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Economic experiments reveal that humans value cooperation and fairness.Punishing unfair behavior is therefore common, and according to the theory ofstrong reciprocity, it is also directly related to rewarding cooperativebehavior. However, empirical data fail to confirm that positive and negativereciprocity are correlated. Inspired by this disagreement, we determine whetherthe combined application of reward and punishment is evolutionary advantageous.We study a spatial public goods game, where in addition to the three elementarystrategies of defection, rewarding and punishment, a fourth strategy combiningthe later two competes for space. We find rich dynamical behavior that givesrise to intricate phase diagrams where continuous and discontinuous phasetransitions occur in succession. Indirect territorial competition, spontaneousemergence of cyclic dominance, as well as divergent fluctuations ofoscillations that terminate in an absorbing phase are observed. Yet despite thehigh complexity of solutions, the combined strategy can survive only in verynarrow and unrealistic parameter regions. Elementary strategies, either in pureor mixed phases, are much more common and likely to prevail. Our resultshighlight the importance of patterns and structure in human cooperation, whichshould be considered in future experiments.

Correlation of positive and negative reciprocity fails to confer an evolutionary advantage: Phase transitions to elementary strategies