  We study the evolution of cooperation in spatial public goods games with four
competing strategies: cooperators, defectors, punishing cooperators, and
punishing defectors. To explore the robustness of the cooperation-promoting
effect of costly punishment, besides the usual strategy adoption dynamics we
also apply strategy mutations. As expected, frequent mutations create kind of
well-mixed conditions, which support the spreading of defectors. However, when
the mutation rate is small, the final stationary state does not significantly
differ from the state of the mutation-free model, independently of the values
of the punishment fine and cost. Nevertheless, the mutation rate affects the
relaxation dynamics. Rare mutations can largely accelerate the spreading of
costly punishment. This is due to the fact that the presence of defectors
breaks the balance of power between both cooperative strategies, which leads to
a different kind of dynamics.
