# A Game Theoretic Approach for Balancing Energy Consumption in Clustered Wireless Sensor Networks

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- We present a game theoretic approach for balancing energy consumption in clustered WSNs. Where sensor nodes are modeled as rational and selfish players and the ones in a close neighbor will participate in a clustering game for CH election;
- In our clustering game, we construct a novel convex payoff function for each sensor node that can capture the node’s realistic behaviors. Through convex optimization, the equilibrium solution of this game is derived. Each node makes its own decisions based on the result of this equilibrium solution to achieve a tradeoff between providing data forwarding services and saving energy, and meanwhile, its own payoff can be maximized;
- Considering the energy heterogeneity of sensor nodes in practical scenario, we introduce a penalty mechanism to compel the sensor nodes which hold more energy to compete for the CHs more actively;
- Through extensive simulations under various conditions, we prove that the performance of our protocol outperforms the recent game theoretic clustering protocols CROSS and LGCA.

## 2. Related Works

## 3. System Model

#### 3.1. Network Model

_{i}for any sensor node i as follows:

_{b}is the basic energy; α is the random energy exponent and rand is a number randomly selected within the range of (0, 1).

- All sensor nodes are stationary or nearly stationary after they have been deployed into sensing field;
- A unique ID is labeled on each sensor node that can be used to distinguish data sources;
- Each node has limited energy stored in the battery which cannot be recharged, but no energy limitations are inflicted on the BS;
- Each sensor node has the ability of changing its power level dynamically to adapt to different transmission distances.

#### 3.2. Radio Model

_{TX}consumed by the transmitter to deliver a q-bit packet can be calculated by:

_{f}and ε

_{m}are the amplifier characteristic constants with regard to free-space propagation model and two-ray ground reflection model; d is the distance from the transmitter to the destination and d

_{0}is the distance threshold which is computed by:

_{f}and ε

_{m}are the amplifier characteristic constants.

_{DA}is energy of data aggregating for a one-bit packet.

## 4. Cluster Head Election Game

_{G}, S

_{G}, U

_{G}>, where N

_{G}is the set of players, S

_{G}= {S

_{i}} is the set of all feasible strategies and U

_{G}= {U

_{i}} is the set of utility functions. All sensor nodes within a local area act as players and participate in a CEG. Each player can rationally and selfishly choose the strategy based on its own interests. For any player i, its strategies include “declare myself as CH” (D) or “not declare myself as CH” (ND). Since energy is a kind of precious resource in WSNs and any node acting as a CH needs to take the responsibility of relaying data for other normal sensor nodes, each sensor node prefers to abdicate the responsibility to other nodes for energy conservation. However, if none of the sensor nodes take the responsibility of being a CH to serve the other nodes, then network failure occurs and no sensor nodes can enjoy the benefit of data transmission. Therefore, each sensor node has to consider the knowledge or expectations of other sensor nodes when making its own decision. We define the utility function U

_{i}of any node i when playing the CEG as follows:

_{i}is the payoff when node i chooses the strategy ND while there exists at least another node choosing the strategy D, and e

_{i}is the extra cost when node i chooses D.

#### 4.1. Achieving the Maximum Payoff

_{i}. Then the payoff H

_{i}for node i when playing the CEG can be expressed as follows:

_{i}is the payoff when node i chooses the strategy ND while there exists at least one CH; e

_{i}is the extra cost when node i chooses D; N

_{G}is the set of sensor nodes participated in CEG and b

_{i}is a positive penalty term inflicted on node i.

_{i}. Letting w

_{i}be the coefficient of the quadratic term, we can express it as follows:

_{j}is the probability of being the CH for node j.

_{i}can be simplified to:

_{i}is the payoff when node i chooses the strategy ND while there exists at least one CH; e

_{i}is the extra cost when node i chooses D; b

_{i}is a positive penalty term inflicted on node i and p

_{i}is the probability of being a CH for node i. According to the value of coefficient w

_{i}, the payoff function H

_{i}may have two forms that are listed as follows:

- H
_{i}is a concave or linear function when w_{i}≥ 0. Under this case, the maximum point of H_{i}is either p_{i}= 0 or p_{i}= 1; - H
_{i}is a convex function when w_{i}< 0. Under this case, the maximum point of H_{i}maybe exists within the open interval (0, 1).

