  Patterns by self-organization in nature have garnered significant interest in
a range of disciplines due to their intriguing structures. In the context of
the snowdrift game (SDG), which is considered as an anti-coordination game, but
the anti-coordination patterns are counterintuitively rare. In the work, we
introduce a model called the Two-Agents, Two-Action Reinforcement Learning
Evolutionary Game ($2\times 2$ RLEG), and apply it to the SDG on regular
lattices. We uncover intriguing phenomena in the form of Anti-Coordinated
domains (AC-domains), where different frustration regions are observed and
continuous phase transitions at the boundaries are identified. To understand
the underlying mechanism, we develop a perturbation theory to analyze the
stability of different AC-domains. Our theory accurately partitions the
parameter space into non-anti-coordinated, anti-coordinated, and mixed areas,
and captures their dependence on the learning parameters. Lastly, abnormal
scenarios with a large learning rate and a large discount factor that deviate
from the theory are investigated by examining the growth and nucleation of
AC-domains. Our work provides insights into the emergence of spatial patterns
in nature, and contributes to the development of theory for analysing their
structural complexities.
