An MPS-BNS Mixed Strategy Based on Game Theory for Wireless Mesh Networks

To achieve a valid effect of wireless mesh networks against selfish nodes and selfish behaviors in the packets forwarding, an approach named mixed MPS-BNS strategy is proposed in this paper. The proposed strategy is based on the Maximum Payoff Strategy (MPS) and the Best Neighbor Strategy (BNS). In this strategy, every node plays a packet forwarding game with its neighbors and records the total payoff of the game. After one round of play, each player chooses the MPS or BNS strategy for certain probabilities and updates the strategy accordingly. In MPS strategy, each node chooses a strategy that will get the maximum payoff according to its neighbor's strategy. In BNS strategy, each node follows the strategy of its neighbor with the maximum total payoff and then enters the next round of play. The simulation analysis has shown that MPS-BNS strategy is able to evolve to the maximum expected level of average payoff with faster speed than the pure BNS strategy, especially in the packets forwarding beginning with a low cooperation level. It is concluded that MPS-BNS strategy is effective in fighting against selfishness in different levels and can achieve a preferable performance.


Introduction
In wireless networks such as ad hoc and mesh networks, the steady operation of the networks must rely on the cooperation for which packets should be forwarded sufficiently between nodes, namely, the routers in networks. However, for the sake of saving battery life for their own communications, some nodes in wireless network do not cooperate to the basic network functioning, making the network lack trust between nodes. ese nodes refuse to forward packets by dropping other's packets and trying to use others to send their own. is noncooperative behavior is named sel�sh behavior, and these routers are sel�sh nodes. �sers or nodes that want to maximize their own welfare and do not contribute to the network are de�ned as sel�sh nodes or free riders [1]. Now the sel�sh behavior is an important issue and is being vigorously researched. e simulation study presented in Michiardi and Molva [2] showed that the performance of multihop wireless networks severely degrades in face of sel�sh node's misbehavior. To deal with this problem, many mechanisms have been proposed to detect the sel�sh nodes and restrict their sel�sh behaviors in wireless networks. In terms of the previous work on this problem, there are main approaches based on the credit approach, the reputation approach, network coding approach, game theory, and so on.
In the credit approach [3][4][5][6][7], cooperation is induced by means of payments received every occasion when a node acts as a relay and forwards a packet, and such credit can later be used by these nodes to encourage others to cooperate.
Reputation-based schemes observe the behavior of their neighboring nodes through promiscuous overhearing and accordingly assign them a reputation rating which is used for identifying the sel�sh nodes. In reputation-based schemes [8][9][10][11][12][13][14], a node's behavior is measured by its neighbors using a watchdog mechanism, and nodes can be punished for noncooperation. To deal with the issue, cooperative nodes sometimes are perceived as being sel�sh because of unreliable transmission. Joang et al. [15] proposed a method based on subjective logic to discover the trust networks between speci�c parties. Kane and Browne [16] successfully transplanted and applied subjective logic to a wireless network environment. Due to the unreliability and lack of information of the wireless networks, the pure subjective logic-based model may lead to a high uncertainty value in some cases. To solve the problem, a reputation computation model was proposed to discover and prevent sel�sh behaviors by combining familiarity values with subjective opinions [17]. And to compute the reputation of the nodes in wireless networks, some techniques such as network coding and fuzzy recommendation were proposed [18,19].
For the credit approach and reputation approach, because the system needs a central control requiring an infrastructure, this method cannot be used in the distributed environment. �ame theory can easily cope with sel�sh behaviors and therefore was introduced to the research of sel�sh nodes related to wireless mesh networks. To adapt with the distributed environment, some detecting methods based on game theory were proposed in recent studies [20][21][22][23][24][25].
A distinct and novel approach named best neighbor strategy (BNS) was proposed to stimulate and enforce cooperation among such sel�sh nodes in an ad hoc or wireless mesh network environment, where there is no central authority to monitor their unacceptable behaviors [26]. Komathy and Narayanasamy [26] pointed out that there were four basic methods using game theory to defend sel�sh nodes in an ad hoc network. is paper is to discuss the �rst and third ones. e �rst is to choose the best response as a rational expectation according to the expected behavior of the other players, and this method is called maximum payoff strategy (MPS) in this paper. e third one is to choose the other player's strategy, if it is achieving more than others. Based on the third method, Komathy and Narayanasamy proposed the BNS [26] against sel�sh neighbors. �n BNS, each node records the total payoff of its every neighbor in each round of the packet forwarding game. Once one round is completed, the player changes its current strategy to its neighbor's current strategy, if the neighbor has achieved a higher total return than any other neighbor. BNS is able to converge faster to enforce cooperation among sel�sh nodes and is robust against sel�shness when invaded by sel�sh behaviors. In our paper, beginning with high cooperation level, BNS is able to provide a superior performance; however, it displays inferior behaviors beginning with low cooperation level, which will be discussed in the simulation sections in this paper. To improve the performance of BNS in low cooperation level, we then combine MPS and BNS to propose a scheme identi�ed as mixed MPS-BNS strategy against sel�sh nodes and sel�sh behaviors. In this paper, the basic and complete models of mixed MPS-BNS strategy are �rstly discussed and analy�ed, respectively, with two repeated packet forwarding (RPF) games of two player and multiplayer. en, via the simulation of the RPF games, the performance of the true BNS and the mixed MPS-BNS will be compared and discussed on average payoff and cooperation level.  In MPS-BNS strategy, each node has a strategy space , consisting of two policies, namely, MPS-BNS and ALL Drop. e strategy space of a player is shown in Table 1, in which each player always drops others' packets in a certain probability of (0 ≤ ≤ ) and chooses MPS-BNS strategy in a probability of ( − ). e probability represents the sel�shness level and also simulates the mutation proportion. Each player chooses MPS and BNS strategies in probabilities of − 0 ≤ ≤ and − − , respectively. MPS and BNS have two substrategies Forward (F) and Drop (D) as shown in Table 2, which also shows the payoff matrix. In MPS strategy, the player chooses the strategy (F or D) with which they will get the maximum payoff in the next round according to its neighboring player. In BNS strategy, the player chooses the strategy (F or D) with which its neighboring player gets the maximum payoff in the previous round.