_{i}= 0 or p

_{i}= 1 to maximize its own payoff. Which means pure strategy is adopted by sensor nodes under this case. Since we adopt mixed strategies for all nodes as aforementioned, we only consider the second case so that each node decides whether being a CH according to a probability within the range of 0 to 1. Then Θ is positive and we have the following inequality:

_{i}is a positive penalty term inflicted on node i; p

_{i}is the probability of being a CH for node i; v

_{i}is the payoff when node i chooses the strategy ND while there exists at least one CH; e

_{i}is the extra cost when node i chooses D and Θ is the multiple formative of probabilities not to be the CH for all sensor nodes.

_{i}> e

_{i}> 0, we find a qualifying value of b

_{i}which can be expressed as follows:

_{i}is residual energy of node i and E

_{max}is the maximum residual energy of sensor nodes participated in the CEG.

_{i}corresponding to node i can be re-expressed as follows:

_{i}is the payoff when node i chooses the strategy ND while there exists at least one CH; e

_{i}is the extra cost when node i chooses D; Θ is the multiple formative of probabilities not to be the CH for all sensor nodes and p

_{i}is the probability of being a CH for node i.

_{i}with regards to p

_{i}as follows:

_{i}′ of H

_{i}as follows:

_{i}is the payoff when node i chooses the strategy ND while there exists at least one CH; e

_{i}is the extra cost when node i chooses D; E

_{i}is residual energy of node i; E

_{max}is the maximum residual energy of sensor nodes participated in the clustering game and Θ’ is the multiple formative $\left(1-{p}_{i}{}^{\prime}\right){\displaystyle \prod _{j\in {N}_{G},j\ne i}\left(1-{p}_{j}\right)}$.

_{i}not only depends on the strategy of node i, but also relies on the combination of all other sensor nodes that participated in the CEG. A sensor node which has more energy has a bigger probability to be CH at the maximum point. In addition, a sensor node with too little energy may have a negative value at the maximum point that is inconsistent with the term probability. As H

_{i}is a convex function, its reasonable maximum point within the range (0, 1) can be expressed as follows:

_{i}′ is the previous optimal probability of being a CH for node i that can maximize its own payoff H

_{i}and p

_{i}

^{m}is the revised optimal probability of being a CH for node i.

_{i}′.

#### 4.2. Searching for the Equilibrium Strategy

_{G}, P

_{−i}be the strategy combination of sensor nodes excepting node i and N be the total number of sensor nodes. Then we define the NE strategy of the CEG as follows:

**Definition 1.**

_{1}, p

_{2}, …, p

_{N}} is an NE point of the CEG, when inequality H

_{i}(P) ≥ H

_{i}(P

_{−i}, p

_{i}) is satisfied for any sensor node i in set N

_{G}[28].

**Proposition 1.**

^{NE}as shown in Equation (15) is a NE point of the CEG.

_{i}

^{m}is the probability of being the CH for sensor node i which is calculated by Equation (14), and N is the total number of nodes participated in the CEG.

**Proof.**

_{i}, we recalculate the payoff function H

_{i}(P

_{−i}, p

_{i}) for node i as follows:

_{i}is the payoff when node i chooses the strategy ND while there exists at least one CH; e

_{i}is the extra cost when node i chooses D; E

_{i}is residual energy of node i; E

_{max}is the maximum residual energy of sensor nodes participated in the CEG; Θ

_{−i}is the multiple formative of probabilities not to be the CH for all sensor nodes excepting node i and P

