[Paper Review] Reward and cooperation in the spatial public goods game

Attila Szolnoki, Matjaž Perc|SSRN Electronic Journal|Oct 27, 2010

Evolutionary Game Theory and Cooperation20 citations

TL;DR

This study investigates how costly rewards promote cooperation in the spatial public goods game by introducing a third strategy—rewarding cooperators—who bear costs to incentivize other cooperators. Surprisingly, moderate rewards outperform high rewards due to cyclic dominance among strategies, and while rewards can stabilize cooperation, especially at low synergy factors, defection remains viable in large parameter regions, making rewards less efficient than punishment in promoting collective action.

ABSTRACT

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.

Motivation & Objective

• To examine the role of costly rewards in promoting cooperation within structured populations.
• To analyze how rewarding cooperators interact with pure cooperators and defectors in a spatial public goods game.
• To investigate whether reward can overcome the second-order free-rider problem and stabilize cooperation.
• To compare the effectiveness of reward versus punishment in sustaining public cooperation.

Proposed method

• The spatial public goods game is modeled on a square lattice with periodic boundary conditions, where each individual interacts with four neighbors.
• Three strategies are defined: defectors (D), pure cooperators (C), and rewarding cooperators (RC), with RCs paying a cost γ to reward neighboring cooperators.
• Payoffs are calculated based on group contributions, a synergy factor r, and rewards β, with RCs receiving β/k per cooperating neighbor and paying γ/k per reward given.
• Phase diagrams are constructed by varying the synergy factor r and reward parameters β and γ to identify stable coexistence regions and phase transitions.
• Time-series simulations track strategy density evolution to analyze transient dynamics and long-term stability.
• Cyclic dominance and predator-prey-like interactions are identified as key mechanisms explaining counterintuitive outcomes, especially around discontinuous phase transitions.

Experimental results

Research questions

• RQ1 How does the introduction of costly rewarding cooperators affect the stability of cooperation in the spatial public goods game?
• RQ2 Under what conditions do rewarding cooperators outperform pure cooperators and defectors?
• RQ3 Why do moderate rewards sometimes outperform high rewards in promoting cooperation?
• RQ4 What role does the synergy factor r play in determining the dominance of different strategies?
• RQ5 How does the second-order free-rider problem manifest when rewarding is introduced, and can it be overcome?

Key findings

• Moderate rewards promote cooperation more effectively than high rewards due to the emergence of cyclic dominance among defectors, pure cooperators, and rewarding cooperators.
• At low synergy factors (r), rewards can stabilize cooperation even when defectors would otherwise dominate, especially if the reward benefit β is sufficiently high relative to cost γ.
• A discontinuous phase transition separates regions where pure cooperators dominate from those where rewarding cooperators prevail, depending on the β/γ ratio.
• Defection remains viable across large regions of the parameter space, indicating that rewards alone cannot fully eliminate defectors.
• When rewards are costly, pure cooperators (second-order free-riders) outcompete rewarding cooperators, as they benefit from rewards without bearing the cost.
• At high synergy factors, network reciprocity alone suffices to suppress defectors, and the outcome reduces to a competition between pure and rewarding cooperators, with the latter winning only if β/γ is sufficiently large.