Cops or robbers - a bistable society

TL;DR

This paper models societal norm dynamics using a bistable system where societies can stably exist in states of widespread norm-breaking or widespread punishing, showing how societal behaviors can switch abruptly.

Contribution

It introduces a new mechanism where the tendency to punish is suppressed by majority norm-breaking and analyzes the resulting bistability with both equations and network simulations.

Findings

Bistable solutions with societal norm-breaking or punishing.

Discontinuous shifts in societal norm adherence.

Bistability observed in network simulations.

Abstract

The norm game described by Axelrod in 1985 was recently treated with the master equation formalism. Here we discuss the equations, where {\it i)} those who break the norm cannot punish and those who punish cannot break the norm, {\it ii)} the tendency to punish is suppressed if the majority breaks the norm. The second mechanism is new. For some values of the parameters the solution shows the saddle-point bifurcation. Then, two stable solutions are possible, where the majority breaks the norm or the majority punishes. This means, that the norm breaking can be discontinuous, when measured in the social scale. The bistable character is reproduced also with new computer simulations on the Erd{\H o}s--R\'enyi directed network.