_{−i}is the strategy combination excepting node i. □

_{i}> e

_{i}> 0, E

_{max}≥ E

_{i}> 0 and Θ

_{−i}> 0, then H

_{i}(P

_{−i}, p

_{i}) is a quadratic convex function depending on probability p

_{i}. And the maximum point of function H

_{i}(P

_{−i}, p

_{i}) is the probability p

_{i}′ calculated by Equation (13). If p

_{i}′ < 0, then H

_{i}(P

_{−i}, p

_{i}) is monotonically decreasing with the increase of p

_{i}from 0 to 1. That is:

_{i}′ ≥ 0, then H

_{i}(P

_{−i}, p

_{i}) firstly increases with the increase of p

_{i}from 0 to p

_{i}′, and then decreases with the increase of p

_{i}from p

_{i}′ to 1. Hence, we have:

_{G}can maximize its own payoff H

_{i}(P

_{−i}, p

_{i}) when it decides whether being the CH according to the probability p

_{i}

^{m}calculated by Equation (14).

_{G}will decide whether being the CH according to the probability calculated by Equation (14) simultaneously for maximizing its payoff. Then the selected strategy combination is P

^{NE}that calculated by Equation (15). Any sensor node i in set N

_{G}has no incentive to deviate from it unilaterally, because the inequality H

_{i}(P

_{−i}

^{NE}, p

_{i}

^{m}) ≥ H

_{i}(P

_{−i}

^{NE}, p

_{i}) is always true according to Equations (14)–(19). Here P

_{-i}

^{NE}is the strategy combination of sensor nodes in set N

_{G}excepting node i. Hence, the strategy combination P

^{NE}is a NE point of the CEG according to Definition 1.

**Proposition 2.**

^{Z}= {0, 0,…, 0} is not a NE point of the CEG.

**Proof.**

_{−i}

^{Z}be the strategy combination of sensor nodes in set N

_{G}excepting node i, then the payoff of node i for the strategy combination (P

_{−i}

^{Z}, p

_{i}) can be calculated by:

_{i}is the payoff when node i chooses the strategy ND while there exists at least one CH; e

_{i}is the extra cost when node i chooses D; E

_{i}is residual energy of node i; E

_{max}is the maximum residual energy of sensor nodes participated in the CEG and p

_{i}is the probability of being a CH for node i. □

_{i}= 0, that is P

^{Z}is not a NE point according to Definition 1.

^{NE}which is given by Equation (15). In addition, the value of Θ at this NE point is within the range (0, 1) according to Equations (13) and (14). However, how to calculate the probability of being a CH for each sensor node at the NE point of CEG still presents great challenges. We can first calculate the value of Θ at the NE point, expressed as Θ

^{NE}. And then we can calculate the probability to be CH at the NE point for each node based on Equations (13) and (14). For the purpose of searching the root of Θ

^{NE}, we can employ a binary search algorithm or Newton’s method to solve the following optimization problem:

_{i}

^{m}is the optimal probability of being a CH for node i that can maximize its own payoff; Θ is the multiple formative of probabilities not to be the CH for all sensor nodes and N

_{G}is the set of sensor nodes that participated in the CEG.

- First step: Input the minimum value of Θ (we denote it as min), the maximum value of Θ (we denote it as max) and error precision err;
- Second step: Calculate the expression (min + max)/2, and assign the result to Θ;
- Third step: If |max − min| < err, or f’(Θ) < err, terminate the iteration process and output the final result of Θ, which is denoted as Θ
^{NE}; Otherwise, go to the next step; - Forth step: If f’(Θ)f’(min) > 0, assign the value of Θ to min; Otherwise, assign the value of Θ to max;
- Last step: Go to the second step to continue the iteration process until the maximal number of iterations is completed.

^{NE}for our CEG as shown in Table 1. Here we only list the result for a CEG which contains in total 8 sensor nodes. These nodes are distributed in the same local area of the network and act as players to join in the same CEG. Moreover, they hold different values about the distance to BS, number of neighbor nodes and residual energy.

