  In a community-structured population, public goods games (PGG) occur both
within and between communities. Such type of PGG is referred as multilevel
public goods games (MPGG). We propose a minimalist evolutionary model of the
MPGG and analytically study the evolution of cooperation. We demonstrate that
in the case of sufficiently large community size and community number, if the
imitation strength within community is weak, i.e., an individual imitates
another one in the same community almost randomly, cooperation as well as
punishment are more abundant than defection in the long run; if the imitation
strength between communities is strong, i.e., the more successful strategy in
two individuals from distinct communities is always imitated, cooperation and
punishment are also more abundant. However, when both of the two imitation
intensities are strong, defection becomes the most abundant strategy in the
population. Our model provides insight into the investigation of the
large-scale cooperation in public social dilemma among contemporary
communities.
