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Social dilemmas concern a natural conflict between cooperation and selfinterests among individuals in large populations. The emergence of cooperationand its maintenance is the key for the understanding of fundamental conceptsabout the evolution of species. In order to understand the mechanisms involvedin this framework, here we study the Optional Public Good Games with focus onthe effects of diffusive aspects in the emergent patterns of cyclic dominancebetween the strategies. Differently from other works, we showed thatrock-paper-scissors (RPS) patterns occur by introducing a simple kind of randommobility in a lattice sparsely occupied. Such pattern has been revealed to bevery important in the conservation of the species in ecological and socialenvironments. The goal of this paper is to show that we do not need moreelaborated schemes for construction of the neighbourhood in the game to observeRPS patterns as suggested in the literature. As an interesting additionalresult, in this contribution we also propose an alternative method to quantifythe RPS density in a quantitative context of the game theory which becomespossible to perform a finite size scaling study. Such approach can be veryinteresting to be applied in other games generically.

Towards Rock-paper-scissors patterns in the Optional Public Goods Game under random mobility