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Patterns by self-organization in nature have garnered significant interest ina range of disciplines due to their intriguing structures. In the context ofthe snowdrift game (SDG), which is considered as an anti-coordination game, butthe anti-coordination patterns are counterintuitively rare. In the work, weintroduce a model called the Two-Agents, Two-Action Reinforcement LearningEvolutionary Game ($2\times 2$ RLEG), and apply it to the SDG on regularlattices. We uncover intriguing phenomena in the form of Anti-Coordinateddomains (AC-domains), where different frustration regions are observed andcontinuous phase transitions at the boundaries are identified. To understandthe underlying mechanism, we develop a perturbation theory to analyze thestability of different AC-domains. Our theory accurately partitions theparameter space into non-anti-coordinated, anti-coordinated, and mixed areas,and captures their dependence on the learning parameters. Lastly, abnormalscenarios with a large learning rate and a large discount factor that deviatefrom the theory are investigated by examining the growth and nucleation ofAC-domains. Our work provides insights into the emergence of spatial patternsin nature, and contributes to the development of theory for analysing theirstructural complexities.

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