# A Hybrid Energy Equating Game for Energy Management in the Internet of Underwater Things
## 4 Hybrid Energy Efficient Clustering Game
### Cluster Head Selection Game
#### Equilibrium Strategy Calculation  
The CH selection process is an important step of our algorithm because it is performed in each round. This process is completed in two steps; in the first step, Tentative Cluster Heads (TCHs) are selected in the network and clusters are formed when the member nodes join these TCHs. In the second step, these selected TCHs are confirmed as Final Cluster Heads (FCHs) by limiting the number of member nodes to equally distribute the energy load on the network as shown in Figure 4. At the beginning of the network an expected probability \(EX_{prob}\) is computed through the following equation:  
\[EX_{prob}=\frac{F}{\pi(\varepsilon R)^{2}N} \tag{13}\]  
where, \(F\) and \(R\) are the NSF and communication radius, respectively, while N represents the number of FNs in the NSF and \(\varepsilon\) is a radius adjustment factor in the range of \((0,1)\). After computing the \(EX_{prob}\), each FN in the NSF generates a Random Number \(R_{no}\) and determines its rank for the existing round by comparing \(R_{no}\) with \(EX_{pro}\). In comparing \(R_{no}\) with \(EX_{pro}\) there are the following cases:  
* **Case 1:** If \(R_{no}<EX_{pro}\), the FN will declare its status as TCHs for the existing round. * **Case 2:** If \(R_{no}>EX_{pro}\), the FN will remain the FN for the existing round. * **Case 3:** If \(R_{no}=EX_{pro}\), the FN will declare its status as Expected Cluster Head (ECH) for the existing round. From the above-discussed cases, all the FNs will actively participate in this CSG and each TCH and ECH selected through \(EX_{pro}\) can simply compute its next round equilibrium probabilities. Before moving further, we need to clarify that the number of TCH, ECH, and SMNs are equal to the total number of FNs. Second, the TCH acquired all the residual energies of the participating players and we also need to define the energy consumption per unit area according to Reference [30]. So, whenever a player \(i\) chooses a strategy \(ND\) and another player chooses strategy \(D\) their payoff can be calculated through the following equation:  
\[v_{i}=\frac{k}{E_{cm_{i}}} \tag{14}\]  
where \(E_{cm_{i}}\) is energy spent to convey \(k-bit\) data-packet to the concern TCH and computed using the following equation:  
\[E_{cm_{i}}=kE_{elec}+kE_{amp}\bigg{(}\frac{2}{3}R\bigg{)}^{2} \tag{15}\]  
where \(2R/3\) expresses the average distance between FN and its corresponding CH. However, \(e_{i}\) is the extra cost for FN \(i\) on choosing the strategy \(D\) can be calculated by:  
\[e_{i}=v_{i}-\frac{k}{E_{ch_{i}}} \tag{16}\]  
where \(v_{i}\) is the payoff for FN \(i\) on choosing the strategy \(ND\) when there exists at least one additional FN selecting the strategy \(D\). \(E_{ch_{i}}\) is used to express the energy expend by FN \(i\) in forwarding a data packet of \(k\) bits to the SS while acting as the FCH. The \(E_{ch_{i}}\) can be computed using the following equation:  
\[E_{ch_{i}}=kE_{elec}Nb_{i}+kE_{DA}(Nb_{i}+1)+kE_{amp}d_{i}^{4} \tag{17}\]  
Putting the value of \(v_{i}\) as:  
\[e_{i}=\frac{k}{E_{cm_{i}}}-\frac{k}{E_{ch_{i}}} \tag{18}\]  
Figure 4: The tentative and expected CHs selection process of the proposed model on the basis of expected probability. \[\frac{e_{i}}{v_{i}}=1-\frac{E_{cm_{i}}}{E_{ch_{i}}} \tag{19}\]  
Let \(w_{i}=e_{i}/v_{i}\), we have the following equation:  
\[w_{i}=1-\frac{E_{cm_{i}}}{E_{ch_{i}}} \tag{20}\]  
In Equation (20), \(E_{ch_{i}}\) is used to express the energy expended by FN \(i\) in forwarding a data packet of \(k\) bits to the SS while acting as the FCH. Sometimes \(w_{i}\) acquires negative values; to avoid such a case, we add another variable \(w_{min}\).