  Economic experiments reveal that humans value cooperation and fairness.
Punishing unfair behavior is therefore common, and according to the theory of
strong reciprocity, it is also directly related to rewarding cooperative
behavior. However, empirical data fail to confirm that positive and negative
reciprocity are correlated. Inspired by this disagreement, we determine whether
the combined application of reward and punishment is evolutionary advantageous.
We study a spatial public goods game, where in addition to the three elementary
strategies of defection, rewarding and punishment, a fourth strategy combining
the later two competes for space. We find rich dynamical behavior that gives
rise to intricate phase diagrams where continuous and discontinuous phase
transitions occur in succession. Indirect territorial competition, spontaneous
emergence of cyclic dominance, as well as divergent fluctuations of
oscillations that terminate in an absorbing phase are observed. Yet despite the
high complexity of solutions, the combined strategy can survive only in very
narrow and unrealistic parameter regions. Elementary strategies, either in pure
or mixed phases, are much more common and likely to prevail. Our results
highlight the importance of patterns and structure in human cooperation, which
should be considered in future experiments.
