The Tragedy of the Commons

Set how many people share the resource. The tool solves each user's own best effort, compares the total against what a single owner would choose, and shows what the difference costs.

Worked example — The Tragedy of the Commons

Each person's effort	3.64
Total effort	36.36
What one owner would use	20.00
Effort against the optimum	1.82×
Total value created	132.2
What one owner would create	400.0
Value left	33.1%
Each person earns	13.22

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Lesson

The theory — The Tragedy of the Commons

A commons here is one resource that several people draw from at once, where the return on your own effort falls as everybody's total effort rises. Each user picks the effort that is best for them given what the others are doing — a Nash equilibrium — and the gap between that and what a single owner of the whole resource would choose is the tragedy. Nobody has to be greedy or stupid for it to appear; it comes from each person keeping all of their own catch and paying only a share of the crowding.

What each symbol means

n: how many people draw on the resource
E: total effort, the sum of everyone's
a, b: the pasture: a is what the first unit of effort returns, b how fast that return falls

Assumes: Identical users choosing at the same moment, with no property rights, no agreement and no way to pay each other. Those are the assumptions Elinor Ostrom spent a career showing real communities break on purpose, and the tragedy on this page is what happens when they cannot.

One owner has no tragedy at all 🖖

Press One owner. The effort is 20.00 units, the optimum is 20.00 units, and the ratio is exactly 1.00×. A single herder on Hardin's pasture does precisely the right thing — not because they are virtuous, but because every blade of grass their extra cow eats is a blade their own other cows do not. The whole cost of crowding lands on the person causing it. Now press Two herders and watch Each person's effort: each of them uses 13.33 units, which is less than the sole owner did. Nobody is working harder as an individual. But each now bears only half the damage they cause while keeping all of the gain, so each uses more than half of 20.00 — and the total goes to 26.67. Every one of them is at their own best effort given what the others are doing.

The overuse stops at twice; the damage does not stop anywhere 🖖

Drag the slider and watch the two ratios come apart. Total effort goes 1.00×, 1.33×, 1.50×, 1.60× — it is 2n/(n+1), so it climbs towards 2 and never arrives. Twenty-four people work the pasture 1.92× too hard; a thousand would manage 1.998×. The share of the value that survives tells a different story: 100.0%, then 88.9%, then 75.0%, then 33.1% at ten people and 15.4% at twenty-four, because it is 4n/(n+1)², which falls towards zero with no floor at all. A thousand herders would work the commons very nearly twice as hard as one owner and leave 0.4% of what it was worth. The limit is worth stating exactly: as the crowd grows the total effort approaches 40 units, and 40 is precisely the effort at which this pasture is worth nothing. The tragedy converges on the point where there is no longer anything to argue about.

The ratios do not care how good the pasture is 🖖

Press Ten herders, then A richer pasture. The second is a different resource in every respect: the first unit of effort returns 90 instead of 50, it crowds twice as fast, and effort costs 5 instead of 10. Every number with units on it moves. Both ratios do not — 1.82× and 33.1%, exactly as before, and the working on Effort against the optimum and Value left comes out character for character identical, because 2n/(n+1) and 4n/(n+1)² contain no a, no b and no c. So the share of a commons that gets wasted is not a fact about how valuable it is, how quickly it degrades, or how expensive it is to exploit, as long as a unit of effort returns more than it costs. It is a fact about how many people get to decide. Which is why enriching a commons cannot rescue it: the share destroyed does not move, and because the value depends on the square of the surplus, doubling what the pasture is worth quadruples what gets thrown away — the plain pasture wastes 267.8 of its 400.0, and one with twice the surplus wastes 1,071.1 of its 1,600.0.

Problem solved in full

What would close the gap? The tool shows the overuse and stops there, because a tax is not something any of the players can choose. Work out the per-unit charge on effort that would bring the crowd back to the optimum, and then look at who ends up holding the money.

Open this problem in the tool
1. Put a charge of t on every unit of effort. Nothing about the game changes except each person's cost, which becomes c + t, so the surplus per unit falls from a − c to a − c − t.
  
  $π_{i} = e_{i} (a - b E) - (c + t) e_{i}, s_{t} = a - c - t$
2. The equilibrium is the same formula with the smaller surplus. Set that total equal to the optimum a planner would choose.
  
  $E^{*} (t) = \frac{n s _{t}}{b ( n + 1 )} =! \frac{a - c}{2 b} = E^{opt}$
3. Cancel the surplus that appears on both sides and solve. The charge depends only on how many people share the commons.
  
  $\frac{n ( s - t )}{n + 1} = \frac{s}{2} ⟹ t = s \cdot \frac{n - 1}{2 n}$
4. With ten herders on this pasture that is 18 per unit. Check it: the surplus after tax is 22, so each person uses 22/11 = 2 units, ten of them use 20 — the optimum exactly, and the pasture yields its full 400 again.
  
  $t = 40 \cdot \frac{9}{20} = 18, s_{t} = 22, e^{*} = \frac{22}{11} = 2, E^{*} = 20$
5. Now look at what the herders keep. They pay 28 per unit on 20 units, so of the 600 they bring in they retain 40. The charge collects 360.
  
  $600 20 (50 - 20) - 560 28 \cdot 20 = 40 vs t E = 18 \cdot 20 = 360$
6. Unregulated, those same ten kept 132.2 between them — ten shares of 13.22. The tax has tripled what the pasture produces and left the people working it a third as well off.
  
  $40 < 132.23, 40 + 360 = 400 = Π^{opt}$
Answer

The corrective charge is t = (a − c)(n − 1)/(2n). It is zero for a single owner, which is the same statement as insight block 1, and it climbs towards half the surplus as the crowd grows — 10 at two herders, 18 at ten, 19.17 at twenty-four. It restores the efficient effort exactly, at every n. But it is not a solution to the herders' problem unless the 360 comes back to them: total value rises from 132.2 to 400.0 while their own share falls from 132.2 to 40.0. Hand the revenue back in equal shares and each of the ten holds 40.0 where they held 13.22, which is the version that works. This is why Elinor Ostrom's fieldwork matters to a page like this one: the commons that survived for centuries were not taxed into efficiency by an outside authority, they were governed by rules the users wrote, monitored and enforced themselves. And note what the model assumes away, because it is not conversation — this is a one-shot simultaneous game, so herders who agree to cut back still each face the same first-order condition the moment they choose. What is missing is any agreement they could be held to. That absence, not the arithmetic, is what makes the tragedy inevitable.

$t = \frac{( a - c ) ( n - 1 )}{2 n} n \to \infty \frac{a - c}{2}$

Learning path

The best reply, and what happens when nobody computes one

References (3)

The essay the phrase comes from, and the pasture the tool is a model of: G. Hardin, "The Tragedy of the Commons." Science 162(3859), 1968, pp. 1243–1248.
Where the arithmetic actually comes from — the common-property fishery, fourteen years before Hardin named it: H. S. Gordon, "The Economic Theory of a Common-Property Resource: The Fishery." Journal of Political Economy 62(2), 1954, pp. 124–142.
And the reason the word "tragedy" is doing too much work. Ostrom's fieldwork found commons that are governed successfully, for centuries, without either privatisation or the state: E. Ostrom, Governing the Commons: The Evolution of Institutions for Collective Action. Cambridge University Press, 1990.

By the 42Math team · Reviewed against published sources. Corrections welcome.