One of the most direct human mechanisms of promoting cooperation is rewarding it. We study the effect of sharing a reward among cooperators in the most stringent form of social dilemma, namely the Prisoners Dilemma. Specifically, for a group of players that collect payoffs by playing a pairwise Prisoners Dilemma game with their partners, we consider an external entity that distributes a fixed reward equally among all cooperators. Thus, individuals confront a new dilemma: on the one hand, they may be inclined to choose the shared reward despite the possibility of being exploited by defectors; on the other hand, if too many players do that, cooperators will obtain a poor reward and defectors will outperform them. By appropriately tuning the amount to be shared a vast variety of scenarios arises, including traditional ones in the study of cooperation as well as more complex situations where unexpected behavior can occur. We provide a complete classification of the equilibria of the $n$-player game as well as of its evolutionary dynamics.
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