The propensity for disobedience: Rule-breaking, compliance and social phase transitions

Abstract

We develop a mathematical model to describe the persistence of rule-breaking behaviors in societies, such as traffic violations, disregard for legal restrictions and other forms of noncompliance. Using a replicator-type dynamics with utility functions incorporating individual benefits, institutional punishment and social sanctions, we first built a general formulation of the system. Within this framework, we analyze two distinct models differing in the nature of social feedback. In the presence of positive feedback, the system exhibits bistability, with widespread compliance and widespread violation as stable equilibria, and the transition between these states occurs discontinuously once a critical threshold is crossed, resembling a first-order phase transition. By contrast, when negative feedback is present, the population undergoes a continuous phase transition between compliant and noncompliant collective states, driven by an increasing collective cost of rule-breaking. Numerical simulations and analytical results illustrate how changes in enforcement, social tolerance or perceived benefits can shift the system across tipping points. The results provide a theoretical explanation for the fragility of social order under weak institutions and highlight possible pathways to promote compliance.

Figures (4)

• Figure 1: Time evolution of the fraction $d(t)$ of rule-breakers for Model I for different initial conditions $d_0=d(t=0)$ and representative values of the punishment probability $p$. Panels (a)-(c) illustrate the three possible dynamical regimes of the Model I: (a) $p=0.03$, corresponding to $K>L$, where the population converges to full rule-breaking ($d=1$); (b) $p=0.15$, corresponding to $0<K<L$, where the system exhibits bistability depending on the initial condition; (c) $p=0.30$, corresponding to $K<0$, where the population converges to full compliance ($d=0$). The parameters are $B=1.0$, $F=5.0$, $C=0.2$, $s=0.3$, and $R=0.6$.
• Figure 2: Stationary behavior of Model I. (a) Stationary fraction of rule-breakers $d$ as a function of the punishment probability $p$. The numerical results (squares) display the two stable branches $d=1$ and $d=0$, as well as the bistable region for $p_1 < p < p_2$, where $p_1=0.06$ and $p_2=0.24$ are obtained from Eqs. \ref{['eq8a']} and \ref{['eq9a']} and are indicated by dotted and dashed vertical lines, respectively. In the bistable region we also exhibit the analytical solution $d^* = 1 - K/L$, Eq. \ref{['eq7']} (solid line), which corresponds to the unstable intermediate branch for $0<K<L$. The figure clearly illustrates the coexistence of the two stable absorbing states in the interval $p_1 < p < p_2$. (b) Basin of attraction for $p=0.15$, showing the stationary value of $d$ as a function of the initial condition $d_0=d(t=0)$. The critical threshold $d^* = 1 - K/L = 0.5$ separates the two basins of attraction: for $d_0 < d^*$ the system converges to $d=0$, whereas for $d_0 > d^*$ it converges to $d=1$. The parameters are $B=1.0$, $F=5.0$, $C=0.2$, $s=0.3$ and $R=0.6$.
• Figure 3: Time evolution of the fraction $d(t)$ of rule-breakers for Model II for different initial conditions $d_0=d(t=0)$ and representative values of the punishment probability $p$. The parameters are $B=1.0$, $F=5.0$, $C=0.2$, $R=0.6$ and $\alpha=0.20$, which leads to $p_c=0.12$ obtained from Eq. \ref{['eq14']}. Panel (a) shows results for $p=0.05$, where the population converges to full rule-breaking ($d=1$), panel (b) exhibits results for $p=0.10$, where the population reaches the stationary state $d=0.50$ given by Eq. \ref{['eq13']}, for any initial condition, and panel (c) shows results for $p=0.30$, where the population converges to full compliance ($d=0$).
• Figure 4: Stationary fraction of rule-breakers $d^{*}$ for Model II as a function of the punishment probability $p$, obtained from the numerical integration of Eq. \ref{['eq12']} (squares). The analytical result of Eq. \ref{['eq13']} is also show for comparison (full line). The parameters are $B=1.0$, $F=5.0$, $C=0.2$, $R=0.6$ and $\alpha=0.20$, which leads to $p_c=0.12$ obtained from Eq. \ref{['eq14']}, indicated by an arrow. The continuous nature of the phase transition is clearly observed.