Cops or robbers - a bistable society

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.

💡 Research Summary

The paper revisits Robert Axelrod’s “norm game” and studies it with two complementary approaches: a mean‑field master‑equation formulation and stochastic simulations on directed Erdős–Rényi networks. The author introduces two explicit assumptions that differ from many earlier works. First, an individual who breaks the norm cannot simultaneously act as a punisher, and a punisher cannot break the norm; the two roles are mutually exclusive. Second, the propensity to punish diminishes as the fraction of defectors grows – a “exhaustion” effect modeled by a factor (1 − z) that multiplies the term responsible for the increase of the punishing probability.

In the mean‑field description the state of the population is described by three probabilities: z (defection or “boldness”), y (punishing or “vengeance”), and x = 1 − y − z (pure compliance). The dynamics are given by

dz/dt = a x − b y z,
dy/dt = −c y z + e x z (1 − z),

where a is the temptation to defect, b the deterrent effect of punishment, c the cost of punishing, and e (set to 1) fixes the time scale. Fixed‑point analysis shows that the trivial state (x*,y*,z*) = (0,0,1) (all punish) is always stable, while (0,1,0) (all defect) is always unstable. However, when the ratio φ = b/a satisfies φ < 4c, two non‑trivial fixed points appear. At φ = 4c the two merge in a saddle‑point (or “fold”) bifurcation, after which the system jumps abruptly from a low‑z (high‑punishment) regime to a high‑z (high‑defection) regime. The “+” branch of the solution is unstable, the “−” branch is stable, producing a classic bistable situation: depending on initial conditions the population can settle in either of the two extreme states. This discontinuous transition is a direct consequence of the mutual exclusion of roles and the exhaustion factor; models that allow simultaneous defection and punishment exhibit only a transcritical bifurcation.

The author then implements an agent‑based simulation. A directed Erdős–Rényi graph of N = 10³–5 × 10³ nodes is generated with mean out‑degree λ. Each node i receives an initial boldness z_i drawn uniformly from (0.9 ρ, ρ) and an initial vengeance y_i = (1 − z_i)/μ (μ > 1). At each Monte‑Carlo step a random node is selected; it either complies (probability 1 − z_i) or defects (probability z_i). If it defects, its boldness is set to 1 and vengeance to 0, and all its outgoing neighbours are given the chance to punish. A neighbour k punishes with probability v_k = y_k; if it does, the defector’s boldness is reduced by a factor (1 − β) and the punisher’s vengeance by (1 − γ). If the neighbour abstains, it adopts the defector’s state (z_k = 1, y_k = 0). The parameters β (punishment strength) and γ (cost of punishing) control the dynamics.

Simulation results reproduce the mean‑field bistability when λ ≥ 2. Plotting the stationary average boldness ⟨z⟩ versus the initial average boldness ρ shows a sharp jump at a critical ρ_c that depends on γ and on the initial vengeance level μ. For λ = 1 the transition is smooth, confirming the role of network connectivity in sustaining the feedback loop. The boundary ρ_c(γ) separating the “all‑punish” basin (⟨z⟩ ≈ 0) from the “all‑defect” basin (⟨z⟩ ≈ 1) is mapped in Fig. 2; increasing γ (higher punishment cost) shrinks the punish‑dominant region, eventually eliminating it. These findings align with the saddle‑point bifurcation predicted analytically.

The discussion interprets the two conditions as socially plausible: many norms indeed forbid a defector from simultaneously acting as an enforcer, and collective defection tends to erode willingness to enforce. When both hold, the social system exhibits hysteresis: a gradual increase in the cost of punishment or in the prevalence of defection can push the society past a tipping point, after which the norm collapses abruptly. Conversely, restoring the norm may require a substantial reduction of the cost or an external “amnesty” that temporarily relaxes sanctions, allowing the punished to re‑enter the system without immediate retaliation. The author suggests that rewarding the few remaining punishers (e.g., making them public heroes) could help maintain the norm.

In summary, the paper contributes a novel exhaustion term to the norm‑game framework, demonstrates analytically that this term together with role exclusivity yields a saddle‑point bifurcation and bistability, and validates the prediction with extensive Monte‑Carlo simulations on directed random networks. The work bridges evolutionary game theory, statistical physics, and social dynamics, and opens avenues for future extensions such as meta‑games (punishing non‑punishers), heterogeneous network topologies, and adaptive network rewiring.