MPS-BNS Strategic Game of Two Player
According to different prime strategies, there exits four states in Markov process as follows.
(i) State (1,1): both player 1 and player 2 have MPS-BNS as their prime strategies for the current game. (ii) State (1,2): player 1 has MPS-BNS as its prime strategy, and player 2 has ALL-D as its prime-strategy for the current game. (iii) State (2,1): player 1 has ALL-D as its prime strategy for the current game, and player 2 has MPS-BNS as its prime strategy. (iv) State (2,2): both player 1 and player 2 have ALL-D as their prime strategy for the current game.
e expected payoff for player 1 in state (1,1) is simulated using Table 3 with parameters , , and , which represent different payoffs gained by the nodes in different strategies. In MPS strategy, every node chooses the strategy which will get the maximum payoff corresponding to the strategies of its neighbors; while in BNS strategy, every node follows the strategy of its neighbor having the maximum payoff in previous round. Accordingly, the payoff of MPS is higher than that of BNS; thus, 1 = 5.5, 2 = 5, 3 = 4.5, 4 = 4 and 1 = 3.5, 2 = 3, = 1, = 1. e 1 and 2 are the proportions of ALL-D chosen by player 1 and player 2, respectively.

Evolution with MPS-BNS as Pure
Strategy. e pure strategy means that every node in network has a uniform strategy pro�le (�orward or Drop) against all its neighbors. When MPS-BNS is implemented as pure strategy, as shown in Table 4, each node in topology has MPS and BNS strategies with the probability of (0 1) and (1 − ), respectively, against all its neighbors ( = 1, 2, , ), where is the number of neighbors. Scheme 2 is a �ow chart to illustrate the strategic game of multiplayer using MPS-BNS as a pure strategy. e computing method of average payoff per player in Scheme 1 is given as follows: where is the total payoff of all nodes, and × is the size of topology. e proposed model MPS-BNS is implemented using tool MATLAB. e topology size ranges from 10 × 10 to 100 × 100, and the simulation runs for 100 generations. random initial strategies and probability of choosing MPS (0 ≤ ≤ ). In the strategy without MPS-BNS, all nodes have random strategies Forward or Drop, choosing each strategy with probability of 0.5. In Scheme 3, the outcome of MPS-BNS as pure strategy is much higher than that without MPS-BNS and is able to evolve to the maximum expected level at about 32. e average payoff of the strategy without MPS-BNS remains the same at 16 in all topology grid sizes. When the size of topology increases to 50 × 50, the evolution speed slows down, and the maximum average payoff per player remains at the maximum expected level.

