# 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  
#### 3.2.1 Resource Allocation Strategy for Cluster-based WSNs  
[PERSON] et al. designed a fair resource management scheme for WSNs based on the Nash bargaining solution (NBS) [36]. NBS is a game theoretic concept that provides a solution to game theory by maximizing the product of utilities that a user can obtain through cooperation compared to that achieved through noncooperation. The authors of [36] also modeled the energy consumption for the cluster members and data transmission. In this routing scheme, the focus was on the resource allocation for frequency interference between clusters. A static cluster based WSN was assumed. Each cluster had the same number of members and they used identical frequency and bandwidth. CMs were assumed to communicate with the CH in a time division multiple access (TDMA) fashion. The sensor nodes with omni-directional antennas were used. The number of members in a certain cluster was equal to the time slot length (\(N\)) of a frame. The scheduling order was pre-assumed for the designed routing scheme. In [36], an additive white Gaussian noise channel was used for each link. An energy consumption model was presented that only considered the energy consumption of the CMs. The energy consumption of the CHs and sink node were ignored. The minimum transmission power required to transmit data packets can be given by:  
\[p_{i}(n)=I_{i}(n)\bigg{(}2^{\frac{r_{i}(n)}{W_{i}(n)}}-1\bigg{)}/|\mu_{ii}(n)|^ {2}, \tag{10}\]  
where \(n\) is a particular time slot, \(i\) is any node in a certain cluster, \(r_{i}(n)\) is the rate of node \(i\) in time slot \(n\), \(h_{ii}(n)\) denotes the channel gain of a link from a cluster member to its CH, \(W_{i}(n)\) is the bandwidth of operation, and \(I_{i}(n)\) is the total amount of interference and noise power. The lifetime of a cluster member \(i\) was formulated as,  
\[\tau_{i}(n)=\frac{E_{i}}{P_{i}(n)}, \tag{11}\]  
where \(\tau_{i}(n)\) is the network lifetime, \(P_{i}(n)\) is the minimum transmission power needed at a certain data rate, and \(E_{i}\) is the energy of the cluster member \(i\). A set of achievable utility \(U\), and a disagreement vector \(u^{c}=\big{[}u^{c}_{i},\ldots,u^{c}_{L}\big{]}^{T}\) were used to present NBS. The disagreement vector was designed using a noncooperative game. The NBS was applied for resource allocation between clusters through the utility of the CMs. The utility was modeled based on the lifetime of the CMs. The utility function was designed as,  
\[U=\Big{\{}u_{i}(W_{i}(n))|W_{i}(n)\in W_{n},u_{i}(W_{i}(n))\geq u^{c}_{i}\Big{\}}, \tag{12}\]  
where \(W_{n}\) is the frequency vector, \(W_{i}(n)\) is the bandwidth of operation, and \(u^{c}_{i}\) is the disagreement vector for node \(i\). When there was a failure between the clusters while bargaining, the disagreement vector of the introduced model was obtained. The model assumed that the disagreement vector consisted of constants to put an emphasis on resource allocation for successful bargaining. The utility function (\(U\)) was determined to be convex and NBS was unique. It was obtained through the following equation:  
\[\rho=arg\ max_{u_{i}\in\cup}\prod_{i=1}^{L}\big{(}u_{i}-u^{c}_{i}\big{)}, \tag{13}\]  
where \(u_{i}\) is the utility of node \(i\), \(U\) is the overall utility, \(L\) is the number of nodes in a certain cluster, and \(u^{c}_{i}\) is the disagreement vector for node \(i\). Finally, for a certain slot \(n\), the optimal power control and resource allocation problem was solved in a joint manner using the following formulation:  
\[max_{u_{i}\in\cup}\sum_{i=1}^{L}\log\Big{(}u_{i}(W_{i}(n),p_{i}(n))-u^{c}_{i}(n )\Big{)}, \tag{14}\]where \(W_{i}(n)\) is the fraction of bandwidth in slot \(n\), \(W\) is the total bandwidth of operation, and \(u_{i}^{c}\) is the disagreement vector for node \(i\).