  The promise of punishment and reward in promoting public cooperation is
debatable. While punishment is traditionally considered more successful than
reward, the fact that the cost of punishment frequently fails to offset gains
from enhanced cooperation has lead some to reconsider reward as the main
catalyst behind collaborative efforts. Here we elaborate on the "stick versus
carrot" dilemma by studying the evolution of cooperation in the spatial public
goods game, where besides the traditional cooperators and defectors, rewarding
cooperators supplement the array of possible strategies. The latter are willing
to reward cooperative actions at a personal cost, thus effectively downgrading
pure cooperators to second-order free-riders due to their unwillingness to bear
these additional costs. Consequently, we find that defection remains viable,
especially if the rewarding is costly. Rewards, however, can promote
cooperation, especially if the synergetic effects of cooperation are low.
Surprisingly, moderate rewards may promote cooperation better than high
rewards, which is due to the spontaneous emergence of cyclic dominance between
the three strategies.