## 5. The Protocol Details

#### 5.1. CHs Election

^{2}. Then the expected number of CHs in the network is S/(πR

^{2}). However, CHs are randomly selected based on this expected probability, so we need to select more CHs for further reselection and the radius adjustment factor ε is introduced to update the expected number of CHs. Finally, the expected probability of being the CH can be expressed by Equation (22).

_{k}, it checks the number n

_{k}of member nodes that are still alive after the cluster topology is constructed. If n

_{k}< 1, ch

_{k}still has to campaign for the CH in the next round according to the expected probability, since there is no other CH candidate within the cluster. If n

_{k}= 1, the probability to be CH of ch

_{k}in the next round is set to be zero to avoid continuous working as CH, and the only CM node will campaign for the CH in the next round according to the expected probability. If n

_{k}> 1, there are more than one CH candidate within the cluster, then the probability to be CH for ch

_{k}in the next round is also set to be zero. Moreover, a CEG and probability transformation mechanism will be introduced by ch

_{k}for the purpose of calculating the probabilities to be CH for its member nodes.

_{k}will act as players to join a CEG. By solving the Optimization Problem (21), ch

_{k}can calculate the absolute equilibrium probabilities to be CH in the next round for its member nodes according to Equations (13)–(15). However, several parameters need to be determined in CEG. Firstly, the number of players that participate in the CEG is exactly equal to the number n

_{k}of active member nodes of ch

_{k}. Secondly, the residual energy of each player can be acquired by ch

_{k}during the process of clusters formation phase. At last, to determine v

_{i}and e

_{i}for player i, we define the payoff of a player as the number of data bits per unit of energy consumed to transmit the data to CH or BS [30]. Then when player i chooses the strategy ND while there exists at least another player choosing the strategy D, its payoff v

_{i}can be calculated by:

_{i}is the amount of energy consumed to transmit the data packet to the corresponding CH that can be expressed as follows:

_{f}is the amplifier characteristic constant corresponding to free-space propagation model; R is the communication radius of each node, and the average distance between a member node and its corresponding CH is 2R/3.

_{i}can be calculated by:

_{i}is the payoff when node i chooses the strategy ND while there exists at least one CH; Ech

_{i}is the amount of energy consumed by player i to transmit the packet with length q to BS when it acts as the CH, and Ech

_{i}can be expressed as follows:

_{i}is the number of neighbor nodes of player i; E

_{DA}is the energy consumed for aggregating a one-bit data packet; ε

_{m}is the amplifier characteristic constant regard to two-ray ground reflection model and d

_{i}is the distance between player i and BS.

_{k}can get the absolute equilibrium probability p

_{i}

^{m}to be the CH in the next round for any member node i. In fact, sensor nodes usually are unevenly distributed in the sensing field which results in uneven clustering. Some clusters may have more member nodes joining the CEG than other clusters so that the nodes in different clusters may have different levels of absolute equilibrium probability. To avoid this case, a probability transformation mechanism is adopted to transform absolute equilibrium probability into relative equilibrium probability. For any member node i, its relative equilibrium probability p

_{i}

^{r}to be CH in the next round can be expressed as follows:

_{i}

^{m}is the absolute equilibrium probability of being a CH for node i; n

_{k}is the number of active member nodes in the cluster to which node i belongs and p

_{ept}is the expected probability calculated by Equation (22). After this transformation is finished, sensor nodes in different clusters have the same average probability to be CH.

#### 5.2. Clusters Formation

## 6. Performance Evaluations

^{2}, node density is 1 node/100 m

^{2}and communication radius is 35 m.

^{2}. From Table 3 we can find that compared with LEACH, the routing scheme that only utilizing our proposed clustering model has improved the network lifetime by 15%. After the game theory is introduced into our clustering model, the network lifetime has been improved by 26%.

^{2}, but the opposite is true when the network size is larger than 150 × 150 m

^{2}. This condition indicates that the LGCA protocol is more suitable for use in larger scale networks than CROSS since its implementation is completely distributed.

