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A fishing ground feeds four villages. If each takes a modest catch, the stock renews and everyone eats for years. If each grabs all it can, the stock collapses and everyone starves. Each village decides alone.

Public goods and the commons

A public good is non-excludable and non-rival: people can benefit without paying, and one person’s use does not leave less for another. A fishery is different. It is hard to exclude users, but one person’s catch leaves fewer fish for others, making it a common-pool resource. Both can create free-riding, through different physical constraints.

From earlier lessons

Two quick ones before we start.

A strategy that beats your alternatives whatever others do is called...

In the Prisoner's Dilemma, both players defect because...

Predict, then play

You and three others each start with $10. Whatever the group puts in is multiplied by 1.6 and split evenly four ways. One of your neighbours is a free-rider; two start out cooperative but watch what everyone else does. Set your contribution and play a few rounds.

Why did that happen?

Take a guess, then reveal the reason

Put in a dollar and the pot grows by $1.60, but that $1.60 is split four ways, so you personally get 40 cents back on your dollar: a loss of 60 cents. That is true no matter what anyone else does, so contributing nothing is a dominant strategy. Yet the same arithmetic run for the group says every dollar contributed creates $1.60 of value. Private incentive and group interest point in opposite directions, and watching the cooperators give up round by round is the commons collapsing in slow motion.

The math

Four players, each endowed with $10. Let $g_i$ be $i$ ‘s contribution. The pot $\sum_j g_j$ is multiplied by $1.6$ and split evenly, so

u_i = 10 - g_i + \frac{1.6}{4}\sum_j g_j = 10 - g_i + 0.4\sum_j g_j.

The marginal per-capita return is $0.4$ : each dollar you contribute returns $0.40$ to you. Since $0.4 < 1$ , raising $g_i$ lowers $u_i$ whatever the others do, so $g_i = 0$ is a dominant strategy and the unique Nash equilibrium is no contribution at all.

Now the group. Total welfare is $\sum_i u_i = 40 + (1.6 - 1)\sum_j g_j = 40 + 0.6\sum_j g_j$ , which rises in every contribution. So the social optimum is full contribution, $10 each, and everyone ends with $16 rather than $10.

Definition, public-goods game

In this public-goods game, the benefit of every contribution reaches the whole group. When the private return to contributing (here $0.4$ ) is below $1$ but the social return (here $1.6$ ) is above $1$ , self-interest gives zero and efficiency demands full contribution. A common-pool resource can produce a related social dilemma through excessive withdrawal rather than insufficient contribution.

A note from the real world

Reality check

Theory predicts zero contribution. Real people do better than that. In public-goods experiments, most players start out giving around half, and many are conditional cooperators: they contribute if others do. But without a way to punish free-riders, contributions decay over repeated rounds, and in the lab they head for zero. The conditional cooperators in the play above do the same thing, sliding down to match the group. How far they fall depends on you: keep giving and they settle around your level, free-ride alongside them and the whole pot empties. The Behavioral unit takes up why cooperation starts high, why it unravels, and what sustains it.

Work it out

Private loss, social gain

Contribute $1. The pot grows by $1.60, split four ways, so you get $0.40 back. Netting the dollar you put in, you are down $0.60. Because this holds regardless of the others, the self-interested contribution is $0.

Now suppose all four contribute $1. The pot grows by $6.40, and each person’s share is $1.60. Netting each person’s own dollar, everyone is up $0.60. The move that is privately bad is collectively good, which is why the equilibrium (all zero) sits so far below the optimum (all ten).

Key idea

When a benefit is shared but the cost is private, contributing nothing is each person’s dominant move, and the group ends up far poorer than it could be. The commons fails not from malice but from ordinary self-interest.

Check yourself

Check yourselfapply

Each dollar contributed returns 40 cents to you but creates $1.60 of total value. What does self-interest prescribe?

Check yourselfanalyse

A larger group shares the same multiplied pot among more people. What happens to the incentive to free-ride?

Check yourselfbrings back: dominance

In what sense is the commons problem the Prisoner's Dilemma with more than two players?

Sources

Hardin, G. (1968). The tragedy of the commons. Science, 162, 1243-1248.
Ledyard, J. O. (1995). Public goods: a survey of experimental research. In Kagel, J. and Roth, A. (eds.), The Handbook of Experimental Economics. Princeton University Press.