  Evaluation relationships are pivotal for maintaining a cooperative society. A
formation of the evaluation relationships has been discussed in terms of
indirect reciprocity, by modeling dynamics of good or bad reputations among
individuals. Recently, a situation that individuals independently evaluate
others with errors (i.e., noisy and private reputation) is considered, where
the reputation structure (from what proportion of individuals in the population
each receives good reputations, defined as goodness here) becomes complex, and
thus has been studied mainly with numerical simulations. The present study
gives a theoretical analysis of such complex reputation structure. We formulate
the time change of goodness of individuals caused by updates of reputations
among individuals. By considering a large population, we derive dynamics of the
frequency distribution of goodnesses. An equilibrium state of the dynamics is
approximated by a summation of Gaussian functions. We demonstrate that the
theoretical solution well fits the numerical calculation. From the theoretical
solution, we obtain a new interpretation of the complex reputation structure.
This study provides a novel mathematical basis for cutting-edge studies on
indirect reciprocity.