^{2}and the node density is 1 node/100 m

^{2}. We first show the detail of number of nodes that still alive in each round when communication radius is 35 m. Then we gives comparison of network lifetime versus different communication radiuses among these protocols, for the purpose of analyzing the effect of the communication radius on network lifetime.

^{2}. For a sensor node in CROSS, once it has been selected as the CH, it will not be the CH again until all other nodes have served as the CHs in the network. Then once a node has been the CH in one round, it will have no chance again to be the CH in the following 24 rounds. In other words, a sensor node will be selected as the CH once every 25 rounds in CROSS, and this mechanism of sensor nodes taking turns to be CHs is very similar to that used in LEACH.

^{2}, communication radius is 35 m and node density increases from 0.2 to 1 node/100 m

^{2}. Figure 10 gives the comparison of network lifetime versus different node densities among LEACH, CROSS, LGCA and GTAB. It shows that our proposed protocol GTAB outperforms other three protocols for all cases of the node density. All four protocols display their shortest network lifetimes simultaneously when the node density is the minimum value 0.2. This is because sensor nodes become CHs more frequently for a network with lower node density. Moreover, with the increase of node density, the network lifetimes of the four protocols first increase and then decrease gradually. This is because the average data burden per CH becomes heavier for a network with bigger node density.

## 7. Conclusions and Future Work

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Yick, J.; Mukherjee, B.; Ghosal, D. Wireless sensor network survey. Comput. Netw.
**2008**, 52, 2292–2330. [Google Scholar] [CrossRef] - Di Pietro, R.; Guarino, S.; Verde, N.V.; Domingo-Ferrer, J. Security in wireless ad-hoc networks—A survey. Comput. Commun.
**2014**, 51, 1–20. [Google Scholar] [CrossRef] - Erdelj, M.; Król, M.; Natalizio, E. Wireless sensor networks and multi-UAV systems for natural disaster management. Comput. Netw.
**2017**, 124, 72–86. [Google Scholar] [CrossRef] - Wu, M.; Tan, L.; Xiong, N. Data prediction, compression, and recovery in clustered wireless sensor networks for environmental monitoring applications. Inf. Sci.
**2016**, 329, 800–818. [Google Scholar] [CrossRef] - Peng, Y.; Wang, X.; Guo, L.; Wang, Y.; Deng, Q. An efficient network coding-based fault-tolerant mechanism in WBAN for smart healthcare monitoring systems. Appl. Sci.
**2017**, 7, 817. [Google Scholar] [CrossRef] - Lersteau, C.; Rossi, A.; Sevaux, M. Robust scheduling of wireless sensor networks for target tracking under uncertainty. Eur. J. Oper. Res.
**2016**, 252, 407–417. [Google Scholar] [CrossRef] [Green Version] - Byun, J.; Jeon, B.; Noh, J.; Kim, Y.; Park, S. An intelligent self-adjusting sensor for smart home services based on ZigBee communications. IEEE Trans. Consum. Electron.
**2012**, 58, 794–802. [Google Scholar] [CrossRef] - Zinonos, Z.; Chrysostomou, C.; Vassiliou, V. Wireless sensor networks mobility management using fuzzy logic. Ad Hoc Netw.
**2014**, 16, 70–87. [Google Scholar] [CrossRef] - Lee, J.-H.; Moon, I. Modeling and optimization of energy efficient routing in wireless sensor networks. Appl. Math. Model.
**2014**, 38, 2280–2289. [Google Scholar] [CrossRef] - Ruitao, X.; Xiaohua, J. Transmission-efficient clustering method for wireless sensor networks using compressive sensing. IEEE Trans. Parallel Distrib. Syst.
**2014**, 25, 806–815. [Google Scholar] [CrossRef] - Akkaya, K.; Younis, M. A survey on routing protocols for wireless sensor networks. Ad Hoc Netw.
**2005**, 3, 325–349. [Google Scholar] [CrossRef] - Abbasi, A.A.; Younis, M. A survey on clustering algorithms for wireless sensor networks. Comput. Commun.
**2007**, 30, 2826–2841. [Google Scholar] [CrossRef] - Liu, X. A survey on clustering routing protocols in wireless sensor networks. Sensors
**2012**, 12, 11113–11153. [Google Scholar] [CrossRef] [PubMed] - Afsar, M.M.; Tayarani-N, M.-H. Clustering in sensor networks: A literature survey. J. Netw. Comput. Appl.
