Theoretical analysis reveals conditions for Nash equilibrium stability in networked games, indicating that second-order reasoning preserves stability under random information access.
This article examines the stability of finite networked evolutionary games (NEGs) considering randomly available second-order neighbors' information. A new strategy update rule (SUR) called SUR with randomly mixed 2-level reasoning is proposed. The idea behind 2-level reasoning is that each player predicts his neighbors' strategies based on the second-order neighbors' information if available, and then updates his own strategy accordingly. Since this information is only available randomly, the induced SUR is a mixed SUR based on 1- and 2-level reasoning. The algebraic form of the profile dynamical system is derived, and the profile's transition probability matrix (TPM) is calculated. Based on these, necessary and sufficient conditions for a Nash equilibrium (NE) stability are proposed. Additionally, it is proved that under mild probability conditions, if an NE of an NEG with 1-level reasoning is stable, then it is also stable for the NEG with the induced randomly mixed 2-level reasoning. An illustrative example is provided to demonstrate the proposed results.
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Li et al. (2025) studied this question.
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