The mathematical journey of spatial reciprocity

Abstract

Over the past three decades, evolutionary game theory in structured populations has witnessed explosive progress. Counterintuitive cooperative phenomena discovered in simulations have stimulated the development of rigorous mathematical theories to describe the underlying dynamics. In this paper, we review these theoretical advancements, including pair approximation and coalescent theory, which offer analytical solutions to evolutionary games on networks. We summarize the key results obtained by these theories, while also discussing their limitations, such as high computational complexity and constrained flexibility when applied to complex models. We therefore suggest that mathematical approaches provide a complementary tool for understanding cooperation in structured populations alongside simulations.