Effect of assessment error and private information on stern-judging in indirect reciprocity

• We analyze a model of indirect reciprocity, a key mechanism to sustain cooperation.
• We incorporate assessment error and private information in the model.
• We show that stern-judging, one of the best-known assessment models, collapses.
• The proportion of “good” individuals in a population with stern-judging is 1/2.

Stern-judging is one of the best-known assessment rules in indirect reciprocity. Indirect reciprocity is a fundamental mechanism for the evolution of cooperation. It relies on mutual monitoring and assessments, i.e., individuals judge, following their own assessment rules, whether other individuals are “good” or “bad” according to information on their past behaviors. Among many assessment rules, stern-judging is known to provide stable cooperation in a population, as observed when all members in the population know all about others’ behaviors (public information case) and when the members never commit an assessment error. In this paper, the effect of assessment error and private information on stern-judging is investigated. By analyzing the image matrix, which describes who is good in the eyes of whom in the population, we analytically show that private information and assessment error cause the collapse of stern-judging: all individuals assess other individuals as “good” at random with a probability of 1/2.