  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.
