# A Game Theoretic Approach for Balancing Energy Consumption in Clustered Wireless Sensor Networks
## 4 Cluster Head Election Game
### Searching for the Equilibrium Strategy  
In this subsection, we search for the equilibrium strategy under which no sensor node has incentives to enhance its own payoff by unilaterally changing the selected strategy. Letting \(P\) be the strategy combination of all sensor nodes in set \(N_{G}\), \(P_{-i}\) be the strategy combination of sensor nodes excepting node \(i\) and \(N\) be the total number of sensor nodes. Then we define the NE strategy of the \(CEG\) as follows:  
**Definition 1**. : _A strategy combination \(P=\{p_{1},p_{2},\ldots,p_{N}\}\) is an NE point of the CEG, when inequality \(H_{i}(P)\)\(\geq H_{i}(P_{-i},p_{i})\) is satisfied for any sensor node \(i\) in set \(N_{G}\)[28]._  
**Proposition 1**. : _The strategy combination \(P^{\rm NE}\) as shown in Equation (15) is a NE point of the CEG._  
\[P^{NE}=\{{p_{1}}^{m},{p_{2}}^{m},\cdots,{p_{i}}^{m},\cdots,{p_{N}}^{m}\},\;s.t.\;P^{NE}\in R^{N}\mbox{\rm and}\forall i\in N_{G},0\leq{p_{i}}^{m}\leq 1 \tag{15}\]  
_where \({p_{i}}^{m}\) is the probability of being the CH for sensor node \(i\) which is calculated by Equation (14), and N is the total number of nodes participated in the CEG._  
Proof. : Since each sensor node is modeled as a rational and selfish individual, it expects to abdicate the duty of serving as a CH to other nodes for energy saving. However, each node also is afraid that no profits can be acquired from data transmission, as it cannot know other nodes' decisions in advance. Then any node \(i\) decides whether being the CH according to a probability \(p_{i}\), we recalculate the payoff function \(H_{i}(P_{-i},p_{i})\) for node \(i\) as follows:  
\[H_{i}(P_{-i},p_{i})=\bigg{(}e_{i}\frac{E_{i}}{E_{\rm max}}-v_{i}\bigg{)}{ \Theta_{-i}}^{2}(1-p_{i})^{2}-e_{i}\Theta_{-i}(1-p_{i})+v_{i}-e_{i}p_{i} \tag{16}\]  
where \(v_{i}\) is the payoff when node \(i\) chooses the strategy \(N\!D\) while there exists at least one CH; \(e_{i}\) is the extra cost when node \(i\) chooses \(D\); \(E_{i}\) is residual energy of node \(i\); \(E_{\rm max}\) is the maximum residual energy of sensor nodes participated in the CEG; \(\Theta_{-i}\) is the multiple formative of probabilities not to be the CH for all sensor nodes excepting node \(i\) and \(P_{-i}\) is the strategy combination excepting node \(i\). \(\square\)  
Since \(v_{i}>e_{i}>0\), \(E_{max}\geq E_{i}>0\) and \(\Theta_{-i}>0\), then \(H_{i}(P_{-i},p_{i})\) is a quadratic convex function depending on probability \(p_{i}\). And the maximum point of function \(H_{i}(P_{-i},p_{i})\) is the probability \({p_{i}}^{\prime}\) calculated by Equation (13). If \({p_{i}}^{\prime}<0\), then \(H_{i}(P_{-i},p_{i})\) is monotonically decreasing with the increase of \(p_{i}\) from 0 to 1. That is:  
\[\forall p_{i}\in[0,1]s.t.H_{i}(P_{-i},p_{i})\leq H_{i}(P_{-i},0). \tag{17}\]  
If \({p_{i}}^{\prime}\geq 0\), then \(H_{i}(P_{-i},p_{i})\) firstly increases with the increase of \(p_{i}\) from 0 to \({p_{i}}^{\prime}\), and then decreases with the increase of \(p_{i}\) from \({p_{i}}^{\prime}\) to 1. Hence, we have:  
\[\forall p_{i}\in[0,1]s.t.H_{i}(P_{-i},p_{i})\leq H_{i}\big{(}P_{-i},{p_{i}}^{ \prime}\big{)}. \tag{18}\]  
Based on Equations (14), (17) and (18), we have:  
\[\forall i\in N_{G},\forall p_{i}\in[0,1]s.t.H_{i}(P_{-i},p_{i})\leq H_{i}(P_{ -i},{p_{i}}^{m}). \tag{19}\]  
which means that any sensor node \(i\) in set \(N_{G}\) can maximize its own payoff \(H_{i}(P_{-i},p_{i})\) when it decides whether being the CH according to the probability \({p_{i}}^{m}\) calculated by Equation (14). As the selfish behaviors of sensor nodes, each node in set \(N_{G}\) will decide whether being the CH according to the probability calculated by Equation (14) simultaneously for maximizing its payoff. Then the selected strategy combination is \(P^{NE}\) that calculated by Equation (15). Any sensor node \(i\) in set \(N_{G}\) has no incentive to deviate from it unilaterally, because the inequality \(H_{i}(P_{-i}^{NE},p_{i}^{m})\geq H_{i}(P_{-i}^{NE},p_{i})\) is always true according to Equations (14)-(19).