Controlling sternness in judging a good person who helps the bad

Abstract: Recent studies on indirect reciprocity with private assessment on complete graphs suggest the possibility that one can continuously modulate the degree of segregation by controlling how to judge a good person helping a bad one. A well-known social norm called L6 judges it as bad, which eventually segregates the society into two antagonistic clusters, but if it is judged as good, the system reaches paradise where everyone likes each other. In this work, we numerically study this transition between segregation and paradise in two different settings. Firstly, in a uniform population of size $N$ where everyone regards such a donor as good with probability $p$ and bad with $1-p$, we observe paradise when $Np$ is sufficiently greater than $O(1)$. In contrast, in a heterogeneous setting where only $k$ individuals judge such a donor as good, the size difference of the clusters increases almost linearly as $k$ increases, so paradise can only be reached as $k \to N$ in a large population. Therefore, when an urgent change is needed to overcome the segregation due to L6, a small change in each and every individual's behavior is more efficient than a radical change in a fraction of the population.