  Social dilemmas concern a natural conflict between cooperation and self
interests among individuals in large populations. The emergence of cooperation
and its maintenance is the key for the understanding of fundamental concepts
about the evolution of species. In order to understand the mechanisms involved
in this framework, here we study the Optional Public Good Games with focus on
the effects of diffusive aspects in the emergent patterns of cyclic dominance
between the strategies. Differently from other works, we showed that
rock-paper-scissors (RPS) patterns occur by introducing a simple kind of random
mobility in a lattice sparsely occupied. Such pattern has been revealed to be
very important in the conservation of the species in ecological and social
environments. The goal of this paper is to show that we do not need more
elaborated schemes for construction of the neighbourhood in the game to observe
RPS patterns as suggested in the literature. As an interesting additional
result, in this contribution we also propose an alternative method to quantify
the RPS density in a quantitative context of the game theory which becomes
possible to perform a finite size scaling study. Such approach can be very
interesting to be applied in other games generically.
