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Evolutionary dynamics of a probabilistic punishment mechanism with environmental feedback in regular networked Prisoner's Dilemma

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  • Liu, Jiaqi
  • Zhang, Qianwei

Abstract

In this paper, we investigate the interconnection between players' strategies and the probability of effective punishment. We propose to integrate the environmental feedback and a punishment mechanism to construct a Prisoner's Dilemma game model within a regular network. Based on our analysis, we identify three unique system states: a single stable state where defection predominates and the probability of effective punishment is low, a single stable state where cooperation prevails and the probability of effective punishment is high, and a bi-stable condition characterized by the coexistence of two stable states. We deduce that alterations in the speed of the environmental feedback can influence the system's convergence behavior under bi-stable conditions. The conclusion could be drawn that accelerating the rate at which the probability of effective punishment is updated can be beneficial for enhancing cooperation within social systems. The findings highlight the critical role of the proposed punishment mechanism in fostering long-term stability and sustainable development within societies.

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  • Liu, Jiaqi & Zhang, Qianwei, 2025. "Evolutionary dynamics of a probabilistic punishment mechanism with environmental feedback in regular networked Prisoner's Dilemma," Chaos, Solitons & Fractals, Elsevier, vol. 195(C).
    • Handle: RePEc:eee:chsofr:v:195:y:2025:i:c:s0960077925003364
    DOI: 10.1016/j.chaos.2025.116323
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