How “punishing evil and promoting good” promotes cooperation in social dilemma

•When only the role of punishing evil is considered, increasing punishment strength alone is not enough to weed out defectors.•When only the effect of promoting good is considered, under the larger temptation to defection, the increase in reward is not sufficient for cooperators to survive.•When considered simultaneously, there exists combinations of minimum values of punishment and reward that bring the system to a full cooperation state.