Impact of Decision Feedback on Networked Evolutionary Game with Delays in Control Channel

Cooperative behavior exists widely in nature and social life, and the delays generated during the transmission of information have an important effect on the dynamic evolution of cooperation. To this end, the effect of delays in the control channel (from decision-making to action) on network evolutionary games is explored, and the strategy synthesis problem is also investigated using semi-tensor products of matrices. Firstly, the dynamics of network evolutionary games with delays in control transmission can be converted into an algebraic expression. Secondly, a reachable set approach is introduced to analyze the closed-loop dynamics with control delay, and some sufficient and necessary conditions for the existence of strategy convergence are derived. Meanwhile, an improved algorithm is developed to design the feedback control law, which guarantees that all the strategy trajectories are stable to the desired profile. Finally, two illustrative examples are provided to demonstrate the effectiveness of the results obtained. The current method will help to devise a favorable framework for designing just-in-time strategies in practice.