Evolution with MPS-BNS as Mixed Strategy.
Instead of having a uniform strategy pro�le in pure strategy, in mixed strategy, every node in topology maintains a strategy pro�le with different strategies against neighbors. Scheme 4 depicts a size of × topology grid, in which the corner node has three neighbors, the edge node (excluding corner node) has �ve neighbors, and each of the other nodes has eight neighbors. �very node in grid maintains a strategy pro�le, Table 5, where ( is the coordinate of node, is the strategy adapted against neighbor node , and is the number of neighbors. = 0, representing the true BNS strategy, BNS behaves advanced performance of evolving to the maximum level in fast convergence speed with niceness more than 70 ; however, as niceness decreases, the convergence slows down; thus with niceness less than 25 , the outcomes are unable to evolve to the maximum level but a lower average payoff instead. e mixed MPS-BNS ( 0) behaves better performance than the true BNS in low niceness. When 0 0 , representing the proportion of MPS to be equal to or greater than 0.01 in mixed MPS-BNS, in all proportion of niceness, all the outcomes of MPS are able to converge to the maximum average payoff 31, and as increases, the convergence converses faster, especially when = 0 convergences are all within the �rst �ve generations in all proportion of niceness. e evolution of percentage of cooperation in Scheme 7 is similarly the same as the average payoff in Scheme 6. As the proportion of MPS ( ) increases, the network experiences better cooperation from nodes and converges faster. From the simulation, we �nd that though the probability of choosing MPS (0 ≤ ≤ ) is just slightly more than zero and much less than the probability of choosing BNS, the outcomes of MPS-BNS behave much better than the true BNS especially with low level of niceness, that is because in MPS the maximum payoff strategy is chosen in a more direct way than BNS, bringing a much faster convergence speed.
high level of average payoff and cooperation at all levels of sel�shness with niceness ≥ 50%; however, it behaves much lower performance with niceness of 25% than with niceness ≥ 50%, especially at a low level of sel�shness. e mixed MPS-BNS ( , representing the choice of MPS and BNS with the same probability) behaves approximately the same high level of performance in various proportions of niceness. MPS-BNS especially with the niceness of 25% gives relatively a distinctly better share of average payoff at all levels of sel�shness. �n addition, MPS-BNS with various proportions of niceness achieves higher average payoff and cooperation than BNS with 75% and 90% percentage of sel�sh nodes. Schemes 8 and 9 conclude that MPS-BNS has better networking performance and better robustness than BNS in �ghting against sel�shness.

Conclusion
e paper has proposed a mixed MPS-BNS strategy based on game theory against sel�sh nodes and sel�sh behaviors in the process of packets forwarding in wireless mesh networks. rough our research, the true BNS is able to provide a superior network performance with high initial cooperation levels but behaves inferior with low initial cooperation levels. To overcome this problem, we combine BNS with MPS strategy, each of which is chosen with respective probability. In MPS strategy, every node selects the strategy which will get the maximum payoff corresponding to the strategies of its neighbors. In BNS strategy, every node follows the strategy of its neighbor having the maximum payoff. A basic MPS-BNS strategic game of two players is discussed and is extended to a complicated strategic game involving multiplayer. e simulation and discussions of the proposed strategy as pure strategy and mixed strategy are carried out on performance of average payoff and cooperation level. e results conclude that MPS-BNS is able to converge to the expected maximum level with various proportions of the initial percentage of cooperation and converge more rapidly than BNS. e simulation results of MPS-BNS against sel�shness conclude that MPS-BNS behaves superior robustness than BNS defending against sel�sh nodes. us, the proposed MPS-BNS strategy is much more effective and efficient in defending against sel�shness in wireless networ�s.