Reward and cooperation in the spatial public goods game

The promise of punishment and reward in promoting public cooperation is debatable. While punishment is traditionally considered more successful than reward, the fact that the cost of punishment frequently fails to offset gains from enhanced cooperation has lead some to reconsider reward as the main catalyst behind collaborative efforts. Here we elaborate on the “stick versus carrot” dilemma by studying the evolution of cooperation in the spatial public goods game, where besides the traditional cooperators and defectors, rewarding cooperators supplement the array of possible strategies. The latter are willing to reward cooperative actions at a personal cost, thus effectively downgrading pure cooperators to second-order free-riders due to their unwillingness to bear these additional costs. Consequently, we find that defection remains viable, especially if the rewarding is costly. Rewards, however, can promote cooperation, especially if the synergetic effects of cooperation are low. Surprisingly, moderate rewards may promote cooperation better than high rewards, which is due to the spontaneous emergence of cyclic dominance between the three strategies.

💡 Research Summary

The paper revisits the classic “stick versus carrot” problem by embedding a rewarding strategy into a spatial public‑goods game (PGG) and systematically exploring how the cost and magnitude of rewards affect the evolution of cooperation. In the model, agents occupy the sites of a two‑dimensional lattice and repeatedly play a PGG with their nearest neighbours. Three strategies are possible: pure cooperators (C) who contribute but do not reward, defectors (D) who contribute nothing, and rewarding cooperators (RC) who both contribute and pay a personal cost γ to give a benefit β to each cooperating neighbour. Because RC bears an extra cost, ordinary cooperators become second‑order free‑riders: they enjoy the reward without paying for it.

The authors vary three key parameters: the synergy factor r (which determines how much the public good is amplified), the reward benefit β, and the reward cost γ. Strategy updates follow a stochastic imitation rule based on payoff differences with a randomly chosen neighbour. Long‑run simulations (up to 10⁶ Monte‑Carlo steps) reveal several robust patterns.

First, when rewards are either too cheap or too expensive, the system collapses to the classic C–D competition. If β is small relative to γ, RC cannot offset its expense and is quickly eliminated by D; the remaining C‑D dynamics behave as in the standard spatial PGG. Conversely, if β is very large while γ remains moderate, RC initially suppresses D, but the high reward cost eventually makes RC vulnerable, allowing D to re‑emerge. Thus, the mere presence of a rewarding strategy does not guarantee higher cooperation.

Second, and most strikingly, an intermediate regime of reward intensity (β≈γ) generates a self‑organizing cyclic dominance among the three strategies: RC outcompetes D by rewarding cooperators, D outcompetes C by exploiting them, and C outcompetes RC by avoiding the extra cost of rewarding. This rock‑paper‑scissors‑like loop is sustained by spatial clustering; each strategy forms compact domains that invade the next in the cycle. The cyclic dominance stabilises a mixed‑state where all three strategies coexist, and the overall fraction of cooperators (C + RC) reaches its maximum. The authors term this the “paradox of reward”: stronger rewards do not necessarily produce more cooperation; moderate rewards can be more effective because they enable the cyclic mechanism.

Third, the impact of rewards is strongly modulated by the synergy factor r. When r is low—meaning the public good yields little extra benefit—rewarding becomes a crucial auxiliary mechanism that can rescue cooperation. When r is high, cooperation is already profitable, and additional rewards provide diminishing returns, sometimes even destabilising the cooperative cluster because of the extra cost borne by RC.

A methodological contribution of the study is the explicit treatment of second‑order free‑riding. By distinguishing C from RC, the model captures the realistic situation where individuals who benefit from a reward system do not contribute to its maintenance. This distinction clarifies why costly reward schemes can be fragile: if the reward cost outweighs the benefit, the system reverts to a D‑dominated state.

From a policy perspective, the findings suggest that designing incentive schemes requires careful calibration. Over‑generous subsidies or bonuses may backfire by creating unsustainable fiscal burdens and encouraging exploitation, whereas modest, well‑targeted rewards can foster robust cooperative networks, especially in environments where the intrinsic returns to cooperation are modest. The spatial aspect of the model underscores that local interactions and clustering matter: incentives that promote the formation of cooperative neighbourhoods can amplify their effectiveness.

In summary, the paper demonstrates that rewards are not a simple lever for increasing public‑goods contributions. Their efficacy depends on a delicate balance between benefit and cost, the underlying productivity of the public good, and the spatial structure of interactions. Moderate rewards can trigger a cyclic dominance that maximises overall cooperation, while extreme reward levels either collapse to the traditional defector‑cooperator dilemma or become self‑defeating. These insights enrich the theoretical understanding of incentive mechanisms and provide actionable guidance for real‑world institutions seeking to promote collective action.