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<p>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 this action as bad, which eventually segregates the society into two antagonistic clusters. However, if it is judged as good, the system reaches paradise state 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 <i>N</i>,&#xa0; where everyone regards such an action as good with probability <i>p</i> and bad with <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(1-p\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>-</mo> <mi>p</mi> </mrow> </math></EquationSource> </InlineEquation>, we observe a relaxation to paradise when <i>Np</i> is sufficiently greater than <i>O</i>(1). In contrast, in a heterogeneous setting where only <i>k</i> individuals judge such an action as good, the size difference of the clusters increases almost linearly as <i>k</i> increases, meaning that the paradise can only be reached only when <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(k \rightarrow N\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo stretchy="false">→</mo> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation> in a large population. Therefore, when an urgent change is needed to overcome the segregation due to L6, the intervention should be made uniformly on the entire population to help small change in every individual’s judgment, rather than intensively on a specific fraction of the population.</p>

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Controlling sternness in judging a good person who helps the bad

  • 期刊论文

  • 发表日期：

    2026年1月1日

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  • Quang Anh Le,
  • Minwoo Bae,
  • Takashi Shimada,
  • Seung Ki Baek

摘要

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 this action as bad, which eventually segregates the society into two antagonistic clusters. However, if it is judged as good, the system reaches paradise state 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 an action as good with probability p and bad with \(1-p\) 1 - p , we observe a relaxation to paradise when Np is sufficiently greater than O(1). In contrast, in a heterogeneous setting where only k individuals judge such an action as good, the size difference of the clusters increases almost linearly as k increases, meaning that the paradise can only be reached only when \(k \rightarrow N\) k → N in a large population. Therefore, when an urgent change is needed to overcome the segregation due to L6, the intervention should be made uniformly on the entire population to help small change in every individual’s judgment, rather than intensively on a specific fraction of the population.