# Game theory-based Routing for Wireless Sensor Networks: A Comparative Survey
## 2 Preliminaries
### Games in Routing for WMSNs
It is not necessary to know what offers and counter offers they present. A coalition is a subset of the set of players. Coalitions are formed such that the players can coordinate strategies and agree on how the total pay-off will be divided among the members. This type of game consists primarily of two elements. One is the set of players and the other is the characteristic function specifying the value created by the different subsets of the players in the game. Let us consider \(P\) to be the set of playersand the number of players is \(N\). In fact, a cooperative game is a pair \(\{N,v\}\), where \(N\) is a finite set and \(v\) is a function that maps subsets of \(N\) to numbers. If no cooperation is allowed among the players, we consider the paradigm of noncooperative games. When a solution point exists where none of the players can improve their pay-off by a unilateral move, it is known as a noncooperative equilibrium or the Nash equilibrium. According to [51], for the precise formulation of a noncooperative game, we have to specify: (i) the number of players, (ii) the possible actions available to each player and any constraints that may be imposed on them, (iii) the objective function of each player that he attempts to optimize (minimize or maximize, as the case may be), (iv) any imposed time requirements for the execution of the actions if the players are allowed to act more than once, (v) any information acquisition that occurs and how the information available to a player at each point in time depends on the past actions of other players, and (vi) whether the action of a player (nature) is the outcome of a probabilistic event with a fixed (known) distribution. Let us assume a WSN where a flat routing protocol is implied, which uses relay nodes to send the data packet from the source to the destination. The source sends the data to four of the relay nodes that are within its range. The relay nodes will have to decide among themselves whether to transmit the data and choose a node among themselves to do it. This will require a complex decision-making process to find an optimal relay node. In noncooperative games, the players (nodes in the WSN) attempt to maximize their own benefit by increasing their pay-off through optimal strategy adaptation. The goal is to reach an equilibrium point where nodes will behave selfishly to obtain a strategy to overwhelm the benefit, which would result in a decrease in the pay-off of other nodes. If the four relay nodes have different energy levels, it is not advisable for the low energy node to forward the data packet. Conversely, even nodes with high energy will be reluctant to aggregate the data packet to save their energy for a longer survival time in the network. In that case, not sending the data will result in a decrease in pay-off of the other nodes and eventually reduced functionalities of the network. Alternatively, other scenarios are possible, such as if a node has a low energy or if it is very near to the destination node, it may decide to send the data packet for reduced delay and overall network lifetime increase. There are many strategies that can be specified. The target is to find strategies that will uniformly distribute the benefit of the entire performance by finding out the Nash equilibrium and using carefully designed pay-off functions. In evolutionary game theory, the behavior of large populations of agents who repeatedly engage in strategic interactions is studied. The key concepts of evolutionary games are the numerous behaviors involving the interaction of multiple organisms in a population. The success of any of these organisms is highly dependent on how it interacts with others. Therefore, the fitness of an organism should not be measured individually, instead, it must be measured considering the full population where it resides. To elaborate evolutionary game theory with an example related to WSNs, let us assume a cluster where there are nodes that have different energy levels.