###### Abstract
For instance, LEACH (Low energy adaptive clustering hierarchy) protocol generates CHs uniformly through the node bootstrapping mechanism [6]. [PERSON] et al. proposed an energy-balanced clustering algorithm based on distance to the base station (BS) and neighbor distribution to determine the optimal CHs according to residual energy and neighborhood distribution of nodes [7]. Moreover, [PERSON] et al. suggested an energy-balanced routing protocol to divide the network into several clusters by using the \(K\)-means++ algorithm and select CH by using the fuzzy logical system [8]. However, the current methods of CH selection usually consider the residual energy and location of nodes and ignore the impact of the selection on network invulnerability, leading to inferior invulnerability performance in the face of network attacks. Therefore, combining CH election with invulnerability improvement and energy consumption optimization, selecting CHs with balanced invulnerability and average residual energy is undoubtedly beneficial to extending the network lifetime. At present, some studies have further changed from the traditional invulnerability quantification analysis to focus on invulnerability enhancement [9]. For example, [PERSON] et al. presented a sink-oriented cascading model for WSNs to resist the cascading failures by topology optimization, and this model can construct a topology with high robustness in a short time [10]. Furthermore, [PERSON] et al. recommended the fireworks and particle swarm optimization algorithm to optimize the topology structure and elevate the network's dynamic and static invulnerability [11]. Nevertheless, the invulnerability evaluation indicators used in these existing studies are insufficient and cannot effectively quantify the network invulnerability to provide powerful support for promoting network quality of service (QoS). In addition, due to the complexity of topology, modeling, and link quality, the success rate of mathematical modelling and competitive analysis for WSNs is unsatisfactory, and game theory has already demonstrated remarkable performance in the optimization techniques for WSNs [12]. Game theory is widely used in the design of routing protocols, including optimal CH selection for hierarchical routing, cluster formation, network topology control, etc. [13]. [PERSON] et al. proposed a game-theory-based energy-efficient routing algorithm for the CH election in hierarchical routing to decrease network energy consumption [14]. Moreover, the topology control game algorithm of Markov lifetime prediction model developed by [PERSON] et al. has acceptable capability in ensuring the connectivity and robustness of network [15]. Nonetheless, the existing routing protocols for WSNs based on game theory barely regard the capability of prolonging the network lifetime cycle as a performance criterion and rarely take into account the changes in network invulnerability. Compared with network coverage and connectivity indicators, invulnerability can comprehensively reflect QoS. Hence, the game between invulnerability and average residual energy in the CH election process can balance the energy consumption and improve the ability against external attacks. To solve the predicaments mentioned above, the objective of this study is to evaluate the effectiveness of an invulnerability-aware clustering routing algorithm (IACRA) using game-theoretic approach for lifetime optimization of WSNs. Firstly, the integration metric describing the invulnerability of the local network and the whole network is proposedto define the profit model, and the energy cost model of nodes becoming CHs or cluster members is established. Secondly, the utility function is set up according to the strategy set to gain the optimal probability of each node elected as CH and then the dominance function of candidate CHs is developed to ensure that the CH with high residual energy and short distance from the BS is obtained.