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Evolutionary Dynamics with Randomly Distributed Benevolent Individuals

Understanding the evolution of cooperation is pivotal in biology and social science. Public resources sharing is a common scenario in the real world. In our study, we explore the evolutionary dynamics of cooperation on a regular graph with degree $k$, introducing the presence of a third strategy, namely the benevolence, who does not evolve over time, but provides a fixed benefit to all its neighbors. We find that the presence of the benevolence can foster the development of cooperative behavior and it follows a simple rule: $b/c > k - p_S(k-1)$. Our results provide new insights into the evolution of cooperation in structured populations.