  We study the evolution of cooperation in spatial public goods games where,
besides the classical strategies of cooperation (C) and defection (D), we
consider punishing cooperators (PC) or punishing defectors (PD) as an
additional strategy. Using a minimalist modeling approach, our goal is to
separately clarify and identify the consequences of the two punishing
strategies. Since punishment is costly, punishing strategies loose the
evolutionary competition in case of well-mixed interactions. When spatial
interactions are taken into account, however, the outcome can be strikingly
different, and cooperation may spread. The underlying mechanism depends on the
character of the punishment strategy. In case of cooperating punishers,
increasing the fine results in a rising cooperation level. In contrast, in the
presence of the PD strategy, the phase diagram exhibits a reentrant transition
as the fine is increased. Accordingly, the level of cooperation shows a
non-monotonous dependence on the fine. Remarkably, punishing strategies can
spread in both cases, but based on largely different mechanisms, which depend
on the cooperativeness (or not) of punishers.
