# 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
#### 3.3.1 Game Theoretic Energy Balanced (GTEB) Routing Protocol
To make this process easy, the sending node incorporates the angle that encompasses the desired subregion, location, and the number of sensor nodes in that region. Subsequently, one of the nodes in the selected subregion is selected, using CGT as a forwarding node. The selected node again deduces a forwarding zone using EGT and a forwarding node is chosen from the newly selected forwarding zone. This process continues until the destination node is determined. In this section of the protocol description, the formulation of EGT in GTEB will be briefly discussed. In the case of EGT, a sender node is supposed to distribute \(A\) packets over \(K\) forwarding subregions. This \(A\) parameter can be represented as:  
\[\lambda=\sum_{k=1}^{k}\lambda_{k}, \tag{25}\]  
where \(\lambda_{k}\) is the number of packets to be transmitted to the subregion \(k\). The selection of a subregion by the sender node depends on the adoption of a particular strategy from the strategy set. The sender node plays the evolutionary game for the packets. The proportion of the packets to be forwarded through a certain region can be formulated as:  
\[X_{k}=\frac{\lambda_{k}}{\lambda}. \tag{26}\]  
The vector \(X\) presents the population state over all the subregions. It is presented as:  
\[X=X_{1},\ X_{2},\ \ldots,X_{k}. \tag{27}\]  
This vector can be calculated using the replicator dynamics discussed later in this section of the paper. Before digging deep into the replicator dynamics, the fitness function and switching probability must be designed for EGT. The fitness function \(F_{k}(X)\) can be presented as:  
\[F_{k}(X)=E_{k}-\lambda X_{k}(2N_{k}E_{TR}+E_{TX}), \tag{28}\]  
where \(E_{TR}\) is the energy consumed by a node while receiving a packet, \(E_{TX}\) is the energy consumed while transmitting the same packet, \(N_{K}\) is the number of nodes in the \(k^{th}\) subregion, and \(E_{K}\) is the remaining energy in subregion \(k\) for a packet. The fitness-function expresses that if the remaining energy per packet in a subregion is greater than the energy needed for data reception and forwarding, the data packet will be forwarded. The replicator dynamics in EGT are used to capture the variation in the proportions of the population. Here, the population refers to the data packets. This packet population opts for different region forwarding strategies. It is the decision of the sending node that up to which proportion the data packets will be transmitted to different subregions. At every single game-interval, the sender attempts to evolve new proportions of packets by switching them to variant subregions. The switching probability \(P_{kl}(X)\) from any subregion \(l\) with fitness \(F_{l}(X)\) to the selected subregion \(k\) with fitness \(F_{k}(X)\) is calculated in GTEB [40]. A converse option (from subregion \(k\) to subregion \(l\)) is calculated too. The replicator dynamics that capture the net change in the number of packets for region \(k\) can be given as:  
\[X_{k}=\sum_{l\
eq k}X_{l}p_{kl}(X)-\sum_{l\
eq k}X_{k}p_{lk}(X), \tag{29}\]  
where \(X_{l}\) is the vector presenting the population state over the subregion \(l\), \(X_{k}\) is the vector presenting the population state over the subregion \(k\), \(P_{kl}(X)\) is the switching probability from any subregion \(l\) with fitness \(F_{l}(X)\) to the selected subregion \(k\) with fitness \(F_{k}(X)\), \(P_{lk}(X)\) is the switching probability from any subregion \(k\) to subregion \(l\). To determine the forwarding node within the selected forwarding region (selected using EGT), a noncooperative game is formulated. The main goal is to establish the energy balance in the region. In this energy balanced noncooperative game, \(N\) is a set of nodes (players) in the selected subregion and it is mandatory that \(N\) at least had two elements for playing the game. If there is one element, it would transfer the data packet directly to the destination.