Evolutionary dynamics of public goods games with diverse contributions in finite populations

The public goods game is a powerful metaphor for exploring the maintenance of social cooperative behavior in a group of interactional selfish players. Here we study the emergence of cooperation in the public goods games with diverse contributions in finite populations. The theory of stochastic process is innovatively adopted to investigate the evolutionary dynamics of the public goods games involving a diversity of contributions. In the limit of rare mutations, the general stationary distribution of this stochastic process can be analytically approximated by means of diffusion theory. Moreover, we demonstrate that increasing the diversity of contributions greatly reduces the probability of finding the population in a homogeneous state full of defectors. This increase also raises the expectation of the total contribution in the entire population and thus promotes social cooperation. Furthermore, by investigating the evolutionary dynamics of optional public goods games with diverse contributions, we find that nonparticipation can assist players who contribute more in resisting invasion and taking over individuals who contribute less. In addition, numerical simulations are performed to confirm our analytical results. Our results may provide insight into the effect of diverse contributions on cooperative behaviors in the real world