Indirect punishment can outperform direct punishment in promoting cooperation in structured populations.

Indirect punishment traditionally sustains cooperation in social systems through reputation or norms, often by reducing defectors' payoffs indirectly. In this study, we redefine indirect punishment for structured populations as a spatially explicit mechanism, where individuals on a square lattice target second-order defectors-those harming their neighbors-rather than their own immediate defectors, guided by the principle: "I help you by punishing those who defect against you". Using evolutionary simulations, we compare this adapted indirect punishment to direct punishment, where individuals punish immediate defectors. Results show that within a narrow range of low punishment costs and fines, adapted indirect punishment outperforms direct punishment in promoting cooperation. However, outside this cost-fine region, outcomes vary: direct punishment may excel, both may be equally effective, or neither improves cooperation, depending on the parameter values. These findings hold even when network reciprocity alone does not support cooperation. Notably, when adapted indirect punishment outperforms direct punishment in promoting cooperation, defectors face stricter penalties without appreciably increasing punishers' costs, making it more efficient than direct punishment. Overall, our findings provide insights into the role of indirect punishment in structured populations and highlight its importance in understanding the evolution of cooperation.