Analysis of cooperative stability for reputation evaluation rules in spatial prisoner's dilemma game

Research on reputation-based indirect reciprocity has found profound achievements, elucidating its role in promoting cooperation over selfish actions. However, some evaluation methodologies have limitations, such as the image scoring model, a classic first-order paradigm. Several studies have suggested that higher-order rules with more individual information can enhance the stability of cooperation. In this study, we introduce a reputation incentive mechanism to explore the cooperative differences among various evaluation rules. Specifically, players evaluate their opponents' actions following first-order and second-order evaluation rules, respectively. Given that players possess varying degrees of social influence, the evaluative intensity is influenced by the neighbor environment and updated in each round. This resultant fluctuations in reputation exhibit heterogeneity and dynamism. Numerical simulations based on the spatial prisoner's dilemma game demonstrate that under stringent conditions, the first-order rule can sustain cooperation, while the second-order rule may fail, leading to complete group defection. Under more relaxed conditions, the second-order rule proves more effective in promoting full cooperation than the first-order rule. Our research contributes to understanding the guidance and influence of reputation on collective behavior.