Public-Goods Game

A game in which individuals contribute to a shared good; the efficient outcome is not individually rational.

Ask the Game theory assistant 1 min read · Updated September 9, 2026

Definition

Each of $n$ players contributes $c_i\in[0,w]$ ; public-good value is $\alpha\sum_j c_j/n$ for $\alpha\in(1,n)$ ; payoff $u_i=w-c_i+\alpha\sum_j c_j/n$ .

Nash equilibrium has $c_i=0$ ; the efficient outcome has $c_i=w$ for all.

Intuition

Free-riding is individually rational but collectively suboptimal.

A multiplayer generalization of the Prisoner's Dilemma.

Worked example

Voluntary-contribution experiments (Isaac-Walker) typically show $40$ - $60$ % contribution decaying toward zero without punishment.

Punishment opportunities sustain cooperation (Fehr-G\"achter).

The math

Socially optimal contribution: $\sum c_i^*=nw$ ; Nash: $c_i=0$ . The efficiency gap scales with $\alpha$ and $n$ .

Linter-Samuelson condition characterizes optimal public-good provision: sum of MRS equals MRT.

Where it is used

Climate change mitigation is the canonical public-goods problem: each country benefits from global emission reductions but has incentives to free-ride on others' efforts. International agreements like the Paris Accord attempt to coordinate contributions.

Open-source software development exemplifies voluntary public-goods provision: developers contribute code that everyone can use, facing a temptation to free-ride while hoping others will maintain the project.

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Assembled from the ReLU.chat curated knowledge base. These explanations are concise on purpose; check the sources for anything important.