Incomplete reputation information and punishment in indirect reciprocity

Indirect reciprocity promotes cooperation by allowing individuals to help others based on reputation rather than direct reciprocation. Because it relies on accurate reputation information, its effectiveness can be undermined by information gaps. We examine two forms of incomplete information: incomplete observation, in which donor actions are observed only probabilistically, and reputation fading, in which recipient reputations are sometimes classified as "Unknown". Using analytical frameworks for public assessment, we show that these models with incomplete reputation information yield qualitatively different outcomes. Under incomplete observation, the conditions for cooperation are unchanged, because less frequent updates are exactly offset by higher reputational stakes. In contrast, reputation fading hinders cooperation, requiring higher benefit-to-cost ratios as the identification probability decreases. We then evaluate costly punishment as a third action alongside cooperation and defection. Norms incorporating punishment can sustain cooperation across broader parameter ranges without reducing efficiency in the reputation fading model. This contrasts with previous work, which found punishment ineffective under a different type of information limitation, and highlights the importance of distinguishing between types of information constraints. Finally, we review past studies to identify when punishment is effective and when it is not in indirect reciprocity.