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

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 critical thresholds separating distinct collective regimes. The results provide a theoretical explanation for the fragility of social order under weak institutions and highlight possible pathways to promote compliance.