Cooperation dynamics in a two-state spatial prisoner's dilemma game with reputation-driven transitions

In reality, dynamic environments often result in players interacting sequentially across various scenarios. Previous studies have primarily focused on the impact of players' single-round actions on these environments, neglecting the cumulative effects of their historical behavior. To address this gap, we introduce reputation-driven game transitions to model the bidirectional feedback between environments and players' long-term behavior. Environmental states are classified into favorable and less favorable types. Compared to two low-reputation players, two high-reputation players are more likely to receive positive feedback from the environment, transitioning to the favorable state for higher payoffs. Within a standard structured network, we examine how state-independent and state-dependent transition mechanisms influence the evolutionary dynamics of populations. Simulation experiments show that under state-independent transitions, stricter positive feedback conditions are more conducive to the evolution of cooperation, indicating that transitions to the favorable state rely more heavily on the reputation of lower-performing players. Under state-dependent transitions, tightening positive feedback conditions in any preceding state enhances cooperation, thus supporting the same conclusion. Additionally, larger differences between the two environmental states lead to stronger support for cooperation, although this effect diminishes as the conditions for positive feedback are relaxed. Notably, as the reputation upper limit increases, cooperation weakens even under strict conditions, suggesting that the longer the cumulative behavioral trajectory of players, the weaker the positive incentives brought by game transitions. Our findings highlight the critical role of players' overall long-term behavior in shaping environmental states, thereby extending the theoretical understanding of game transitions.