  According to the public goods game (PGG) protocol, participants decide freely
whether they want to contribute to a common pool or not, but the resulting
benefit is distributed equally. A conceptually similar dilemma situation may
emerge when participants consider if they claim a common resource but the
related cost is covered equally by all group members. The latter establishes a
reversed form of the original public goods game (R-PGG). In this work, we show
that R-PGG is equivalent to PGG in several circumstances, starting from the
traditional analysis, via the evolutionary approach in unstructured
populations, to Monte Carlo simulations in structured populations. However,
there are also cases when the behavior of R-PGG could be surprisingly different
from the outcome of PGG. When the key parameters are heterogeneous, for
instance, the results of PGG and R-PGG could be diverse even if we apply the
same amplitudes of heterogeneity. We find that the heterogeneity in R-PGG
generally impedes cooperation, while the opposite is observed for PGG. These
diverse system reactions can be understood if we follow how payoff functions
change when introducing heterogeneity in the parameter space. This analysis
also reveals the distinct roles of cooperator and defector strategies in the
mentioned games. Our observations may hopefully stimulate further research to
check the potential differences between PGG and R-PGG due to the alternative
complexity of conditions.
