A game theoretic approach to detect and co-exist with malicious nodes in wireless networks

Abstract

Identification and isolation of malicious nodes in a distributed system is a challenging problem. This problem is further aggravated in a wireless network because the unreliable channel hides the actions of each node from one another. Therefore, a regular node can only construct a belief about a malicious node through monitoring and observation. In this paper, we use game theory to study the interactions between regular and malicious nodes in a wireless network. We model the malicious node detection process as a Bayesian game with imperfect information and show that a mixed strategy perfect Bayesian Nash Equilibrium (also a sequential equilibrium) is attainable. While the equilibrium in the detection game ensures the identification of the malicious nodes, we argue that it might not be profitable to isolate the malicious nodes upon detection. As a matter of fact, malicious nodes can co-exist with regular nodes as long as the destruction they bring is less than the contribution they make. To show how we can utilize the malicious nodes, a post-detection game between the malicious and regular nodes is formalized. Solution to this game shows the existence of a subgame perfect Nash Equilibrium and reveals the conditions that are necessary to achieve the equilibrium. Further, we show how a malicious node can construct a belief about the belief held by a regular node. By employing the belief about the belief system, a Markov Perfect Bayes–Nash Equilibrium is reached and the equilibrium postpones the detection of the malicious node. Simulation results and their discussions are provided to illustrate the properties of the derived equilibria. The integration of the detection game and the post-detection is also studied and it is shown that the former one can transit into the latter one when the malicious node actively adjusts its strategies.