## 4 Cell Selection Games
### Congestion Game with ICI  
The deployment of heterogeneous cells decreases the inter-BS distance on average, which severely increases ICI. The congestion model in the previous subsection is simple, but it does not take ICI into account. In this subsection, we consider the game with a cost function combining congestion and ICI linearly, called the congestion game with ICI (CGI). Figure 3: Two-dimensional case of the traditional congestion game (TCG) for the proof of Theorem 1. Since the device that interferes with the device \(D\!V_{j}\) is highly related to the RA strategy and can be any device selecting adjacent cells, the estimation of the ICI for a general CS problem is quite difficult. The study in this article focuses on the scenario with densely-deployed devices, where the average ICI from devices in other cells can be used to represent the ICI that \(D\!V_{j}\) undergoes. Therefore, a device's uplink transmission to the picocell \(B\!S_{i}\) undergoes an average ICI from devices using the macrocell \(B\!S_{0}\), which is a function of \(n_{i}\), given by:  
\[I_{0i}(n_{i})=\frac{GP_{0}}{N_{0}-\sum_{\alpha}n_{\alpha}}\big{(}\sum_{j\in \mathbf{D}\mathbf{V}_{all}}\frac{1}{d^{\gamma}_{ij}}-\sum_{\alpha}\sum_{j\in \mathbf{D}\mathbf{V}_{\alpha}}\frac{1}{d^{\gamma}_{\alpha j}}\big{)} \tag{12}\]  
where \(P_{0}\) is the transmission power of the devices to \(B\!S_{0}\), \(\mathbf{\gamma}\) is the path loss exponent, \(G=G_{t}G_{r}\lambda^{2}/4\pi^{2}\) is a constant related to transmission antenna gain \(G_{t}\), reception antenna gain \(G_{r}\) and wavelength \(\lambda\). Note that we do not consider shadowing or multipath fading due to the fact that our model and analysis are statistical for densely-deployed sensor and mobile devices, so the instantaneous signal strength of a particular device does not affect the integral results much. Compared to \(R_{0}\), the radius of the picocell \(R_{i}\) is small, so we ignore the position difference of the devices in it when calculating the average interference to \(B\!S_{0}\), given by:  
\[I_{i0}=\frac{GP_{i}}{D^{\gamma}_{0i}} \tag{13}\]  
where \(P_{i}\) is the transmission power of the devices to \(B\!S_{i}\). Therefore, the cost of \(D\!V_{k}\) using \(B\!S_{i}\) is given by:  
\[C_{k}(B\!S_{i},p_{-k})=\frac{n_{i}}{B_{i}}+wI_{0i}(n_{i}) \tag{14}\]  
where \(w\) is the weight of ICI, which is required, because ICI and congestion have different magnitudes and are difficult to normalize. \(w\) should not be too large, otherwise it changes the monotonicity of Equation (14), and the PSNE will collapse to the macrocell without any equilibrium between \(N_{i}\) and zero. Figure 4a shows that, once a picocell device decides to select the macrocell, it leads to a much faster increment of the cost for other devices still preferring the picocell. Therefore, devices will change their preferences one by one until an equilibrium is reached where only a few devices or even no device still prefers the picocell. Similarly, the cost of \(D\!V_{l}\) using \(B\!S_{0}\) is given by:  
\[C_{l}(B\!S_{0},p_{-l})=\frac{N_{0}-\sum_{\alpha}n_{\alpha}}{B_{0}}+wI_{i0} \tag{15}\]  
After \(D\!V_{k}\) changes the CS preference to \(B\!S_{0}\), its cost becomes:  
\[C_{k}(B\!S_{0},p_{-k})=\frac{N_{0}-\sum_{\alpha}n_{\alpha}+1}{B_{0}}+wI_{i0} \tag{16}\]  
Since ICI is related to the positions of devices, we assume that the device that has the strongest willingness to change the CS preference to \(B\!S_{i}\) is the one closest to the circular boundary. Therefore, after \(D\!V_{l}\) changes the CS preference to \(B\!S_{i}\), its cost becomes:  
\[C_{l}(B\!S_{i},p_{-l})=\frac{n_{i}+1}{B_{i}}+wI_{0i}(n_{i}+1) \tag{17}\]  
**Theorem 2**.