# A Game Theoretic Approach for Balancing Energy Consumption in Clustered Wireless Sensor Networks
## 5 The Protocol Details
### CHs Election  
To select the appropriate CHs in the network, each node holds a probability to be CH, and decides whether being a CH based on this probability. In the first round of CHs election, each node decides whether being a CH according to the expected probability which is expressed as follows:  
\[p_{ept}=\frac{S}{\pi(\varepsilon R)^{2}n} \tag{22}\]  
where \(S\) is sensor field size; \(R\) is communication radius of each node; \(n\) is the total number of sensor nodes in the network and \(\varepsilon\) is the radius adjustment factor within the range of \((0,1)\). We provide some explanations about the significance of the expected probability: as the communication radius of each node is \(R\), the area covered by a CH is exactly equal to \(\pi R^{2}\). Then the expected number of CHs in the network is \(S/(\pi R^{2})\). However, CHs are randomly selected based on this expected probability, so we need to select more CHs for further reselection and the radius adjustment factor \(\varepsilon\) is introduced to update the expected number of CHs. Finally, the expected probability of being the CH can be expressed by Equation (22). For a sensor node, if it is finally selected as the CH, then it will take the responsibilities of receiving, aggregating and forwarding data for its member nodes after the cluster formation phase is finished. Most importantly in our protocol, it will calculate the probabilities to be CH in the next round for its member nodes. Figure 3: Procedure of the proposed game theoretic clustering protocol. For any CH node \(ch_{k}\), it checks the number \(n_{k}\) of member nodes that are still alive after the cluster topology is constructed. If \(n_{k}<1\), \(ch_{k}\) still has to campaign for the CH in the next round according to the expected probability, since there is no other CH candidate within the cluster. If \(n_{k}=1\), the probability to be CH of \(ch_{k}\) in the next round is set to be zero to avoid continuous working as CH, and the only CM node will campaign for the CH in the next round according to the expected probability. If \(n_{k}>1\), there are more than one CH candidate within the cluster, then the probability to be CH for \(ch_{k}\) in the next round is also set to be zero. Moreover, a \(CEG\) and probability transformation mechanism will be introduced by \(ch_{k}\) for the purpose of calculating the probabilities to be CH for its member nodes. As for the third case, all active member nodes of \(ch_{k}\) will act as players to join a \(CEG\). By solving the Optimization Problem (21), \(ch_{k}\) can calculate the absolute equilibrium probabilities to be CH in the next round for its member nodes according to Equations (13)-(15). However, several parameters need to be determined in \(CEG\). Firstly, the number of players that participate in the \(CEG\) is exactly equal to the number \(n_{k}\) of active member nodes of \(ch_{k}\). Secondly, the residual energy of each player can be acquired by \(ch_{k}\) during the process of clusters formation phase. At last, to determine \(v_{i}\) and \(e_{i}\) for player \(i\), we define the payoff of a player as the number of data bits per unit of energy consumed to transmit the data to CH or BS [30]. Then when player \(i\) chooses the strategy \(ND\) while there exists at least another player choosing the strategy \(D\), its payoff \(v_{i}\) can be calculated by:  
\[v_{i}=q/Ecm_{i} \tag{23}\]  
where \(q\) is the length of data packet, and \(Ecm_{i}\) is the amount of energy consumed to transmit the data packet to the corresponding CH that can be expressed as follows:  
\[Ecm_{i}=q\zeta+q\varepsilon_{f}\left(\frac{2}{3}R\right)^{2} \tag{24}\]  
where \(\zeta\) is the energy consumed by the transmitting or receiving circuitry; \(\varepsilon_{f}\) is the amplifier characteristic constant corresponding to free-space propagation model; \(R\) is the communication radius of each node, and the average distance between a member node and its corresponding CH is \(2R/3\).