  A good group reputation often facilitates more efficient synergistic teamwork
in production activities. Here we translate this simple motivation into a
reputation-based synergy and discounting mechanism in the public goods game.
Specifically, the reputation type of a group, either good or bad determined by
a reputation threshold, modifies the nonlinear payoff structure described by a
unified reputation impact factor. Results show that this reputation-based
incentive mechanism could effectively promote cooperation compared with linear
payoffs, despite the coexistence of synergy and discounting effects. Notably,
the complicated interactions between reputation impact and reputation threshold
result in a sharp phase transition from full cooperation to full defection. We
also find that the presence of a few discounting groups could increase the
average payoffs of cooperators, leading to an interesting phenomenon that when
the reputation threshold is raised, the gap between the average payoffs of
cooperations and defectors increases while the overall payoff decreases. Our
work provides important insights into facilitating cooperation in social
groups.
