Mixed strategy under generalized public goods games

The relationship between group's contribution and public goods produced often exhibits nonlinearity, which constitutes the generalized public goods game. Far less attention has been paid to how the mixed strategy evolves in such generalized games. Here, we study the effects of nonlinear production functions on the evolution of the mixed strategy in finite populations for the first time. When the group size and the population size are comparable, cooperation is doomed irrespective of the production function. Otherwise, nonlinear production functions may induce a convergent evolutionary stable strategy (CESS) or a repeller, but cannot yield the evolutionary branching. Moreover, we particularly consider three representative families of production functions, intriguingly which all display the hysteresis effect. For two families of production functions including concave and convex curves, a unique CESS or a unique repeller may occur even if the group size is two. Whereas for the third class encompassing symmetrically sigmoidal and inverse sigmoidal curves, the coexistence of a CESS and a repeller only occurs if group size is above two, and two saddle-node bifurcations appear. Our work includes some evidently different results by comparing with the evolution of continuous investment or binary strategy.