The evolution of large-scale cooperation among genetic strangers is a fundamental unanswered question in the social sciences. Behavioral economics has persuasively shown that so called ‘strong reciprocity’ plays a key role in accounting for the endogenous enforcement of cooperation. Insofar as strongly reciprocal players are willing to costly sanction defectors, cooperation flourishes. However, experimental evidence unambiguously indicates that not only defection and strong reciprocity, but also unconditional cooperation is a quantitatively important behavioral attitude. By referring to a prisoner’s dilemma framework where punishment (‘stick’) and rewarding (‘carrot’) options are available, here we show analytically that the presence of cooperators who don’t punish in the population makes altruistic punishment evolutionarily weak. We show that cooperation breaks down and strong reciprocity is maladaptive if costly punishment means ‘punishing defectors’ and, even more so, if it is coupled with costly rewarding of cooperators. In contrast, punishers don’t perish if cooperators, far from being rewarded, are sanctioned. These results, based on an extended notion of strong reciprocity, challenge evolutionary explanations of cooperation that overlook the ‘dark side’ of altruistic behavior.