**2014**, 46, 198–226. [Google Scholar] [CrossRef] - Guravaiah, K.; Leela Velusamy, R. Energy efficient clustering algorithm using RFD based multi-hop communication in wireless sensor networks. Wirel. Pers. Commun.
**2017**, 95, 3557–3584. [Google Scholar] [CrossRef] - Shi, H.Y.; Wang, W.L.; Kwok, N.M.; Chen, S.Y. Game theory for wireless sensor networks: A survey. Sensors
**2012**, 12, 9055–9097. [Google Scholar] [CrossRef] [PubMed] - Harré, M. Utility, revealed preferences theory, and strategic ambiguity in iterated games. Entropy
**2017**, 19, 201. [Google Scholar] [CrossRef] - AlSkaif, T.; Guerrero Zapata, M.; Bellalta, B. Game theory for energy efficiency in wireless sensor networks: Latest trends. J. Netw. Comput. Appl.
**2015**, 54, 33–61. [Google Scholar] [CrossRef] - Koltsidas, G.; Pavlidou, F.-N. A game theoretical approach to clustering of ad-hoc and sensor networks. Telecommun. Syst.
**2010**, 47, 81–93. [Google Scholar] [CrossRef] - Xie, D.; Sun, Q.; Zhou, Q.; Qiu, Y.; Yuan, X. An efficient clustering protocol for wireless sensor networks based on localized game theoretical approach. Int. J. Distrib. Sens. Netw.
**2013**, 9, 264–273. [Google Scholar] [CrossRef] - Heinzelman, W.R.; Chandrakasan, A.; Balakrishnan, H. Energy-Efficient Communication Protocol for Wireless Microsensor Networks. In Proceedings of the 33rd Annual Hawaii International Conference on System Sciences, Maui, HI, USA, 4–7 January 2000; pp. 1–10. [Google Scholar]
- Younis, O.; Fahmy, S. Heed: A hybrid, energy-efficient, distributed clustering approach for ad hoc sensor networks. IEEE Trans. Mob. Comput.
**2004**, 3, 366–379. [Google Scholar] [CrossRef] - Pooranian, Z.; Barati, A.; Movaghar, A. Queen-bee algorithm for energy efficient clusters in wireless sensor networks. World Acad. Sci. Eng. Technol.
**2011**, 73, 1070–1073. [Google Scholar] - Bajaber, F.; Awan, I. An efficient cluster-based communication protocol for wireless sensor networks. Telecommun. Syst.
**2013**, 55, 387–401. [Google Scholar] [CrossRef] - Naranjo, P.G.V.; Shojafar, M.; Mostafaei, H.; Pooranian, Z.; Baccarelli, E. P-sep: A prolong stable election routing algorithm for energy-limited heterogeneous fog-supported wireless sensor networks. J. Supercomput.
**2016**, 73, 733–755. [Google Scholar] [CrossRef] - Soleimani, H.; Tomasin, S.; Alizadeh, T.; Shojafar, M. Cluster-head based feedback for simplified time reversal prefiltering in ultra-wideband systems. Phys. Commun.
**2017**, 25, 100–109. [Google Scholar] [CrossRef] - Naserian, M.; Tepe, K. Game theoretic approach in routing protocol for wireless ad hoc networks. Ad Hoc Netw.
**2009**, 7, 569–578. [Google Scholar] [CrossRef] - Lin, X.-H.; Kwok, Y.-K.; Wang, H.; Xie, N. A game theoretic approach to balancing energy consumption in heterogeneous wireless sensor networks. Wirel. Commun. Mob. Comput.
**2015**, 15, 170–191. [Google Scholar] [CrossRef] - Huang, D.; Zhang, Y.; Zheng, Z. Clustering algorithm based on territory game in wireless sensor networks. In Lecture Notes in Electrical Engineering, Proceedings of the International Conference on Information Engineering and Applications (IEA) 2012, Chongqing, China, 26–28 October 2012; Springer: London, UK, 2013; pp. 457–465. [Google Scholar]
- Yang, L.; Lu, Y.-Z.; Zhong, Y.-C.; Wu, X.-G.; Xing, S.-J. A hybrid, game theory based, and distributed clustering protocol for wireless sensor networks. Wirel. Netw.
**2015**, 22, 1007–1021. [Google Scholar] [CrossRef] - Zahedi, Z.M.; Akbari, R.; Shokouhifar, M.; Safaei, F.; Jalali, A. Swarm intelligence based fuzzy routing protocol for clustered wireless sensor networks. Expert Syst. Appl.
**2016**, 55, 313–328. [Google Scholar] [CrossRef] - Yang, L.; Lu, Y.-Z.; Zhong, Y.-C.; Yang, S.X. An unequal cluster-based routing scheme for multi-level heterogeneous wireless sensor networks. Telecommun. Syst.
**2017**. [Google Scholar] [CrossRef] - Qasem, A.A.; Fawzy, A.E.; Shokair, M.; Saad, W.; El-Halafawy, S.; Elkorany, A. Energy efficient intra cluster transmission in grid clustering protocol for wireless sensor networks. Wirel. Pers. Commun.
**2017**, 97, 915–932. [Google Scholar] [CrossRef]

