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The exploration of social dilemmas is being considered a major foundation for encountering the enforced necessities of cooperation in self-organizing environments. Such environments are characterized by self-interested parties and the absence of trusted third parties. Recent approaches apply evolutionary socio-inspired games to formally prove the existence and further prolongation of cooperation patterns within communities. For instance, the Prisoner's Dilemma game has thus provided a rich opportunity to examine self-interested behaviors in pure peer-to-peer networks. However, assuming a total absence of coalitions, incentives and punishment mechanisms, several works argue against a durable maintenance of cooperation neither at single-shot nor repeated-scenarios. In this article, we formally and experimentally demonstrate a counterexample for the latter by applying evolutionary game theory and a particular instance of the Rock-Scissors-Paper game. Our framework proves that the cyclic dominance of certain type of nodes within a P2P system has an impact and introduces a strategic aspect to the evolution of the overall community.