Title: Incomplete Reputation Information and Punishment in Indirect Reciprocity Show Chinese title

Title: 不完全的声誉信息与间接互惠中的惩罚

Abstract: 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 seemingly similar models 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. Show Chinese abstract

Abstract: 间接互惠通过允许个体根据声誉而非直接互惠来帮助他人来促进合作。 由于它依赖于准确的声誉信息，其有效性可能因信息缺口而受到损害。 我们研究了两种不完整信息的形式：不完全观察，其中捐赠者的行为仅以概率方式被观察到，以及声誉消退，其中接收者的声誉有时被归类为“未知”。 使用公共评估的分析框架，我们表明这些看似相似的模型产生了定性不同的结果。 在不完全观察下，合作的条件保持不变，因为更新频率较低恰好被更高的声誉风险所抵消。 相反，声誉消退阻碍了合作，随着识别概率的降低，需要更高的收益与成本比率。 然后，我们将代价高昂的惩罚作为第三种行动，与合作和背叛并列。 包含惩罚的规范可以在不减少声誉消退模型中效率的情况下，在更广泛的参数范围内维持合作。 这与之前的研究结果形成对比，之前的研究发现惩罚在另一种类型的信息限制下无效，并突显了区分信息约束类型的重要性。 最后，我们回顾了以往的研究，以确定在间接互惠中惩罚何时有效以及何时无效。