Public Goods Game: Why Free-Riding Wins (Until Punishment Arrives)

Q: What is the Nash equilibrium of the public goods game?

Zero contribution by everyone. Each dollar you keep is worth 1 to you, but each dollar you contribute is only worth r/N back to you after the multiplier and equal split — and r/N is less than 1 whenever the multiplier r is smaller than the group size N, which is the defining condition of a public goods game. So the individually rational move is always to contribute nothing, regardless of what anyone else does.

Q: Why is zero contribution bad for the group if it is rational for the individual?

Because the multiplier r exceeds 1: money put into the pot grows before being split. If every player contributed everything, the group's total payoff would be r times the endowment pool, strictly more than if everyone kept their money private. The Nash equilibrium (zero) is the worst outcome for total welfare even though it is the best individual response to everyone else's likely strategy — the defining tension of the public goods game.

Q: Does punishment actually work to sustain cooperation?

In lab experiments, yes, and dramatically. Fehr and Gächter's 2002 study found that when players could pay a small cost to reduce a free-rider's payoff by more, contributions — which decayed toward zero over repeated rounds without punishment — instead rose and stayed high across many rounds. The catch is that punishment itself is a public good: punishing costs the punisher and benefits the whole group, so a purely rational agent should not punish either, yet people reliably do.

The setup

N players each start with an equal endowment, say 20 tokens. Privately and simultaneously, each decides how many tokens to keep and how many to put into a shared pot. The pot is multiplied by a factor r, where 1 < r < N, and the multiplied total is split equally among all N players regardless of who contributed what. Contribute nothing and keep your whole endowment plus your equal share of everyone else's multiplied contribution; contribute everything and you get back only 1/N of your own money times r.

live demo · agents deciding how much to contribute● LIVE

The payoff arithmetic

Every token you contribute returns to you as only r/N tokens through the pot, and r/N is always less than 1 because r < N by definition of the game — otherwise a lone individual could profitably fund the whole public good alone, which defeats the point. So each token contributed is a net private loss of 1 - r/N tokens for the contributor, even though it is a net gain of r - 1 tokens for the group as a whole once everyone's share is added up.

payoff_i = (endowment_i - c_i) + (r / N) * sum(c_j for all j)

c_i        = tokens player i contributes
r          = multiplier, 1 < r < N
r / N < 1  = private return on your own contributed token
r > 1      = social return on that same token, summed across everyone

The Nash equilibrium is the tragedy

Because r/N < 1 for every player regardless of what anyone else contributes, contributing zero is a dominant strategy — it is your best response no matter what the rest of the group does. If everyone reasons the same way, the unique Nash equilibrium is universal defection: nobody contributes, and the group is left with only its private endowments, forgoing the multiplied return entirely. This is the reverse image of the tragedy of the commons: instead of overusing a shared resource, players under-provide a shared benefit.

In repeated laboratory games with strangers, contributions typically start around 40-60% of the endowment and decay toward zero over ten or so rounds as participants learn that free-riders are never excluded and cooperators are never rewarded — a well-replicated finding since the 1990s in experimental economics.

Punishment changes the game

Ernst Fehr and Simon Gächter's widely cited 2002 experiment added a second stage after contributions were revealed: any player could pay a small cost to reduce another player's earnings by a larger amount. Purely self-interested players should never pay to punish — punishment is itself a public good, since the deterrent effect benefits the whole group while only the punisher pays the cost — so standard game theory predicts no punishment and, therefore, no change in behaviour. What actually happened was the opposite: people punished free-riders often and severely enough that average contributions, instead of decaying, rose and stayed close to full contribution across many repeated rounds.

What sustains cooperation without punishment

Punishment is not the only lever. Reputation systems that let players choose partners based on past behaviour, repeated interaction with the same fixed group (which turns a one-shot dilemma into a repeated game where retaliation and reciprocity are possible), and communication before play — simply letting players talk, even non-bindingly — all measurably raise contributions above the Nash prediction in experiments. What they share is a mechanism that makes a player's contribution history matter for their own future payoff, breaking the anonymity that makes zero contribution individually costless.

Frequently asked questions

What is the Nash equilibrium of the public goods game?

Zero contribution by everyone. Each dollar you keep is worth 1 to you, but each dollar you contribute is only worth r/N back to you after the multiplier and equal split — and r/N is less than 1 whenever the multiplier r is smaller than the group size N, which is the defining condition of a public goods game. So the individually rational move is always to contribute nothing, regardless of what anyone else does.

Why is zero contribution bad for the group if it is rational for the individual?

Because the multiplier r exceeds 1: money put into the pot grows before being split. If every player contributed everything, the group's total payoff would be r times the endowment pool, strictly more than if everyone kept their money private. The Nash equilibrium (zero) is the worst outcome for total welfare even though it is the best individual response to everyone else's likely strategy — the defining tension of the public goods game.

Does punishment actually work to sustain cooperation?

In lab experiments, yes, and dramatically. Fehr and Gächter's 2002 study found that when players could pay a small cost to reduce a free-rider's payoff by more, contributions — which decayed toward zero over repeated rounds without punishment — instead rose and stayed high across many rounds. The catch is that punishment itself is a public good: punishing costs the punisher and benefits the whole group, so a purely rational agent should not punish either, yet people reliably do.

Try it live

Everything above runs in your browser — open Public Goods Game and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.

▶ Open Public Goods Game simulation