Evolutionary dynamics in stochastic nonlinear public goods games

Understanding the evolution of cooperation in multi-player games is of vital significance for natural and social systems. An important challenge is that group interactions often lead to nonlinear synergistic effects. However, previous models mainly focus on deterministic nonlinearity, where synergy or discounting effects occur under specific conditions, not accounting for uncertainty and stochasticity in real-world systems. Here, we develop a probabilistic framework to study the cooperative behavior in stochastic nonlinear public goods games. Through both analytical treatment and Monte Carlo simulations, we provide a comprehensive understanding of social dilemmas with stochastic nonlinearity in both well-mixed and structured populations. We find that increasing the degree of nonlinearity makes synergy more advantageous when competing with discounting, thereby promoting cooperation. Furthermore, we show that network reciprocity loses effectiveness when the probability of synergy is small. Moreover, group size exhibits nonlinear effects on group cooperation regardless of the underlying structure. Our findings thus provide insights into how stochastic nonlinearity influences the emergence of prosocial behavior. Cooperation in multi-player games is influenced by nonlinear interactions and randomness found in natural and social systems. The authors develop a probabilistic framework and find that stronger nonlinear effects enhance cooperation by boosting the collective benefits of working together, and that network reciprocity loses effectiveness when synergistic interactions are rare.