## 4 Cell Selection Games
### Congestion Game with Capacity  
The game model in the previous subsection focuses on the linear combination of congestion and ICI, so the precision of such a game depends on the weight of ICI. We look for a more realistic payoff function to represent the impact of ICI, so channel capacity is considered in this subsection. We first consider the case where FFR for ICI coordination is adopted by picocells, called the congestion game with capacity under the condition of FFR (CGC-FFR), so the utility functions can be written as follows. The channel capacity of \(D\!V_{k}\) selecting the picocell \(B\!S_{i}\) is:  
\[U_{k}(B\!S_{i},p_{-k})=\frac{B_{i}}{n_{i}}\log_{2}\big{(}1+\frac{GP_{i}/d_{ik} ^{\gamma}}{\sigma^{2}+I_{0i}(n_{i})}\big{)} \tag{23}\]  
where \(\sigma^{2}\) represents the variance of additive white Gaussian noise (AWGN). Due to the fact that the device on the edge of the circular boundary has the strongest willingness to change the CS preference, \(d_{ik}\) is used as the distance between \(D\!V_{k}\) and \(B\!S_{i}\). If \(D\!V_{k}\) changes preference to the macrocell \(B\!S_{0}\), its channel capacity becomes:  
\[U_{k}(B\!S_{0},p_{-k})=\frac{B_{0}}{N_{0}-\sum_{\alpha}n_{\alpha}+1}\log_{2} \big{(}1+\frac{GP_{0}/D_{0i}^{\gamma}}{\sigma^{2}+I_{i0}}\big{)} \tag{24}\]  
Figure 4b shows the monotonicity of the above two payoff functions. The curves show that, when the devices gradually change their preferences to the macrocell, the other devices' willingness (_i.e._, the capacity) to select the picocell increases, while their willingness to select the macrocell decreases. When the two curves meet after 716 iterations, an equilibrium is reached. Similarly, the channel capacity of \(D\!V_{l}\) selecting \(B\!S_{0}\) is:  
\[U_{l}(B\!S_{0},p_{-l})=\frac{B_{0}}{N_{0}-\sum_{\alpha}n_{\alpha}}\log_{2} \big{(}1+\frac{GP_{0}/D_{0i}^{\gamma}}{\sigma^{2}+I_{i0}}\big{)} \tag{25}\]  
If \(D\!V_{l}\) hands over to \(B\!S_{i}\), its channel capacity is:  
\[U_{l}(B\!S_{i},p_{-l})=\frac{B_{i}}{n_{i}+1}\log_{2}\big{(}1+\frac{GP_{i}/d_{ il}^{\gamma}}{\sigma^{2}+I_{0i}(n_{i}+1)}\big{)}. \tag{26}\]  
**Theorem 3**. : _For CGC-FFR, define:_  
\[g_{3}(n_{i})=\log_{2}\big{(}1+\frac{GP_{i}[N_{0}/(n_{i}R_{0}^{2})]^{\gamma/2} }{\sigma^{2}+I_{0i}(n_{i})}\big{)}^{\frac{N_{0}-\sum_{\alpha}n_{\alpha+1}}{n_ {i}}} \tag{27}\]  
\[A_{3,i}=\log_{2}(1+\frac{GP_{0}/D_{0i}^{\gamma}}{\sigma^{2}+I_{i0}})^{\frac{B _{0}}{H_{i}}} \tag{28}\]  
_(1) if \(g_{3}(N_{i})>A_{3,i}\), \(n_{i}^{*}=N_{i}\) forms the PSNE;__(2) if \(g_{3}(1)<A_{3,i}\), \(n_{i}^{*}=0\) forms the PSNE; (3) otherwise, \(n_{i}^{*}\) satisfying \(g_{3}(n_{i}^{*}+1)\leq A_{3,i}\leq g_{3}(n_{i}^{*})\) forms the PSNE._  
**Proof. ** Based on Definition 1, we obtain the conditions for the PSNE of CGC-FFR as follows:  
\[U_{k}(B\!S_{i},p_{-k}^{*})\geq U_{k}(B\!S_{0},p_{-k}^{*}) \tag{29}\]  
\[U_{l}(B\!S_{0},p_{-l}^{*})\geq U_{l}(B\!S_{i},p_{-l}^{*}) \tag{30}\]  
Since all of the devices are assumed identical, the device most preferably changing its preference is the one on the edge of the blue circle in Figure 2, so:  
\[d_{ik}^{\forall}=\big{(}\frac{R_{0}^{2}n_{i}^{*}}{N_{0}}\big{)}^{\forall/2} \tag{31}\]  
\[d_{il}^{\forall}=\big{[}\frac{R_{0}^{2}(n_{i}^{*}+1)}{N_{0}}\big{]}^{\forall/2} \tag{32}\]  
Taking Equations (23)-(26), (31) and (32) into Equations (29) and (30), we get:  
\[U_{k}(B\!S_{0},p_{-k})=\frac{B_{0}}{N_{0}-\sum_{\alpha}n_{\alpha}+1}\log_{2} \big{(}1+\frac{GP_{0}/D_{0i}^{\gamma}}{\sigma^{2}+I_{i0}}\big{)} \tag{24}\]0  
To sum up,  
(1) If \(g_{3}(N_{1})\geq A_{3,i}\), all of the devices under the coverage of \(B\!S_{i}\) select it, so \(n_{i}^{*}=N_{i}\) forms the PSNE. (2) If \(g_{3}(1)<A_{3,i}\), similar to the explanation in the proof of Theorem 2, \(n_{i}^{*}=0\) forms the PSNE. (3) Otherwise, \(\mathbf{n}^{*}\) satisfying Equation (33) forms the PSNE.