Nonlinear Public Goods Game in Dynamical Environments

The evolutionary mechanisms of cooperative behavior represent a fundamental topic in complex systems and evolutionary dynamics. Real-world collective interactions, particularly in multi-agent systems, are often characterized by behavior-dependent mechanism switching where the environmental state is endogenously shaped by group strategies. However, existing models typically treat such environmental variations as static stochasticity and neglect the closed-loop feedback between environmental states and cooperative behaviors. Here, we introduce a dynamic environmental feedback mechanism into a nonlinear public goods game framework to establish a coevolutionary model that couples environmental states and individual cooperative strategies. Our results demonstrate that the interplay among environmental feedback, nonlinear effects, and environmental randomness can drive the system toward a wide variety of steady-state structures, including full defection, full cooperation, stable coexistence, and periodic limit cycles. Further analysis reveals that asymmetric nonlinear parameters and environmental feedback rates exert significant regulatory effects on cooperation levels and system dynamics. This study not only enriches the theoretical framework of evolutionary game theory but also provides a foundation for modeling environmental feedback loops in scenarios ranging from ecological management to the design of cooperative mechanisms in autonomous systems.

💡 Research Summary

The paper addresses a fundamental gap in evolutionary game theory by integrating a continuously evolving environmental feedback loop into a nonlinear public goods game (PGG). Traditional PGG models assume a static or discretely switching environment, which fails to capture the bidirectional influence observed in real ecological and socio‑technical systems where collective actions both shape and are shaped by environmental conditions. To remedy this, the authors construct a co‑evolutionary framework that couples the fraction of cooperators (x) with an environmental state variable (p) that determines the probability of playing either a synergistic PGG (sPGG) or a discounting PGG (dPGG).

Model formulation
The payoff structure is generalized by introducing separate nonlinearity parameters δₛ (synergy) and δ_d (discounting). For a group of size N containing n_C cooperators, the expected payoffs are:

• sPGG: πₛ^C = rN