## 5 SINR-Based TPC as a Potential Game
### Potential Game Formulation  
Initially, the game that is played in this approach consists of \(N\) players, \(A=\{P_{1},P_{2},\ldots,P_{n}\}\) strategies, where each node selects the best receiver power according to the transmission power with which the data are sent, and \(U_{1},U_{2},\ldots,U_{n}\) utility functions. The utility function of each node is given by the formula  
\[U_{i}=SINR(i,j) \tag{8}\]  
The potential function of the approach is given using the following formula,  
\[V=\sum\limits_{i\in N}U_{i}. \tag{9}\]  
We utilise a mechanism, the same one provided with the example above as given in [40], to ensure that the strategies of the remaining utilities of the players are 0. To this end, we can use the following lemma when using the SINR:  
**Lemma 1**. : _If there is a node \(x_{1}\) that changes the transmission power level from \(p_{1}\) to \(p_{2}\), where \(p_{1}<p_{2}\), the interference in neighboring nodes can be mitigated, such as \(\sum\limits_{x_{3}\in N}SINR_{x_{1},x_{3}}^{P_{1}}=\sum\limits_{x_{3}\in N} SINR_{x_{1},x_{3}}^{P_{2}}\)._  
Proof. : The proof follows from the SIC interference cancellation and the fact that the SINR is reconstructed in such a way that the two sums are equal. Essentially, the denominator of the \(SINR_{x_{1},x_{2}}=\frac{P_{Rx}}{I_{N}+S_{N}}\) diminishes the interference; thus making the SINR equal to the one before the change in the transmission power. To that end, the following proposition is provided that follows the aforementioned lemma and is at the heart of the approach, namely the proof of the potential game of the TPC. Figure 1: SIC of two nodes. The A, B, and C circles correspond to nodes, which operate within a transmission range and the boxes the operations that take place during the operation. \[ \begin{split} V(s_{i},s_{-i})-V(s^{\prime}_{i},s_{-i})& =\\ \sum_{i\in N}U_{i}(s_{i},s_{-i})-\sum_{i\in N}U_{i}(s^{\prime}_{i}, s_{-i})&=\\ =U_{i}(s_{i},s_{-i})-U_{i}(s^{\prime}_{i},s_{-i})+\\ +(\sum_{m\in N,m\
eq i}U_{m}(s_{m},s_{-m})-\sum_{m\in N,m\
eq i} U_{m}(s^{\prime}_{m},s_{-m}))\end{split}  \tag{10}\]  
As was shown in the Lemma 1, after the interference cancellation, the term  
\[\sum_{m\in N,m\
eq i}U_{m}(s_{m},s_{-m})-\sum_{m\in N,m\
eq i}U_{m}(s^{\prime} _{m},s_{-m})=0. \tag{11}\]  
hence, the game is a potential game with the potential function given by (9). The algorithm of the approach follows. The Signal-to-Noise Ratio (SNR) for a specified player is computed, taking into account the transmission powers of all players, the channel gains, and the noise power. Similarly, the utility (SINR) for a given player is determined, essentially performing the same computation as the SINR calculation. Thereafter, the potential function is evaluated, by summing up the utilities (SINRs) of all players. Moving forward, the Nash equilibrium is found employing gradient descent to identify the Nash Equilibrium transmission powers. Note that the Nash equilibrium that is found by the potential game approach may not be the global optimum. Hence, we have the fact that comes as a result of the theorem:  
Proposition 1: _Let \(U_{i}(s_{i},s_{-i})\) denote the utility of node \(i\) in the system where the utility is based on the SINR. Let \(s^{*}\) be the strategy profile that maximizes the sum of utilities (the socially optimal strategy), and let \(s^{Nash}\) be the Nash equilibrium strategy profile._  
_If \(\alpha\) is a constant such that for every node \(i\),_  
\[U_{i}(s^{*},s_{-i})\leq\alpha U_{i}(s^{Nash},s_{-i}), \tag{12}\]  
_then PoA is bounded by \(\alpha\), i.e.,_  
\[PoA\leq\alpha. \tag{13}\]  
Proof: Let \(s^{*}\) be the optimal strategy profile that maximizes the social welfare, which is the sum of all utilities:  
\[OPT=\sum_{i\in N}U_{i}(s^{*},s_{-i}). \tag{14}\]  
Similarly, let \(s^{Nash}\) be the Nash equilibrium strategy profile, where no node has an incentive to deviate unilaterally.