  We study the evolution of cooperation in public goods games on the square
lattice, focusing on the effects that are brought about by different sizes of
groups where individuals collect their payoffs and search for potential
strategy donors. We find that increasing the group size does not necessarily
lead to mean-field behavior, as is traditionally observed for games governed by
pairwise interactions, but rather that public cooperation may be additionally
promoted by means of enhanced spatial reciprocity that sets in for very large
groups. Our results highlight that the promotion of cooperation due to spatial
interactions is not rooted solely in having restricted connections amongst
players, but also in individuals having the opportunity to collect payoffs
separately from their direct opponents. Moreover, in large groups the presence
of a small number of defectors is bearable, which makes the mixed phase region
expand with increasing group size. Having a chance of exploiting distant
players, however, offers defectors a new way to break the phalanx of
cooperators, and even to resurrect from small numbers to eventually completely
invade the population.
