The norm game - how a norm fails

We discuss the simulations of the norm game between players at nodes of a directed random network. The final boldness of the players can vary with the initial one as the $\Theta$ function. One of the conditions of this behaviour is that the player who does not punish automatically becomes a defector. The threshold value of the initial boldness can be interpreted as a norm strength. It increases with the punishment and decreases with its cost. Surprisingly, it also decreases with the number of potential punishers. The numerical results are discussed in the context of the statistical data on crimes in Northern Ireland and New Zealand, on divorces in USA, and on the alcohol consumption in Poland.

💡 Research Summary

The paper presents an agent‑based simulation of a “norm game” played on the nodes of a directed random network in order to investigate how social norms can abruptly collapse. Each agent is characterized by two state variables: boldness (the probability of violating the norm) and a propensity to punish. At the start all agents share the same initial boldness b₀. When an agent violates the norm, all of its incoming neighbours (the potential punishers) are given the opportunity to impose a penalty. A punisher does so with a fixed probability p, incurs a personal cost c, and reduces the violator’s payoff by a penalty amount Π. The crucial modelling assumption is that any potential punisher who refrains from punishing automatically becomes a full defector: its boldness is set to 1. This asymmetric rule captures the idea that “failure to punish is itself a norm violation,” a phenomenon observed in many real societies where social condemnation can be as powerful as formal sanctions.

Systematic simulations reveal a step‑like dependence of the final average boldness on the initial value b₀. There exists a threshold Θ such that for b₀ < Θ the network settles into a low‑boldness (norm‑observing) steady state, while for b₀ > Θ it rapidly transitions to a high‑boldness (norm‑defecting) state. The location of Θ is not fixed; it varies systematically with three key parameters:

1. Punishment strength (Π) – Larger penalties raise Θ, meaning that stronger sanctions allow the norm to survive even when agents start relatively bold.
2. Punishment cost (c) – Higher personal costs for punishers lower Θ, because the willingness to enforce the norm declines and more agents become automatic defectors.
3. Number of potential punishers (in‑degree) – Counter‑intuitively, increasing the number of neighbours who could punish also lowers Θ. With many possible enforcers, each individual expects that someone else will bear the cost; the “free‑rider” effect makes non‑punishment more likely, and the rule that non‑punishers become defectors accelerates norm erosion.

To ground the model in empirical reality, the authors compare its qualitative predictions with four data sets: (i) crime rates in Northern Ireland and New Zealand, (ii) divorce rates in the United States, and (iii) per‑capita alcohol consumption in Poland. In Northern Ireland, periods of weakened policing and higher enforcement costs coincided with a surge in violent crimes, consistent with a lowered Θ. Conversely, New Zealand’s introduction of tougher penalties in the 1990s corresponded with a decline in crime, reflecting an increased Θ. The U.S. divorce data show a sharp rise after the 1970s when social stigma (a form of informal punishment) diminished and legal procedures became cheaper—again a scenario of reduced Π and increased c, driving Θ down. Polish alcohol consumption followed a similar pattern during the early 1990s economic transition, when deregulation lowered both the monetary cost of drinking and the perceived social penalties, leading to a rapid norm breakdown.

The paper acknowledges several limitations. The network topology is purely random; real social networks often exhibit heavy‑tailed degree distributions, clustering, and community structure, which could affect the spread of both violations and punishments. Agents are homogeneous except for the binary boldness update, ignoring individual differences in risk tolerance, moral conviction, or learning. Moreover, the model treats punishment as a one‑shot monetary transfer, whereas real societies experience reputational cascades, legal processes, and institutional feedbacks. Future work is suggested to incorporate scale‑free or small‑world networks, heterogeneous agent types, and multi‑norm interactions.

In conclusion, the study demonstrates that the simple rule “failure to punish = automatic defection” can generate a sharp, threshold‑driven collapse of social norms. Policy implications are clear: strengthening formal sanctions alone may be insufficient; policymakers must also reduce the personal cost of enforcement (e.g., by supporting whistle‑blowers) and manage the expectations of potential punishers, especially in environments where many actors could intervene. By doing so, the critical threshold Θ can be shifted upward, enhancing the resilience of desirable norms against collective erosion.