The perceived legitimacy of costly punishment in the evolution of cooperation under private reputation

Indirect reciprocity promotes cooperation by allowing reputation to serve as a social currency for future benefits. However, the role of costly punishment under private reputation remains less understood. Under private reputation, independent assessments may lead observers and punishers to hold different views about the same recipient. This raises the question of how an observer evaluates punishment when the punished recipient appears to have a good reputation. We introduce the perceived legitimacy of costly punishment into a private-reputation model of indirect reciprocity. We first analyze the core dynamics among unconditional cooperators, unconditional defectors, and discriminating punishers, and then test whether the cooperative state is stable against the full set of action-rule mutants, including antisocial punishers. Our results show that costly punishment promotes cooperation only when its perceived legitimacy lies within an appropriate range. In this regime, discriminating punishers and unconditional cooperators can coexist stably, producing high cooperation and social payoff. However, insufficient legitimacy prevents punishers from disciplining defectors, whereas excessive legitimacy allows strategies that punish high-reputation individuals to invade, linking the model to antisocial punishment. Finite-population Markov analyses further confirm that the main qualitative dynamics are robust to small punishment execution and assessment errors. These findings show that, under private reputation, the perceived legitimacy of costly punishment determines whether punishment sustains cooperation or facilitates antisocial punishment.