Social bonds and helping dynamics promote cooperation in nested group populations.

Most studies on the evolution of cooperation focus on single-group settings, whereas real-world interactions often occur within nested group structures and are shaped by pre-existing social relationships. The coexistence of cross-level public-goods allocation, dynamically evolving social bonds, and heterogeneous helping responses creates multiple feedback channels, making the stability of cooperation dependent on both group structure and relationship dynamics. To address these limitations, this study develops a two-stage game model that incorporates social bonds to characterize relationship strength. A helping function is further constructed based on bond sensitivity, bond midpoint threshold, and helping response parameters, capturing how social bonds influence cooperative behavior. Simulation results demonstrate that strong social bonds significantly promote cooperation. When bond sensitivity and midpoint threshold are low, even weak bonds can trigger mutual assistance, facilitating bond accumulation and generating a positive feedback loop that reinforces cooperation. Increasing the helping response enhances the level of assistance, accelerates the formation of strong bonds, and substantially improves cooperation. In contrast, although a higher bond learning rate speeds up bond accumulation, its impact on cooperation remains limited. Furthermore, larger local group sizes amplify the positive effect of social bonds on cooperation. Overall, this study provides a novel theoretical perspective on the mechanisms underlying the maintenance of cooperation in nested group structures.