# 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
#### 3.1.4 Probabilistic Routing Scheme Based on Game Theory (PRGT)  
Opportunistic networks (OppNets) do not always guarantee an end-to-end path from source to destination. In OppNets, routing is quite challenging because of the intermittent network connection. Nodes may act selfishly and show an unwillingness to participate in message forwarding. To address this issue, [PERSON] et al. proposed a probabilistic routing scheme based on game theory (PRGT) to simulate cooperation among the selfish nodes [44]. In PRGT, an incentive probabilistic routing scheme for message transmission in OppNets was proposed. Based on the previous history of the nodes meeting each other, the authors of [44] established a first-order Markov process. Then, a game-theoretic approach was proposed to reduce the selfishness of the relay nodes and increase cooperation among them. All the messages to be transmitted and received were sole identities that were addressed as commodities. Nodes in the network were assumed to have virtual money that could be used to buy and sell the commodities. When nodes did not participate in message forwarding, they would lack enough money to have more profits. To precipitate cooperation among nodes, if a destination node received a message, all the other nodes involved in the transmission process got rewarded. PRGT considers a general opportunistic model, where there is no assurance that an end-to-end path would exist. Nodes in the network communicate with each other via a store-carry-forward mechanism. Initial node energy is represented as \(E_{s}\) and the buffer size as \(B_{s}\). The authors introduced a credit clearance center (CCC) and each node had an account in it. CCC is basically a server connected to the Internet. When a message is forwarded by a node, it is supposed to produce a receipt of a digital signature. After that, the receipt is submitted by both the message carrier and the relay node. When the destined node finally received the message, it would inform the CCC about the reception and the CCC awards all the relay nodes. This phenomenon indicates that the relay node would have enough virtual money to forward more messages so that they could gain more profits. To maximize cooperation among the nodes, the message forwarding process was modeled as a bargaining game. The game consisted of two players: buyer and seller. The message-carrying node was assumed as a seller (S) and the relay as a buyer (\(B\)). It was necessary that both players reach a transaction price for every message in a round. A deal was put forward by seller \(S\) first at the beginning of the bargaining game. Buyer \(B\) must decide if the deal would be accepted. If the deal was accepted, the game ended. Otherwise, it would go to the next round. This process was repeated unless the transaction was done completely. In PRGT, every node is considered as a message storage entity. Therefore, the forwarding process of the message (\(m\)) is regarded as a binary set having a series of discrete positions (\(P_{i}^{m}\)), and time elements (\(T_{i}^{m}\)). The transfer process of the messages is described as:  
\[PT=\Big{\{}(P_{1}^{m},T_{1}^{m}),\ \ldots,\ \Big{(}P_{i}^{m},\ T_{i}^{m} \Big{)},\ \ldots,\ (P_{n}^{m},P_{n}^{m})\Big{\}}, \tag{8}\]  
where \((P_{i}^{m},T_{i}^{m})\) indicates that at the \(T_{i}^{m}\) moment, the message \(m\) is in the position \(P_{i}^{m}\). The process was also described as a Markov chain [44]. Nodes in the network represent the set of states (S): \(S=\{1,2,\)\(3,\)\(\ldots,\)\(i,\)\(\ldots,\)\(n\}\). Here, \(i\) denotes a message in a certain node \(N_{i}\). The transfer probability matrix is denoted as \(P\). \(P_{i},j\in P\) denotes the transition probability between states \(i\) and \(j\). In other words, it is the probability of the message \(m\) to be forwarded from \(N_{i}\) to \(N_{j}\). The selfishness of the nodes participating in the routing process is divided into two types. One is self-centered selfishness (SCS), which is related to residual energy and buffer the nodes.