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We study the evolution of cooperation in spatial public goods games where,besides the classical strategies of cooperation (C) and defection (D), weconsider punishing cooperators (PC) or punishing defectors (PD) as anadditional strategy. Using a minimalist modeling approach, our goal is toseparately clarify and identify the consequences of the two punishingstrategies. Since punishment is costly, punishing strategies loose theevolutionary competition in case of well-mixed interactions. When spatialinteractions are taken into account, however, the outcome can be strikinglydifferent, and cooperation may spread. The underlying mechanism depends on thecharacter of the punishment strategy. In case of cooperating punishers,increasing the fine results in a rising cooperation level. In contrast, in thepresence of the PD strategy, the phase diagram exhibits a reentrant transitionas the fine is increased. Accordingly, the level of cooperation shows anon-monotonous dependence on the fine. Remarkably, punishing strategies canspread in both cases, but based on largely different mechanisms, which dependon the cooperativeness (or not) of punishers.

Punish, but not too hard: How costly punishment spreads in the spatial public goods game