  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.

Populations and Evolution;Physics and Society