Nodes Participated in the CEG | Distances to BS (m) | Number of Neighbors | Residual Energy (J) | NE Solution (P^{NE}) |
---|---|---|---|---|

Node 1 | 78.8692 | 9 | 0.2934 | 0 |

Node 2 | 98.5443 | 4 | 0.2781 | 0 |

Node 3 | 93.2947 | 1 | 0.3720 | 0 |

Node 4 | 86.7700 | 5 | 0.4303 | 0.0522 |

Node 5 | 96.0795 | 5 | 0.3040 | 0 |

Node 6 | 91.2931 | 7 | 0.3965 | 0.0319 |

Node 7 | 74.0076 | 14 | 0.4711 | 0.2370 |

Node 8 | 87.9603 | 7 | 0.2531 | 0 |

Parameters | Values |
---|---|

Basic energy E_{b} (J) | 0.5 |

Random energy exponent α | 0.1 |

Packet size q (bits) | 3000 |

Control packet size (bits) | 300 |

ζ (nJ/bit) | 50 |

ε_{m} (pJ/bit/m^{4}) | 0.0013 |

ε_{f} (pJ/bit/m^{2}) | 10 |

E_{DA} (nJ/bit/message) | 5 |

Radius adjustment factor ε | 0.8 |

Routing Scheme | Network Lifetime |
---|---|

LEACH | 856 |

Only-clustering | 986 |

GTAB | 1081 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yang, L.; Lu, Y.; Xiong, L.; Tao, Y.; Zhong, Y.
A Game Theoretic Approach for Balancing Energy Consumption in Clustered Wireless Sensor Networks. *Sensors* **2017**, *17*, 2654.
https://doi.org/10.3390/s17112654

**AMA Style**

Yang L, Lu Y, Xiong L, Tao Y, Zhong Y.
A Game Theoretic Approach for Balancing Energy Consumption in Clustered Wireless Sensor Networks. *Sensors*. 2017; 17(11):2654.
https://doi.org/10.3390/s17112654

**Chicago/Turabian Style**

Yang, Liu, Yinzhi Lu, Lian Xiong, Yang Tao, and Yuanchang Zhong.
2017. "A Game Theoretic Approach for Balancing Energy Consumption in Clustered Wireless Sensor Networks" *Sensors* 17, no. 11: 2654.
https://doi.org/10.3390/s17112654