Integrated strong reciprocity enables productive punishment and protective defection

Abstract: Cooperation in large groups and one-shot interactions is often hindered by freeloading. Punishment can enforce cooperation, but it is usually regarded as wasteful because the costs of punishing offset its benefits. Here, we analyze an evolutionary game model that integrates upstream and downstream reciprocity with costly punishment: integrated strong reciprocity (ISR). We demonstrate that ISR admits a stable mixed equilibrium of ISR and unconditional defection (ALLD), and that costly punishment can become productive: When sufficiently efficient, it raises collective welfare above the no-punishment baseline. ALLD players persist as evolutionary shields, preventing invasion by unconditional cooperation (ALLC) or alternative conditional strategies (e.g., antisocial punishment). At the same time, the mixed equilibrium of ISR and ALLD remains robust under modest complexity costs that destabilize other symmetric cooperative systems.