# Game-Theoretic Power and Rate Control in IEEE 802.11\(p\) Wireless Networks
## 3 System Model
### Utility Function and Game Formulation  
We consider the wireless devices as selfish players, whose interest is to maximise their rewards in a similar manner as we see in [17]. To this end, we consider a game \(G\langle N,A,u\rangle\) with \(N\) players, \(A\) set of actions, and \(u\) payoffs in the form of utility functions in order to consider a non-cooperative game. Let us define the utility function for the \(i\)-th device which communicates with the \(j\)-th device, as follows  
\[u_{i,j}(p_{i})=R_{i,j}PRR_{i,j} \tag{3}\]  
where \(p_{i}\) is the transmission power for the \(i\)-device with \(p_{i}\in[p^{min},p^{max}]\), which also represents the set of actions. Transmission rate \(R_{i,j}\) is defined as:  
\[R_{i,j}=B\log_{2}(1+\gamma_{i,j})\in[0,R^{max}] \tag{4}\]  
which is a function of green transmission power adjustment, the interference, and the noise. \(B\) is the bandwidth, and \(R^{max}\) is the maximum achievable rate. Recall that the  
Figure 2: PRR as a function of SINR. packet reception rate at the physical layer \(PRR_{i,j}\) has been defined in Equation (1), which is a function of the SINR \(\gamma_{i,j}\). Essentially, the introduced utility function is the product of the transmission rate \(R_{i,j}\) and the \(PRR_{i,j}\), which both depend on the transmission power \(p_{i}\). In our game formulation, each game attempts to maximise its utility by selecting the transmission power that achieves the best PRR and rate, which with the rate it is considered to be the throughput of a link \((i,j)\). Therefore, we want to perform the following optimisation procedure  
\[\max_{p_{i}}\ u_{i,j}(p_{i})\] subject to \[PRR_{i,j}\geq PRR_{w}\] \[R_{i,j}\geq B\log_{2}(1+SINR_{i,j}^{w}), \tag{5}\]  
where \(SINR_{i,j}^{w}\) is the SINR which obtains \(PRR_{w}\), \(PRR_{w}\) is a value of PRR that shows a highly reliable link (\(\geq\)90 %), and \(R_{i,j}\) is given as the rate node that \(i\) can transmit with when \(PRR_{w}\) is satisfied. Thereafter, we formulate this non-cooperative game into a potential game. In order to use exact potential games, it is essential to have a potential function that has the same behaviour as the individual utility function when a player unilaterally deviates. More formally, a game \(G\langle N,A,u\rangle\), with \(N\) players, \(A\) strategy profiles and \(u\) as the payoff function, is an exact potential game if there exists a potential function  
\[\Phi: A\rightarrow\mathbb{R}\] subject to \[\forall i\in G,\forall\sigma_{-i}\in A_{-i},\forall\sigma_{i}, \sigma_{i}^{{}^{\prime}}\in A_{i} \tag{6}\]  
where \(\sigma_{i}\) is the strategy of player \(i\), \(\sigma_{i}^{{}^{\prime}}\) is the deviation of player \(i\), \(\sigma_{-i}\) is the set of strategies followed by all the players except player \(i\), and \(A_{-i}\) is the set of strategy profiles of all players except \(i\) such as  
\[V(\sigma_{i},\sigma_{-i})-V(\sigma_{i}^{{}^{\prime}},\sigma_{-i})=u_{i}( \sigma_{i},\sigma_{-i})-u_{i}(\sigma_{i}^{{}^{\prime}},\sigma_{-i}) \tag{7}\]  
We define the potential function \(\Phi_{i,j}\) as  
\[\Phi(p)=\sum_{i,j\in S_{i}}^{N}R_{i,j}PRR_{i,j}^{phy} \tag{8}\]  
**Lemma 1**. : _The \(G\langle N,A,u\rangle\) is a potential game with the potential function given by (8)._  
Proof. : We have  
\[R_{i,j}=B\log_{2}(1+\gamma_{i,j})\in[0,R^{max}] \tag{4}\]0  
where \(S_{k}\) is the neighbourhood of node \(k\). By the definition of the exact potential game, one node can deviate to accomplish a Nash equilibrium. If we allow node \(i\) transmitting to node \(j\) to deviate, the remaining nodes remain with their current strategies. Hence, it has to be shown that  
\[R_{i,j}=B\log_{2}(1+\gamma_{i,j})\in[0,R^{max}] \tag{4}\]1From (2), which gives the SINR expression, the reader can see that the data rate as it comes from the upper bound of [PERSON]'s formula is  
\[R_{i,j}=B\log_{2}(1+\gamma_{i,j})\in[0,R^{max}] \tag{4}\]2  
where \(B\) is the bandwidth.