Indirect reciprocity is one of the major mechanisms for the evolution of cooperation in human societies. There are two types of indirect reciprocity: upstream and downstream. Cooperation in downstream reciprocity follows the pattern, 'You helped someone, and I will help you'. The direction of cooperation is reversed in upstream reciprocity, which instead follows the pattern, 'You helped me, and I will help someone else'. In reality, these two types of indirect reciprocity often occur in combination. However, upstream and downstream reciprocity have mostly been studied theoretically in isolation. Here, we propose a new model that integrates both types. We apply the standard giving-game framework of indirect reciprocity and analyze the model by means of evolutionary game theory. We show that the model can result in the stable coexistence of altruistic reciprocators and free riders in well-mixed populations. We also found that considering inattention in the assessment rule can strengthen the stability of this mixed equilibrium, even resulting in a global attractor. Our results indicate that the cycles of forwarding help and rewarding help need to be established for creating and maintaining diversity and inclusion in a society.

翻译：间接互惠是人类社会中合作演化的主要机制之一。间接互惠分为两种类型：上游互惠和下游互惠。下游互惠中的合作遵循“你帮助了某人，我将帮助你”的模式。而上游互惠中的合作方向则相反，遵循“你帮助了我，我将帮助其他人”的模式。在现实中，这两种间接互惠常常结合发生。然而，上游互惠和下游互惠在理论上大多被孤立研究。在此，我们提出一个整合两种类型的新模型。我们采用间接互惠的标准捐赠博弈框架，并通过演化博弈理论对模型进行分析。我们证明，该模型在充分混合的群体中能够实现利他互惠者与搭便车者的稳定共存。我们还发现，在评估规则中考虑疏忽因素可以增强这种混合均衡的稳定性，甚至形成全局吸引子。我们的结果表明，传递帮助与回报帮助的循环对于创建和维护社会的多样性与包容性至关重要。