How the indirect reciprocity with co-evolving norm and strategy for 2 ×× 2 prisoner’s dilemma game works for emerging cooperation

•A new indirect reciprocity model with coevolution of agent’s strategy and norm is made.•A rather larger dilemma shows robust cooperation establishing compared with a smaller dilemma.•A norm punishing a so-called second-order free-rider is prompted.•For socially healthy norm, small number of defectors are needed.

We built a new indirect reciprocity model based on binary image scores, where an agent’s strategy and norm co-evolve. The norm, meaning what behavior is evaluated as “good” or “bad,” stipulates how image scores of two agents playing a game is altered, which has been presumed to be a fixed value in most previous studies. Also, unlike former studies, our model allows an agent to play with an agent who has a different norm. This point of relaxing the freedom of the model pulls down cooperation level vis-à-vis the case where an agent always plays with another one having same norm. However, it is observed that a rather larger dilemma shows robust cooperation establishing compared with a smaller dilemma, since a norm that punishes a so-called second-order free-rider is prompted. To encourage the evolution of norms to be able to punish second-order free-riders, a society needs a small number of defectors. This is elucidated by the fact that cases with action error are more cooperative than those without action error.