Public Goods Game & Cooperation

Public Goods Games

A public goods game models situations where individuals can contribute to a common resource that benefits everyone, but cooperation is costly and defection is tempting. This captures the essence of many real-world dilemmas: climate action, team projects, public infrastructure, and environmental conservation.

The Basic Game

• N players interact in a group
• Cooperators contribute cost c to the public good
• Defectors contribute nothing
• All contributions are multiplied by r and distributed equally
• Payoff to cooperator: r·(nC/N) - c, where nC is the number of cooperators
• Payoff to defector: r·(nC/N)

The Tragedy of the Commons

The dilemma arises when 1 < r < N. The public good is valuable (r > 1 means contributions increase total wealth), but each individual benefits more from defecting (r/N < 1 means you get back less than you contribute). Defection dominates cooperation, yet universal defection gives everyone a payoff of zero, while universal cooperation gives everyone r - c > 0 when r > N·c.

Evolution of Cooperation

• Without punishment: Defection typically dominates. Cooperators are exploited by free-riders.
• With punishment: Cooperators can punish defectors at cost k, imposing fine f. If f > c and punishment is strong enough, cooperation can be maintained.
• Finite populations: Mutation and selection interact. Even slight advantages compound over time.
• Phase transitions: Cooperation frequency can exhibit bistability—both all-cooperate and all-defect can be stable.

Real-World Examples

• Climate change: Nations face costs to reduce emissions but benefits are shared globally (r < N)
• Team projects: Individual effort is costly but group success benefits everyone
• Commons resources: Fisheries, forests, and shared resources are depleted by overuse
• Open source: Contributing code is costly but everyone benefits from the project

Parameters to Explore

• r/N ratio: When r/N > 1, cooperation is actually favored (no dilemma). When r/N < 1, we have the tragedy.
• Group size N: Larger groups make cooperation harder (per-capita return r/N decreases).
• Punishment: Sanctioning defectors can stabilize cooperation if punishment is strong enough to outweigh the temptation to defect.
• Selection strength: Weak selection (β → 0) gives drift; strong selection amplifies fitness differences.

Historical Context

The tragedy of the commons was popularized by Garrett Hardin (1968), though the concept dates back to Aristotle's observation that "what is common to the greatest number has the least care bestowed upon it." Public goods games have been studied experimentally since the 1980s, revealing that humans cooperate more than game theory predicts, especially when punishment is available. Elinor Ostrom won the Nobel Prize in Economics (2009) for showing how communities can self-organize to manage commons without top-down regulation.