition, namely, the utility of the coalition is larger than zero. ## 5 Auction Game

After making a balance among WiFi APs, the second layer of CAGT algorithm, an auction game is performed between LAA and WiFi coalitions to coexist more friendly and boost the spectrum utilization of two systems simultaneously. We assume that there are \(L\) coalitions totally after the coalition game. In the auction game, the role of the auctioneer is played by the LAA BS, while the AP-coalition plays the role of the bidder. A second-price sealed-bid auction, namely Vickrey auction in our work is applied. Firstly, LAA determines a base price \(d_{i}=\omega_{i}(i=1,\dots,N)\), which means LAA would participate in the auction only when \(d_{i}\geq\omega_{i}\), where \(d_{i}\) is slots in which WiFi compensates for LAA in helping reduce the traffic burden and \(w_{i}\) means the actual slots which were used in WiFi data transmitting by the LAA network. When the auction begins, AP coalitions have \(m_{1},m_{2},\dots,m_{N}\) bits data waiting for transferring to the LAA network respectively. Next, LAA choose the AP coalition which could maximize the value of \((d_{i}-\omega_{i})\) to cooperate. If \((d_{i}-\omega_{i})<0\), WiFi and LAA would continue working in their original way without offloading between two networks and the utility of players is zero. For the second-price sealed-bid auction, the auctioneer selects the highest bidder, but only needs to pay the second high price to the bidder which wins the game. An advantage of this type of auction over others is that the dominant strategy for each bidder is to bid his true valuation. In addition, the game is repeated for \(l\) rounds and because the price is sealed, the information in auction progress cannot be known by their AP opponents.