# Game theory-based Routing for Wireless Sensor Networks: A Comparative Survey
## 3 Game Theory-Based Routing Protocols in WSNs
### Routing Protocols Based on Cooperative Games  
1.1 Cluster Head Cooperative Trustworthy Energy-efficient Multi-input Multi-output (CH-C-TEEM) Routing Protocol  
WSNs are energy constrained and they require energy-efficient routing protocols for a fading environment. Many of the state-of-the-art works utilized a multi-input multi-output (MIMO) scheme to increase energy efficiency in a fading environment. [PERSON] et al. designed the cluster head cooperative trustworthy energy-efficient MIMO (CH-C-TEEM) routing algorithm to manage the fading environment in a WSN [31]. This protocol was based on game theory and its performance was analyzed based on energy consumption. Channel fading and interference may cause a reduced network lifetime of the sensor nodes. To increase the lifetime, a CH-based cooperative MIMO scheme was designed. In the MIMO scheme, it was possible to create small groups among the sensor nodes participating in routing. The cooperative game theory was used to form such small groups for cooperative data transmission, as discussed in [31]. The CHs cooperate among themselves for data transmission and each node becomes a CH consecutively. The CH-C-TEEM also incorporates a trust-based framework through a game-theoretic model to reduce packet loss and routing overhead by eliminating malicious nodes, which tend to drop useful data packets. Sensors in the game-theoretic model are intelligent routing agents. A CH election game is designed in the following way:  
\(N=\) A set of players (sensor nodes in this case);  
\(A=\) A set of available actions for a sensor node to finalize a decision;  
\(L=\{l_{1},l_{2},l_{3},\ldots,l_{n}\}\), where \(L\) is a set of strategies. If any node \(n_{1}\) chooses a strategy \(l_{1}\), the value of \(l_{1}\) is \"1\" if the node chooses to be a CH and \"0\" otherwise;  
The pay-off set is \(\{u_{1},u_{2},u_{3},\ldots,u_{n}\}\), a set containing numerical values generated from the strategy profile. The pay-off function for the CH election game is:  
\[U=\alpha\frac{E_{i}}{E_{init}}+\beta R_{i}-\gamma n, \tag{2}\]  
where \(\alpha\) is the weight parameter of the node's residual energy level, \(\beta\) is the weight parameter of the node's trust level, \(\gamma\) is the weight parameter corresponding to average path loss, \(E_{init}\) is the node's initial energy level, \(E_{i}\) is the node's current residual energy level, and \(R_{i}\) is node's trust level. The cooperative nodes that participate in the MIMO communication are chosen based on a coalitional game designed using the residual energy level of the sensor nodes that act as players, and the game is played to sort out a particular set of nodes for cooperative data transmission and reception. The aim of this kind of grouping is to decrease the total power consumption. The game is modeled as \((N,v,S)\), where \(N=\) a set of CH nodes, \(v=\) characteristic function, and \(S\) is the partition of \(N\) (\(S\subseteq N\)). The characteristic function of the system was based on network lifetime and it is presented by,  
\[vS_{j}=x^{S_{j}}\in R^{|S_{j}|}|x_{i}^{S_{j}}=T_{net},\ \forall i\in S, \tag{3}\]  
where \(x_{i}^{S_{j}}\) is the utility of the node within the coalition \(S_{j}\), \(|S_{j}|\) is the number of sensor nodes in \(S_{j}\), \(T_{net}\) is the network lifetime. The network lifetime in this protocol is defined as the time taken for the first sensor node to run out of energy. We assume \(T_{co1}\) and \(T_{co2}\) to be the lifetime of the first two nodes that are in coalition and are cooperatively participating in data communication. Then, \(T_{non-co1}\) and \(T_{non-co2}\) are the lifetimes of the other two nodes that are not in coalition and are not participating in data communication.