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Indirect reciprocity is one of the major mechanisms driving the evolution of cooperation in human societies. There are two types of indirect reciprocity: upstream and downstream reciprocity. Cooperation in downstream reciprocity follows the pattern ‘You helped someone, and I will help you’, while the direction of cooperation is reversed in upstream reciprocity, which follows the pattern ‘You helped me, and I will help someone else’. These two types of indirect reciprocity often occur in combination. However, upstream and downstream reciprocity have mostly been theoretically studied in isolation. In this study, we propose a new model that integrates both types of reciprocity. In particular, we apply the standard giving-game framework of indirect reciprocity and analyze the three-strategy model including reciprocal altruists, indiscriminate altruists, and free riders using evolutionary game theory. We show that the model allows reciprocal altruists and free riders to coexist stably in well-mixed populations. We also find that by accounting for inattention in the assessment rule, the stability of this mixed equilibrium can be strengthened to prevent the invasion of infamous indiscriminate altruists and can even be made globally stable.
Sasaki et al. (Thu,) studied this question.