# Game theory-based Routing for Wireless Sensor Networks: A Comparative Survey
## 3 Game Theory-Based Routing Protocols in WSNs
### Routing Protocols Based on Noncooperative Games
#### 3.2.2 Inter Cluster Routing Algorithm for a WSN  
Link quality and node residual energy are two key factors to be considered in routing. [PERSON] et al. designed a routing protocol based on game theory taking into consideration these two key factors [39]. They developed a game model based on the quality of service (QoS) and node residual energy. They claimed that the designed game theoretic model had a unique Nash equilibrium solution. The assumed network in the protocol [39] had sensor nodes with limited energy. A hexagonal cluster topology identical to a cellular network was deployed. The CH was placed at the center and had a fixed position. The ordinary sensor nodes gathered data required by the system in a periodic manner. They sent the data to their own subnet CH. The CH processed this data to further transmit it to the sink node through multiple hops. The CHs had limited range and were bound to collaborate with each other for data forwarding. The clustering algorithm designed in [39] can be regarded as a compromise between the network QoS and energy consumption. From the perspective of game theory, it was a game between the QoS and energy consumption of the nodes. The Nash equilibrium of the game was determined through the game solving process that extended the survival time of the network. The CH nodes that forwarded the data to the sink nodes were the players of the game. The CHs had to collaborate with each other for an extended range. There were two strategies, from which one could be chosen: (i) being a relay CH to help in the data forwarding process to the sink node, and (ii) not being a relay. The utility function was defined with the QoS demand (\(M_{i}\)) and the residual energy of the nodes,  
\[U(M)=\sum_{i=1}^{k}M_{i}e_{i}-\frac{1}{2\Big{(}\sum_{i=1}^{k}M_{i}^{2}+2\rho \sum_{i\
eq j}M_{i}M_{j}\Big{)}}-\sum_{i=1}^{k}E_{i}M_{i}, \tag{15}\]  
where \(M=\{M_{1},\,M_{2},\,\ldots,\,M_{K}\}\) is the QoS of \(k\) different nodes, \(i\) and \(j\) are CH nodes where one node wants to transfer data to the sink node and another has to relay it, \(\rho\) is the competitive factor between different values of QoS, and \(e_{i}\) is the efficiency factor of \(M_{i}\). Normally, the CH node would select a route with good QoS for packet forwarding. If the QoS requirements were higher, the energy consumption of the node would also be higher. Assuming that a node \(i\) decided to use node \(j\) as the next hop node and both \(i\) and \(j\) had the same amount of maximum energy, the decision function to select \(j\) as a next hop node was expressed as,  
\[u_{j}=\frac{\pi_{j}}{C_{j}}, \tag{16}\]where \(\pi_{j}\) is the revenue function for node \(j\) and \(C_{j}\) is the cost function for node \(j\), if it is selected as a relay node. \(\pi_{j}\) and \(C_{j}\) were expressed by the following two equations:  
\[\pi_{j}=M_{ji}\rho+\frac{E_{j}}{E}(1-\rho), \tag{17}\]  
\[C_{j}=(E_{j}^{\prime}+E_{j}^{t}).n/\Big{(}E_{i}+E_{j}\Big{)}, \tag{18}\]  
where \(M_{ji}\) is the QoS between node \(j\) and \(i\), \(E_{j}\) and \(E_{i}\) are the current surplus energy of nodes \(j\) and \(i\), respectively, E is the initial energy of the node, \(\rho\) is an adjustable parameter, \(E_{j}^{r}\) is the energy consumed by node \(j\) while receiving data, \(E_{j}^{t}\) is the energy consumed by node j while transmitting data, \(n\) is the forwarding data volume to node. The best strategy function in the designed game of this protocol was \(BRF(E_{-i})=argmax\ (E_{-i}\cup\{E_{i}\})\). Here, \(E_{-i}\) is the residual energy of the nodes except for node \(i\). The vector, \(E^{*}=\Big{(}E_{1}^{*},E_{2}^{*},\ldots,E_{i-1}^{*},E_{i}^{*},E_{i+1}^{*},E_{k }^{*}\Big{)}\) was regarded as the Nash equilibrium as it was true for all the nodes. Figure 4 illustrates a network scenario presenting the discussed protocol. Node \(A\) does not have a range to transmit data directly to the sink node. It must collaborate with another CH node, node \(B\) to transfer data.