Reputation structure in indirect reciprocity under noisy and private assessment

TL;DR

This paper provides a theoretical analysis of reputation structures in indirect reciprocity under noisy and private assessments, deriving dynamics and equilibrium states that match numerical simulations, thus offering new insights into complex reputation systems.

Contribution

It introduces a mathematical framework for analyzing reputation dynamics with errors, deriving equilibrium distributions, and interpreting complex reputation structures.

Findings

The equilibrium reputation distribution can be approximated by Gaussian functions.

Theoretical results align well with numerical simulations.

Provides a new mathematical basis for studying indirect reciprocity.

Abstract

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.