Making public reputation out of private assessments

TL;DR

The approach provides a theoretical framework for describing the formation of reputation in mathematical terms and argues that the formation of reputation can be modeled by a bi-stochastic matrix, provided that both assessment and behavior are regarded as continuous variables.

Abstract

Reputation is not just a simple opinion that an individual has about another but a social construct that emerges through communication. Despite the huge importance in coordinating human behavior, such a communicative aspect has remained relatively unexplored in the field of indirect reciprocity. In this work, we bridge the gap between private assessment and public reputation: We begin by clarifying what we mean by reputation and argue that the formation of reputation can be modeled by a bi-stochastic matrix, provided that both assessment and behavior are regarded as continuous variables. By choosing bi-stochastic matrices that represent averaging processes, we show that only four norms among the leading eight, which judge a good person's cooperation toward a bad one as good, will keep cooperation asymptotically or neutrally stable against assessment error in a homogeneous society where every member has adopted the same norm. However, when one of those four norms is used by the resident population, the opinion averaging process allows neutral invasion of mutant norms with small differences in the assessment rule. Our approach provides a theoretical framework for describing the formation of reputation in mathematical terms.

Figures (2)

• Figure 1: The effect of $\theta$ on the high-order excitations of the normative dynamics. We consider $N=50$ individuals who use Simple Standing (L3). The initial condition is close to $m^\ast=1$, except that we introduce perturbation by randomly choosing $20\%$ of the image matrix elements and changing the values to $0.9$. The observation probability is $q=0.5$. The shades mean standard error estimated from $10$ independent runs.
• Figure 2: Invasion analysis of a resident population using Simple Standing (L3). (a) The mutant norm has a small difference $\delta(x,y,z) = \delta_1 (2yz-2z+1)$ with $\delta_1=5 \times 10^{-2}$ from the residents' $\alpha(x,y,z) = yz-z+1$ (see Table \ref{['tab:cont']}). (b) The mutant norm has a small difference $\eta(x,y) = \eta_1 xy$ with $\eta_1 = 5\times 10^{-2}$ from the residents' $\beta(x,y) = y$. The population size is $N=50$, the fraction of mutants is $p=0.1$, and the observation probability is $q=0.5$. We have generated $10^2$ independent samples for each data point to estimate standard error as represented by the shades. Each sample has been simulated for $10^2$ MCS, out of which the last half has been used for calculating the payoffs.