  Understanding the evolution of cooperation in multiplayer games is of vital
significance for natural and social systems. An important challenge is that
group interactions often leads to nonlinear synergistic effects. However,
previous models mainly focus on deterministic nonlinearity where the arise of
synergy or discounting effect is determined by certain conditions, ignoring
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 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. Interestingly, 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
novel insights into how stochastic nonlinearity influences the emergence of
prosocial behavior.
