Supplementary text from Evolution of coordinated punishment to enforce cooperation from an unbiased strategy space

The emergence and maintenance of punishment to protect the commons remains an open puzzle in social and biological sciences. Even in societies where pro-social punishing is common, some individuals seek to cheat the system if they see a chance to do so—and public goods are often maintained in spite of cheaters who do not contribute. We present a model accounting for all possible strategies in a public goods game with punishment. While most models of punishment restrict the set of possible behaviours, excluding seemingly paradoxical anti-social strategies from the start, we show that these strategies can play an important role in explaining large-scale cooperation as observed in human societies. We find that coordinated punishment can emerge from individual interactions, but the stability of the associated institutions is limited due to anti-social and opportunistic behaviour. In particular, coordinated anti-social punishment can undermine cooperation if individuals cannot condition their behaviour on the existence of institutions that punish. Only when we allow for observability and conditional behaviours, anti-social strategies do no longer threat cooperation. This is due to a stable coexistence of a minority supporting pro-social institutions and those who only cooperate if such institutions are in place. This minority of supporters is enough to guarantee substantial cooperation under a wide range of conditions. Our findings resonate with the empirical observation that public goods are resilient to opportunistic cheaters in large groups of unrelated individuals. They also highlight the importance of letting evolution, and not modellers, decide which strategies matter.