Threshold-based punishment mechanism in evolutionary public goods games on random hypergraphs

Free-riding severely undermines the sustainability of cooperation in public goods games, yet existing many mechanisms for curbing such behavior often overlook the role of conditional punishment emerging from higher-order interactions. To bridge this gap, we introduce an evolutionary public goods game on hypergraphs, where individuals engage through group-based interactions (hyperedges), and punishment is conditionally triggered when the number of defectors in a group exceeds a predefined threshold $\varphi$. Once triggered, defectors retain only a fraction $1-\beta$ of their original payoff, while cooperators benefit from the redistributed penalties. We analytically derive exact critical points for the reduced synergy factor that govern the emergence and saturation of cooperation on uniform random hypergraphs. Specifically, lower thresholds enable cooperation to emerge at a critical reduced synergy factor inversely proportional to the group size $g$ and punishment intensity $\beta$. Conversely, higher thresholds impede cooperation by raising both critical points to 1. Furthermore, introducing heterogeneity in group size, threshold, or punishment intensity amplifies cooperation compared to homogeneous counterparts. By unifying higher-order interactions, threshold-based punishment, and structural heterogeneity into a single analytical framework, this work provides new insights for designing robust cooperation-enhancing mechanisms in complex systems beyond pairwise networks.