Emergence of anti-coordinated patterns in snowdrift game by reinforcement learning

Patterns emerging through self-organization in nature have sparked considerable interest across various disciplines, owing to their significance in comprehending collective behaviors. The anti-coordination phenomenon is widespread in both social systems and ecosystems. However, within the context of the SnowDrift Game (SDG), considered as a typical anti-coordination game, the occurrence of anti-coordination patterns is surprisingly rare. In this work, we introduce a model called the Two-Agent, Two-Action Reinforcement Learning Evolutionary Game (2 × 2 RLEG), and utilize it in the SDG on different regular lattices. We unveil intriguing phenomena represented by Anti-Coordinated domains (AC-domains) within spatial patterns, which hold a significant sway over the cooperation preference within populations. The frustration contributes substantially to the diversity of AC-domains within parameter space. To comprehend the underlying mechanism, we have developed a perturbation theory to analyze the stability of different AC-domains comprising action patterns and identify continuous phase transitions within parameter space. Our theory effectively partitions the parameter space into several areas, each characterized by unique features of domains that determine the cooperation preferences within the population. Our work offers insights into the emergence of spatial patterns and their impact on cooperation in nature, contributing to the development of theories that analyze their structural complexities.