RGB algorithm for spatial evolutionary game theory with finite populations

Evolutionary dynamics is captured by replicator equations when populations are well mixed. However, in realistic ecosystems, competitions often occur between neighbors and the spatial structure of the system is of significant importance. In most evolutionary algorithms, the dynamics of local death/birth processes often relies on the effective fitness: a global knowledge of the whole ecosystem. To make the spatial game theory logically consistent, it is desirable to introduce an algorithm where only local information is necessary. Here we resolve the challenge by introducing the three-party Reference-Gamble-Birth (RGB) algorithm. For the well-mixed case, the RGB algorithm reproduces the replicator equations in the large population limit. We also apply the RGB algorithm on the rock-paper-scissor game to demonstrate how the ecological stability sensitively depends on the spatial structures. The proposed RGB algorithm is not limited to cyclically competing systems and can be applied to various spatial games with different network structures.