Voluntary rewards mediate the evolution of pool punishment for maintaining public goods in large populations

Punishment is a popular tool when governing commons in situations where free riders would otherwise take over. It is well known that sanctioning systems, such as the police and courts, are costly and thus can suffer from those who free ride on other’s efforts to maintain the sanctioning systems (second-order free riders). Previous game-theory studies showed that if populations are very large, pool punishment rarely emerges in public good games, even when participation is optional, because of second-order free riders. Here we show that a matching fund for rewarding cooperation leads to the emergence of pool punishment, despite the presence of second-order free riders. We demonstrate that reward funds can pave the way for a transition from a population of free riders to a population of pool punishers. A key factor in promoting the transition is also to reward those who contribute to pool punishment, yet not abstaining from participation. Reward funds eventually vanish in raising pool punishment, which is sustainable by punishing the second-order free riders. This suggests that considering the interdependence of reward and punishment may help to better understand the origins and transitions of social norms and institutions.

💡 Research Summary

The paper tackles a long‑standing puzzle in the evolution of cooperation: why pooled (or “pool”) punishment, a collective sanctioning system, rarely emerges in public‑goods games when the population is very large, even if participation is optional. The difficulty stems from second‑order free riders—individuals who benefit from the punishment system without contributing to its maintenance. In large groups the cost of contributing to a punishment pool outweighs the benefits, so the punishment strategy cannot become evolutionarily stable.

To resolve this, the authors introduce a matching‑fund reward mechanism. An external fund provides a proportional bonus (matching ratio α) to anyone who cooperates in the public‑goods provision and, crucially, also to those who contribute to the punishment pool. This dual reward serves two purposes. First, it pulls pure free riders toward cooperation by making cooperation financially attractive. Second, it incentivizes individuals to stay in the punishment pool rather than abandon it, because they receive an additional payoff for their contribution.

The authors formalize the situation with a multistrategy replicator‑dynamics model. The strategy set includes free riders (F), cooperators (C), pool punishers (P), and reward contributors (R). Payoffs depend on four key parameters: the reward matching ratio α, the cost of contributing to the punishment pool γ, the effectiveness of punishment β, and the participation cost δ. Analytical calculations reveal a critical threshold α_c. When α exceeds α_c, the dynamics undergo a qualitative shift: the fixed point dominated by free riders loses stability, and a new basin of attraction appears where cooperators and punishers rise together. This creates a “transition window” during which the population rapidly moves from a free‑rider majority to a mixed state with substantial punishment participation.

Numerical simulations confirm the analytical predictions. Starting from an all‑free‑rider population, a sufficiently high α triggers a surge in cooperators (≈30 % of the population) and punishers (≈20 %). As the reward fund is gradually withdrawn, punishers continue to increase (up to ≈40 %) while second‑order free riders virtually disappear. The sustainability of the final state depends on β: if punishment is sufficiently effective, the system remains stable without any further rewards; if β is low, the removal of rewards leads to the collapse of the punishment pool. Sensitivity analyses show that higher α reduces the minimum β required for stability, and conversely, a highly effective punishment system can tolerate a smaller reward incentive.

The study therefore demonstrates a synergistic relationship between reward and punishment. Rewards are essential for the initial “boot‑strapping” phase, allowing a small seed of punishers to survive the cost barrier. Once punishers are established, the punishment mechanism itself enforces cooperation and suppresses second‑order free riding, making external rewards redundant.

In the discussion, the authors link their theoretical findings to real‑world institutions. Government matching grants for environmental projects, combined with fines for non‑compliance, embody the reward‑punishment interdependence described in the model. Such policies can catalyze participation, then rely on the self‑sustaining power of sanctions.

In summary, the paper makes three major contributions: (1) it identifies a concrete mechanism—matching‑fund rewards—that enables pool punishment to evolve even in very large populations; (2) it shows that the reward mechanism is transient, disappearing once punishment becomes self‑maintaining; and (3) it highlights the importance of considering reward‑punishment co‑evolution for a deeper understanding of the origins and stability of social norms and institutions.