Stability of Networked Evolutionary Games With Randomly Mixed Two-Level Reasoning

This article examines the stability of finite networked evolutionary games (NEGs) considering randomly available second-order neighbors' information. A new strategy update rule (SUR) with randomly mixed two-level reasoning is proposed. The idea behind two-level reasoning is that each player predicts his/her neighbors' strategies based on the second-order neighbors' information if available, and then updates his/her own strategy accordingly. Since this information is only available randomly, the induced SUR is a mixed SUR based on one-level and two-level reasoning. The algebraic form of the profile dynamical system is derived, and the profile's transition probability matrix is calculated. Based on these, necessary and sufficient conditions for a Nash equilibrium (NE) stability are proposed. In addition, it is proved that under mild probability conditions, if an NE of an NEG with one-level reasoning is stable, then it is also stable for the NEG with the induced randomly mixed two-level reasoning. An illustrative example is provided to demonstrate the proposed results.