Threshold behavior of a social norm in response to error proneness

TL;DR

The paper addresses the stability of social norms under heterogeneous assessment errors in indirect reciprocity, focusing on Simple Standing. It combines a discrete agent-based framework with a continuous analytic model to assess when a rare, more error-prone mutant can invade a resident population. A key finding is a threshold $\epsilon^* = \tfrac{1}{4}\bigl(3-2r-\sqrt{1+4r-4r^2}\bigr)$ (with $r=c/b$) below which residents prevent invasion by higher-error mutants, and above which the norm collapses due to invasion by error-prone individuals. This reveals that treating error proneness as an individual attribute can qualitatively alter norm stability and helps explain why error-prone individuals may thrive when assessment errors are distributed heterogeneously. The results have implications for understanding cultural transmission and the design of norms resilient to observational noise.

Abstract

A social norm defines what is good and what is bad in social contexts, as well as what to do based on such assessments. A stable social norm should be maintained against errors committed by its players. In addition, individuals may have different probabilities of errors in following the norm, and a social norm would be unstable if it benefited those who do not follow the norm carefully. In this work, we show that Simple Standing, which has been known to resist errors and mutants successfully, actually exhibits threshold behavior. That is, in a population of individuals playing the donation game according to Simple Standing, the residents can suppress the invasion of mutants with higher error proneness only if the residents' own error proneness is sufficiently low. Otherwise, the population will be invaded by mutants that commit assessment errors more frequently, and a series of such invasions will eventually undermine the existing social norm. This study suggests that the stability analysis of a social norm may have a different picture if the probability of error itself is regarded as an individual attribute.

Figures (7)

• Figure 1: A mutant's payoff advantage relative to a resident's, with dotted lines showing the contours of $\Delta\pi_0 = 0$. We have run ABM simulations with $N=10^2$, $T=10^4$, and $S=10^3$. The donation game is parametrized by $b=1$ and $c=1/2$, and the observation probability is set to $q=1$. (a) Numerical results for Image Scoring, which is zero everywhere up to statistical noise, in full agreement with the prediction from the continuous model. (b) Stern Judging shows similar behavior as predicted by our analysis.
• Figure 2: A mutant's payoff advantage relative to a resident according to Shunning [see Eq. \ref{['eq:sh']}]. The dotted lines show the contour lines of $\Delta \pi_0 = 0$. (a) Analytic results obtained by solving the continuous model. (b) ABM results with the same simulation parameters as in Fig. \ref{['fig:scsj']}.
• Figure 3: Visualization of Eq. \ref{['eq:ss_dpi0']} in the case of Simple Standing, where the dotted lines show the contour lines of $\Delta \pi_0 = 0$. (a) Analytic results from the continuous model. (b) ABM results with the same simulation parameters as in Figs. \ref{['fig:scsj']} and \ref{['fig:shun']}.
• Figure 4: Mutation-selection dynamics with $N=50$ and $r\equiv c/b=1/2$. The total simulation period is $T=10^6$ time steps, and the periods for selection and mutation are $\tau_s=10^3$ and $\tau_m=10^4$, respectively. The mutation is represented by the Gaussian random variable with width $w=10^{-2}$. (a) Sample trajectories starting from mean values $\bar{\epsilon} = 0.05$, $0.1$, $0.2$, and $0.3$, respectively. The horizontal dotted line indicates the theoretical threshold $\epsilon^\ast =(2-\sqrt{2})/4 \approx 0.15$ [Eq. \ref{['eq:threshold']}]. (b) The distribution of $\left\{ \epsilon_i \right\}$ after the $T=10^6$ time steps, obtained from $10$ sample trajectories for each of the initial mean values. The vertical dotted line indicates $\epsilon^\ast$.
• Figure 5: (a) Equations \ref{['eq:m10ss']} and \ref{['eq:m11ss']} for Simple Standing, when $\epsilon_0=1/2$, at which $m_{01}$ is identically $1/2$ [Eq. \ref{['eq:m01ss']}]. (b) The resulting $\Delta \pi_0$ [Eq. \ref{['eq:ss_dpi0']}] for Simple Standing, when $b=1$ and $c=1/2$. The horizontal dashed line is drawn to show where $\Delta \pi_0$ changes the sign.