Mixed strategy under generalized public goods games

Zhang, Y., Wu, T., Chen, X., Xie, G., & Wang, L. (2013). Mixed strategy under generalized public goods games. IIASA Interim Report. IIASA, Laxenburg, Austria: IR-13-077

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Abstract

The relationship between public goods benefiting all group members irrespective of their contribution and the proportion of cooperators in the group, modeled as the production function, often exhibits nonlinearity. Previous nonlinear game theoretical models assume two discrete strategies or continuous investments. Here, we investigate the effect of nonlinear production functions on the evolution of the mixed strategy in finite populations. When the group size and population size become comparable, cooperation is doomed irrespective of the production function. Otherwise, nonlinear production functions may lead to a convergent evolutionary stable strategy (CESS) or a repeller, but can't yield the evolutionary branching in contrast with the evolution of continuous investments. In particular, we consider three representative families of production functions, in which most previous studied production functions are recovered as special cases. Full defection, full cooperation, a unique CESS or a unique repeller may occur for two families of production functions including concave and convex curves even if the group size is two. However, the parameter region of each evolutionary situation exhibits a great difference since the value of public goods produced is fixed for one class and variable for the other when all group members cooperate. A third class encompassing symmetrically sigmoidal and inverse sigmoidal curves may lead to the coexistence of a CESS and a repeller impossible for the other two classes only when group size exceeds two. Intriguingly, the hysteresis effect is found in all three classes. However, two saddle-node bifurcations appear for the thrid class but not for the other two classes.

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