Dynamics and Convergence of Hyper-Networked Evolutionary Games with Time Delay in Strategies

Abstract Networked evolutionary game theory is an important tool to study the emergence and maintenance of cooperation in natural, social, and economical systems. In this paper, we investigate the dynamics and convergence of a generalized networked evolutionary game, i.e., delayed hyper-networked evolutionary game (HNEG), which considers the multi-players in fundamental network game and time delay in strategies simultaneously. Based on the tool of semi-tensor product (STP) of matrices, the definition of delayed potential HNEG and representation of potential function are given. Moreover, we conclude the steps to analyze the dynamics and convergence of delayed potential HNEGs. Considering the efficiency in updating process, we define a new strategy updating rule based on the myopic best response adjustment rule (MBRAR), which is called delayed group-based sequential MBRAR. Furthermore, we prove that delayed potential HNEG converges to one of the pure Nash equilibrium trajectories under this rule. Finally, public good game is provided to illustrate the realistic application of our results.