# Game theory-based Routing for Wireless Sensor Networks: A Comparative Survey
## 3 Game Theory-Based Routing Protocols in WSNs
### Routing Protocols Based on Evolutionary Games
#### 3.3.3 Game Theory-Based Energy-Efficient Clustering (GEEC)  
WSNs are energy limited networks and significant attention has been focused on increasing the life span of WSNs. Increasing the network lifetime of a WSN can be achieved by designing efficient routing protocols. [PERSON] et al. designed a game theory-based energy efficient clustering routing protocol (GEEC) that used an EGT mechanism to achieve simultaneous energy exhaust equilibrium and lifetime expansion [42]. Sensor nodes deplete their energy through operations like sensing, receiving, and transmitting. Only the energy consumption for data transmission was considered in this protocol [42]. It can be formulated using (28), similar to the previous EGT-based routing protocol. All the nodes in the network are capable of varying their transmission power for different communication ranges using a power control mechanism. The sensor nodes are location aware and have fixed positions. The authors of [42] also assumed that the sink node had an infinite amount of processing power and energy. They were randomly deployed in the network area. Based on the basic principles of EGT, the CH selection game in [42] had three components. The structure of the game model could be presented as _G (P, S, U)_. Here, P was the set of players (nodes), \(S\) was the set of strategies, and \(U\) was the utility function. The players were divided into two classes. Class \(M\) contained the players with more energy than the others and class \(L\) had the nodes with lesser amount of energy. The players in set \(P\) could opt for two strategies: to either become a CH or to remain in the cluster as a CM. Therefore, it can be said that _P = (M, L)_ and _S = (being CH, being CM)_. The utility function consisted of a reward and penalty. It was formulated as:  
\[u=r-p, \tag{32}\]  
where \(r\) is the reward and \(p\) is the penalty. Usually, the nodes with higher energy level send out the information of CH status to other nodes within the transmission range. After receiving the CH announcement, other nodes decide if they will join the cluster or not. It is possible for the nodes with low energy to send CH advertisements repeatedly to other nodes in the transmission range, owing to their limited rationality. Owing to this phenomenon, several control messages are sent, causing energy wastage. The novelty of GEEC lies in the avoidance of this energy wastage by introducing rationality to the nodes through EGT. The utility function is designed in a way that it encourages the nodes in class \(M\) to become CHs. We discussed replicator dynamics (change rate of the players' action) in previously discussed protocols. The authors of [42] formulated equations for replicator dynamics. It was assumed that the rate of class \(M\) nodes in selecting the strategy of becoming CH was \(x\) and \(M\) nodes opting for the strategy of remaining as CMs was (_1-x_). Similarly, the rate of class \(L\) nodes selecting the strategy of becoming CH was assumed to be \(y\) and \(L\) nodes opting for strategy to remain as CMs was (_1-y_). The utility functions of the nodes in class \(M\) choosing the strategy of becoming CHs (\(U_{M-CH}\)) and choosing the strategy of remaining as CMs (\(U_{M-CM}\)) were formulated subsequently by the following two expressions:  
\[U_{M-CH}=ya+(1-y)(a+2\Delta), \tag{33}\]  
\[U_{M-CM}=y(a-\Delta)+(1-y)(-\Delta), \tag{34}\]  
where \(a\) is the reward for the \(M\) nodes in becoming a CH, \(\Delta\) is the step to punish or encourage.